Fifteen Lectures

From LÉVY PROCESSES To SEMIMARTINGALES

With 4 Serious Figures

Fall 02

by Patrik Albin

16th November 2002

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Contents

1 Lecture 1 6/9 - 02 1 1.0 Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Lev́y Processes and Additive Processes . . . . . . . . . . . . . 1 1.2 Markov Property . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Infinitely Divisible Random Variables and Processes …. 6

2 Lecture 2 10/9 - 02 10 2.1 Lévy-Khintchine Formula . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Compound Poisson Processes . . . . . . . . . . . . . . . . . . . 12 2.3 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Lecture 3 13/9 - 02 17 3.1 Stable Random Variables and Processes . . . . . . . . . . . . 17

4 Lecture 4 17/9 - 02 22 4.1 Continuity Properties of id Distributions . . . . . . . . . . . 22 4.2 Selfdecomposable Distributions . . . . . . . . . . . . . . . . . . 24 4.3 Subordination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Lecture 5 20/9 - 02 29 5.1 Ornstein-Uhlenbeck Processes . . . . . . . . . . . . . . . . . . . 29 5.2 Weak Convergence of ID Distributions . . . . . . . . . . . . . 30 5.3 Lévy-Itô Decomposition . . . . . . . . . . . . . . . . . . . . . . . 30 5.4 Sample Function Behaviour . . . . . . . . . . . . . . . . . . . . 32 5.5 Recurrence and Transience . . . . . . . . . . . . . . . . . . . . . 34

6 Lecture 6 27/9 - 02 37 6.1 Outline of Stochastic Integration wrt. Semi-martingales . . 37 6.2 Representation of Additive Processes . . . . . . . . . . . . . . 38

6.3 Some Facts from Measure Theory and Functional Analysis . . . . . . 40

7 Lecture 7 1/10 - 02 45 7.1 Stochastic Integrals wrt. ID Random Measures . . . . . . . . 45 7.2 Stochastic Integrals wrt. α-stable Lévy Motion . . . . . . . . 48 8 Lectures 8-9 4-8/10 - 02: Sections I.1-I.5 in Protter 50

8.1 Basic Definitions and Notation: Section I.1 in Protter . . . . . 50

8.2 Martingales: Section I.2 in Protter . . . . . . . . . . . . . . . . . . 51

8.3 Poisson Process, Brownian Motion: Section I.3 in Protter … 53

8.4 Levy Processes: Section I.4 in Protter . . . . . . . . . . . . . . . . 54 8.5 Local Martingales: Section I.5 in Protter . . . . . . . . . . . . . . 54

10 Lecture 10 15/10 - 02: Sections I.6-7 and II.1-3 in Protter 58 10.1 Stieltjes Integration: Section I.6 in Protter . . . . . . . . . . . . 58 10.2 Naive Stochastic Integration: Section I.7 in Protter . . . . . . 59 10.3 Introduction to Semimartingales: Section II.1 in Protter … 59 10.4 Properties of Semimartingales: Section II.2 in Protter . . . . . 59 10.5 Examples of Semimartingales: Section II.3 in Protter . . . . . 59

11 Lecture 11 18/10 - 02: Sections II.4-6 in Protter 62 11.1 Stochastic Integrals: Section II.4 in Protter . . . . . . . . . . . . 62 11.2 Properties of Stochastic Integrals: Section II.5 in Protter … 62

12 Lecture 12 22/10 - 02: Sections II.6-8 in Protter 65 12.1 Quadratic Variation of Semimartingales: Section II.6 in Protter 65

12.2 Itô’s Formula: Section II.7 in Protter . . . . . . . . . . . . . . . . 66

12.3 Application of Itô’s Formula: Section II.8 in Protter . . . . . . 66

13 Lecture 13 24/10 - 02: Sections III.1-3 in Protter 68

13.1 Introduction: Section III.1 in Protter . . . . . . . . . . . . . . . . 68

13.2 The Doob-Meyer Decompositions: Section III.2 in Protter . . 68

13.3 Quasimartingales: Section III.3 in Protter . . . . . . . . . . . . . 69

14 Lecture 14 6/11 - 02 8 am: Sections III.4-5 in Protter 71 14.1 The Fundamental Theorem: Section III.4 in Protter . . . . . . 71 14.2 Classical Semimartingales: Section III.5 in Protter . . . . . . . 72

15 Lecture 15 8/11 - 02: Sections III.6-8 in Protter 74

15.1 Girsanov’s Theorem: Section III.6 in Protter . . . . . . . . . . . 74

15.2 Bichteler-Dellacherie Theorem: Section III.7 in Protter …. 74 15.3 Natural and Predictable Processes: Section III.8 in Protter . 74

A Appendix. Solutions to Difficult Exercises 76

1 Lecture 1 6/9 - 02

The first five lectures cover classical distribution theory for Lévy processes and infinitely divisible distributions, following “Sato: A Course on Lévy Processes”. We have added proofs when negotiable and of sufficient “general value”. Most of them come from “Sato: Lévy Processes and Infinitely Divisible Distributions”, often slightly modified. We have also added many exercises. They are all intended to be quite

straightforward (and it is a mistake when they are not).

Another example of standard literature on Lévy Processes is “Bertoin: Lévy Processes”. See also e.g., the relevant parts of “Feller: An Introduction to Probability Theory and Its Application”, “Fristedt & Gray: A Modern Approach to Probability Theory” and “Kallenberg: Foundations of Modern Probability”.

1.0 Basic Notation

In Lectures 1-7, X ={X(t)}_{t≥0}, X_{1}, X_{2}, … denote ℝ^{d}-valued stochastic processes, and Y, Y_{1}, Y_{2}, … ℝ^{d}-valued random variables (rv's).

Processes and rv's that feature are assumed defined on a common complete basic probability space \underline{(Ω, 𝔉, P)}, when needed. A \underline{null-event} is a set N ∈𝔉 with P\{N\}=0.

The probability distribution (law) P∘Y^{−1} of Y on ℝ^{d} is denoted \underline{L(Y)}.

We write \underline{Y_{1} =_{d} Y_{2}} when L(Y_{1})=L(Y_{1}), and \underline{X_{1} =_{d} X_{2}} when the fidi's (see Definition 1.9 below) of X_{1} and X_{2} coincide. Further, \underline{→_{d}} denotes weak convergence.

The \underline{Borel sets} in ℝ^{d} (the σ-algebra generated by the open sets) is denoted \underline{B(ℝ^{d})}. For (Σ, S) a measurable space (see Section 6.3), \underline{𝕃^{0}(Σ, S)} ≡ \{measurable f:Σ→ℝ\}.

1.1 Lev́y Processes and Additive Processes

Definition 1.1 X has \underline{independent increments} if, for n∈ℕ and 0≤t_{1} <…<t_{n},

X(t_{1}), X(t_{2})−X(t_{1}), …, X(t_{n})−X(t_{n−1}) are independent.

Definition 1.2 X is \underline{time homogeneous} if , for t≥0,

L(X(t+s)−X(s)) does not depend on s≥0.

Definition 1.3 X is \underline{stochastically continuous} (also called \underline{continuous in probabil}- \underline{ity} or \underline{P-continuous}), if, for s≥0,

X(t+s)−X(s) →_{P} 0 as t→0.

Definition 1.4 X is \underline{càdlàg} (for “continu à droit avec limites à gauche”, also called \underline{rcll}) if X(t)=X(ω; t) is right continuous with left limits, except for ω in a null-event.

Definition 1.5 X is an \underline{additive process in law} (also called \underline{independent increm}- \underline{ent process in law}) if the following conditions hold;

(1) X has independent increments;

(2) X(0)=0 a.s.;

(3) X is stochastically continuous.

Definition 1.6 X is a \underline{Lévy process in law} if an additive process in law such that (4) X is time homogeneous.

Definition 1.7 X is an \underline{additive process} (also called \underline{independent increment pro}- \underline{cess}) if an additive process in law such that

(v) X is càdlàg.

Definition 1.8 X is a \underline{Lévy process} if a Lévy process in law such that

(v) X is càdlàg.

Example 1 Brownian motion (Bm) and a Poisson process (Pp) are Lévy processes. For independent identically distributed (iid) {Y_{k}}_{k=1}^{∞}, X = ∑_{k=1}^{⌊·⌋}Y_{k} is additive. #

Exercise 1 Make computer simulations of sample paths of Bm and Pp. Discuss what sources of errors there are, if any.

Definition 1.9 The \underline{finite dimensional distributions ( fidi's)} of a stochastic process {Z(t)}_{t∈T} are the probability laws

{{L(Z(t_{1}), …, Z(t_{n}))}_{(t_{1},...,t_{n})∈T^{n}}}_{n∈ℕ}.

Exercise 2 Give examples of stochastic processes {Z_{1}(t)}_{t∈T} and {Z_{2}(t)}_{t∈T} such that Z_{1} =_{d} Z_{2}, but P{Z_{1}(t)=Z_{2}(t) for all t∈T } < 1. (In general, this “event” may not be measurable, so that the probability is not even well-defined.)

Definition 1.10 X has \underline{stationary increments} if

{X(t+h)−X(h)}_{t≥0} =_{d} {X(t)−X(0)}_{t≥0} for h≥0.

Exercise 3 Explain how Definition 1.10 connects to the intuitive concept of stationary increments. Show that Lévy processes in law have stationary increments.

Exercise 4 Show that a time homogeneous process X has mean function E{ X(t)} = Kt for t ≥ 0, for some constant K ∈ ℝ^{d}, when that mean is well-defined (and a measurable function of t). Exercise 5 Show that a Lévy process in law X has variance function Var{ X(t)} = Kt for t ≥ 0, for some constant non-negative definite matrix K ∈ ℝ_{d|d}, when that variance is well-defined.

Definition 1.11 Two stochastic processes \{Z_{1}(t)\}_{t∈T} and \{Z_{2}(t)\}_{t∈T} are \underline{modi}- \underline{fications} (also called \underline{indistinguishable} by some authors) of each other, if

P{Z_{1}(t)=Z_{2}(t)} = 1 for all t∈T.

Remark 1.12 Many authors (e.g., Protter) call Z_{1} and Z_{2} indistinguishable if

P{Z_{1}(t)=Z_{2}(t) for all t∈T } = 1.

However, people in “general stochastic processes” do not use this language, since the above probability is not even well-defined without additional assumptions (e.g., cádlág, a.s. continuity, etc.). And under such assumptions, the two possible definitions of indistinguishability typically coincide (see Exercise 6 below). #

Exercise 6 Show that the two alternative definitions of indistinguishability indicated above coincide for cádlág processes X_{1} and X_{2}.

Theorem 1.13 Each Lévy process (additive process) in law has a modification that is a Lévy process (additive process).

Albeit quite straightforward under additional technical conditions (e.g., existence of second moments), the general proof is delicate. It would occupy us for several lectures. The usual approach is by oscillation analysis of martingales or Markov processes.

On http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Levy Paul.html we found

4

1.2 Markov Property

Definition 1.14 A family 𝔽 = \{𝔉_{t}\}_{t≥0} of σ-algebras 𝔉_{t} ⊆ 𝔉 is a \underline{filtration} if it is non-decreasing, i.e., 𝔉_{s} ⊆𝔉_{t} for 0≤s≤t.

Definition 1.15 X is \underline{adapted} to a filtration 𝔽 if X(t) is 𝔉_{t}-measurable for t≥0.

Definition 1.16 X is a \underline{Markov process wrt. a filtration 𝔽}, if adapted to 𝔽 with

P{X(t+s)∈ ·|𝔉_{s}} = P{X(t+s)∈ ·|X(s)} for t+s>s≥0.

Definition 1.17 X is a \underline{Markov process wrt. itself} if it is a Markov process wrt. the filtration {σ(X(s) : s∈[0, t])}_{t≥0}.

Exercise 7 Show that if X is a Markov process, then it is Markov wrt. itself.

Definition 1.18 P (·, t, x, s) is a \underline{transition probability} for a Markov process X if

P{X(t+s)∈·|X(s)=x} = P (·, t, x, s) a.e. (dF_{X(s)}) for x∈ℝ^{d}, for t+s>s≥0.

Definition 1.19 A transition probability P (·, t, x, s) is \underline{time homogeneous} if it does not depend on the last argument s≥0.

Definition 1.20 A Markov process X is \underline{time homogeneous} if it has a time homogeneous transition probability.

Fact 1.21 A Lévy process in law is a time homogeneous Markov process wrt. itself with transition probability

P (·, t, x, s) = P{X(t)∈ · −x}.

Proof. The Markov property follows from that, by independence of increments,

P{X(t+s)∈·|F_{s}} = P{X(t+s)−X(s)∈·−X(s)|F_{s}} = P{X(t+s)−X(s)∈·−X(s)|X(s)}.

We get a transition probability similarly, this time also using time homogeneity,

P{X(t+s)∈·|X(s)=x} = P{X(t+s)−X(s)∈· −x}. □

Exercise 8 Show that an additive process in law is a Markov process wrt. itself.

1.3 Infinitely Divisible Random Variables and Processes

Definition 1.22 Y is \underline{infinitely divisible ( id)} if, for each n∈ℕ,

Y =_{d} Y_{1}+…+Y_{n} for some iid Y_{1}, …, Y_{n}.

The rv's {Y_{k}}_{k=1}^{n} in Definition 1.22 that “divides” Y must have common characteristic function (chf) φ_{Y_{k}}(θ) = E{e^{i⟨θ,Y_{k}⟩}} = φ_{Y} (θ)^{1/n} for θ ∈ℝ^{d}.

Definition 1.23 For Y id, \underline{Y^{∗1/n}} denotes an rv with chf φ_{Y}^{1/n} for n∈ℕ.

Fact 1.24 For a Lévy process in law X, X(t) is id with X(t)^{∗1/n} =_{d} X(\frac{t}{n}) for t>0 and n∈ℕ.

Proof. X(t) = ∑_{k=1}^{n}(X(\frac{k}{n}t)−X(\frac{k−1}{n}t)) with X(\frac{k}{n}t)−X(\frac{k−1}{n}t)=_{d} X(\frac{t}{n}) iid. □

Fact 1.25 For Y id and t>0, there exists an rv \underline{Y^{⋆t}} with φ_{Y^{⋆t}} = φ_{Y}^{t}.

Proof. By weak convergence, it is enough to check that φ_{Y}^{t} is a chf for 0<t= \frac{k}{ℓ} ∈ℚ. But φ_{Y}^{k/ℓ} is chf for a sum of k iid rv's with law L(Y^{∗1/ℓ}), since φ_{Y}^{1/ℓ}=φ_{Y^{∗1/ℓ}}. □

Exercise 9 For Y id, show that Y^{⋆t} is id for t>0.

Fact 1.26 The fidi's of a Lévy process in law X are determined by L(X(t)) for any choice of t>0.

Proof. To check that L(X(t_{1}), …, X(t_{n})) is determined by L(X(t)), it is enough to check that L(X(t_{1}), X(t_{2})−X(t_{1}), …, X(t_{n})−X(t_{n−1})) is, for 0<t_{1} <…<t_{n}. By P-continuity and independence and homogeneity of increments, this holds if L(X(t)) determines L(X(t_{j} −t_{j−1})) for t_{j} −t_{j−1} = \frac{k}{ℓ}t ∈ tℚ (t_{0} ≡0). This follows from

φ_{X(t_{j}−t_{j−1})} = φ_{X(ℓ(t_{j}−t_{j−1}))}^{1/ℓ} = φ_{X(kt)}^{1/ℓ} = φ_{X(t)}^{k/ℓ}. □

Corollary 1.27 For a Lévy process in law X, we have φ_{X(s)} = φ_{X(t)}^{s/t} for s, t>0.

The next theorem is one of the most important in theory for stochastic processes.

Theorem 1.28 (Kolmogorov Consistency) Given distribution functions {{ F_{t}:ℝ^{k} →[0, 1]}_{t∈T^{k}}}_{k∈ℕ}, there exists a stochastic process {Z(t)}_{t∈T} with these distributions as its fidi's, iff. the following two consistency conditions hold

(1) F_{...,t_{i−1},t_{j},t_{i+1},...,t_{j−1},t_{i},t_{j+1},...}(…, x_{i−1}, x_{j}, x_{i+1}, …, x_{j−1}, x_{i}, x_{j+1}, …) = F_{t}(x);

(2) lim_{x_{k+1}→∞} F_{t,t_{k+1}}(x, x_{k+1}) = F_{t}(x).

The proof is not difficult, but requires a basic understanding of probability measures on cylinder sets, together with the general theory of weak convergence.

Fact 1.29 For Y id, there exists a Lévy process X with X(1)=_{d} Y.

Proof. We specify the law of F_{t_{1},...,t_{n}} through that of F_{t_{1},t_{2}−t_{1},...,t_{n}−t_{n−1}}, for 0 < t_{1} < … < t_{n}, as that with chf φ_{Y} (θ_{1})^{t_{1}}φ_{Y} (θ_{2})^{t_{2}−t_{1}} … φ_{Y} (θ_{n})^{t_{n}−t_{n−1}}. These distributions are consistent. Thus Kolmogorov’s Theorem gives us a process X, with these fidi's, that must be a Lévy process in law with X(1) =_{d} Y. By Theorem 1.13, there exists a Lévy process with the same fidi's. □

Exercise 10 Explain why the 1-dimensional Kolmogorov Theorem works for d-dimensional processes.

Exercise 11 For Y id, why is a Lévy process in law X that satisfies X(1)=_{d} Y unique in law? (I.e., why does any other such Lévy process have the same fidi's?)

Also process values of additive processes are id. But this is no longer an elementary observation, and the proof requires some background results and notation.

Definition 1.30 A sequence \{\{Y_{n,k}\}_{k=1}^{r_{n}}\}_{n=1}^{∞} of rv's is a \underline{null-array} if Y_{n,1}, …, Y_{n,r_{n}} are independent for n∈ℕ, with

lim_{n→∞} max_{1≤k≤r_{n}} P{|Y_{n,k}|>ε} = 0 for ε>0.

The three (or so) page proof of ou next theorem is not difficult. But it belongs to basic probability courses, rather than being suitable to give here.

Theorem 1.31 (Khintchine) Y is id iff., for some null-array {{Y_{n,k}}_{k=1}^{r_{n}}}_{n=1}^{∞}, and for some constants b_{n} ∈ℝ^{d},

∑_{k=1}^{r_{n}} Y_{n,k}− b_{n} →_{d} Y as n→∞.

Exercise 12 One implication in Khintchine’s Theorem is trivial: Prove that part.

Lemma 1.32 A P-continuous X is locally uniformly P-continuous, i.e., for t_{0} >0,

lim_{δ↓0} sup_{s,t∈[0,t_{0}], |s−t|≤δ} P{|X(s)−X(t)|>ε} = 0 for ε>0.

Exercise 13 Prove Lemma 1.32.

Fact 1.33 For an additive process in law X, any process value X(t) is id.

Proof. With r_{n} = n and Y_{n,k} = X(\frac{k}{n}t)−X(\frac{k−1}{n}t), we trivially have ∑_{k=1}^{r_{n}} Y_{n,k} = X(t) →_{d} X(t). Thus Khintchine’s Theorem shows that X(t) is id, if {{Y_{n,k}}_{k=1}^{r_{n}}}_{n=1}^{∞} is a null-array. This we get from Lemma 1.32, since, as n→∞,

max_{1≤k≤r_{n}} P{|Y_{n,k}|>ε} ≤ sup_{r,s∈[0,t], |r−s|≤t/n} P{|X(r)−X(s)|>ε} → 0. □

Id processes are defined by an infinite dimensional version of Definition 1.22.

Definition 1.34 A stochastic process \{Z(t)\}_{t∈T} is \underline{id} if, for each n∈ℕ,

{Z(t)}_{t∈T} =_{d} {Z_{1}(t)}_{t∈T} +…+{Z_{n}(t)}_{t∈T} for some iid processes Z_{1}, …, Z_{n}.

Exercise 14 Show that Lévy processes are id processes.

Definition 1.35 An id Y has \underline{cumulant generating function ( cgf)}

\underline{Ψ(−i ·)} ≡ ln(φ_{Y} (·)).

1.4 Something I came to think of this time ...

Vilken eller vilka av dessa fiskar är en 1) gädda, 2) hongädda, 3) horngädda, 4) näbbgädda, 5) skädda, 6) honskädda, 7) hornskädda, 8) näbbskädda?

2.1 Lévy-Khintchine Formula

The following result is one of the most fundamentally important in probability. The proof is not really difficult, but too technical to be worthwhile doing here.

Theorem 2.1 (Lévy-Khintchine) Y is id iff. there exists a \underline{triplet (A, ν, γ)} of

\begin{cases}A a symmetric non-negative definite d|d-matrix (the \underline{Gaussian covariance}) \\ ν a measure on ℝ^{d} with ν(\{0\})=0 and ∫_{ℝ^{d}}|y|^{2}∧1 dν(y)<∞ (the \underline{Lévy measure}) \\ γ ∈ℝ^{d} a constant \end{cases} ,

which in that case is uniquely determined, such that, for θ ∈ℝ^{d},

E\{e^{i⟨θ,Y ⟩}\} = exp\{−\frac{1}{2}⟨θ, Aθ⟩ + i⟨θ, γ⟩ + ∫_{ℝ^{d}}(e^{i⟨θ,y⟩}−1−1_{\{|y|≤1\}}i⟨θ, y⟩) dν(y)\}.

That the integral in the exponent is well-defined follows from the requirements on ν and the fact that |e^{i⟨θ,y⟩}−1−1_{{|y|≤1}}i⟨θ, y⟩| = O(|y|^{2}) [O(1)] as |y|→0 [|y|→∞].

The function 1_{{|y|≤1}} in Lévy-Khintchine Formula may be (often is) replaced with, e.g., y/(1∧|y|), or any other measurable function that is y+O(|y|^{2}) [O(1)] as |y|→0 [|y|→∞]. The only (other) effect of this replacement is that the value of γ changes.

Exercise 15 Explain the claim about replacement of 1_{{|y|≤1}} above.

Corollary 2.2 If Y is id with triplet (A, ν, γ) such that

(the \underline{drift})  γ_{0} ≡ γ − ∫_{|y|≤1} y dν(y)  is well-defined and finite,

then we may rewrite the Lévy-Khintchine Formula with a new triplet \underline{(A, ν, γ_{0})_{0}}, as

E\{e^{i⟨θ,Y ⟩}\} = exp\{−\frac{1}{2}⟨θ, Aθ⟩ + i⟨θ, γ_{0}⟩ + ∫_{ℝ^{d}}(e^{i⟨θ,y⟩}−1) dν(y)\} for θ ∈ℝ^{d}.

Corollary 2.3 If Y is id with triplet (A, ν, γ) such that

(the \underline{center})  γ_{1} ≡ γ + ∫_{|y|>1} y dν(y)  is well-defined and finite,

then we may rewrite the Lévy-Khintchine Formula with a new triplet \underline{(A, ν, γ_{1})_{1}}, as

E\{e^{i⟨θ,Y ⟩}\} = exp\{−\frac{1}{2}⟨θ, Aθ⟩ + i⟨θ, γ_{1}⟩ + ∫_{ℝ^{d}}(e^{i⟨θ,y⟩}−1−i⟨θ, y⟩) dν(y)\} for θ ∈ℝ^{d}.

On http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Khinchin.html we found

11

Definition 2.4 The \underline{generating triplet} of a Lévy process is the triplet of X(1).

Exercise 16 Compute all well-defined generating triplets for Bm and for Pp.

Definition 2.5 A Lévy process with generating triplet (A, ν, γ) is of

\underline{type A} if A=0 and ν(ℝ^{d})<∞,
\underline{type B} if A=0 and ν(ℝ^{d})=∞ but ∫_{|x|≤1} |x| dν(x)<∞,
\underline{type C} if A≠0 or ∫_{|x|≤1} |x| dν(x)=∞.

Many important properties of Lévy processes and id rv's vary with the type.

Exercise 17 Determine the types of Bm and Pp.

2.2 Compound Poisson Processes

Definition 2.6 A \underline{compound Pp} is a Lévy process with generating triplet (0, cσ, 0)_{0}, where c>0 is a constant and σ a probability measure on ℝ^{d} with σ({0})=0.

Compound Pp are crucail in the proof of many theorems on id phenomena.

Exercise 18 Determine the generating triplets (·, ·, ·) and (·, ·, ·)_{1} for a compound Pp. Show that a Lévy process is a compound Pp iff. it has generating triplet (0, ν, 0)_{0} with ν(ℝ^{d})∈(0, ∞).

Theorem 2.7 Let {N(t)}_{t≥0} be a Pp with intensity c, and {Y_{k}}_{k=1}^{∞} iid rv's, independent of N, with L(Y_{k}) = σ where σ({0}) = 0. Denoting S_{n} = ∑_{k=1}^{n} Y_{k} for n∈ℕ, X(t)≡S_{N(t)} is a compound Pp with generating triplet (0, cσ, 0)_{0}.

Proof. Càdlàg sample paths and X(0)=_{d} 0 are immediate. P-continuity follows from

P{|X(t+s)−X(s)|>ε} ≤ P{|N(t+s)−N(s)|>0} = 1−e^{−c|t|} → 0 as t→0.

Independence and homogeneity of increments come by conditioning on the values of N involved, see Exercise 19 below. The generating triplet is the claimed one, since

E\{e^{i⟨θ,X(1)⟩}\} = ∑_{n=0}^{∞} E\{e^{i⟨θ,S_{1}⟩}\}^{n} \frac{c^{n}}{n!} e^{−c} = exp\{c ∫_{ℝ^{d}}(e^{i⟨θ,y⟩}−1) dσ(y)\}. □

Exercise 19 Show that the process S_{N(t)} in Theorem 2.7 has independent and time homogeneous increments.

Exercise 20 The ⇐ part of Lévy-Khintchine Formula is easy: Show that it is enough to prove that φ(θ) = exp{∫_{ℝ^{d}}(e^{i⟨θ,y⟩}−1−1_{{|y|≤1}}i⟨θ, y⟩) dν(y)} is a chf for any Lévy measure ν [i.e., measure with ∫_{ℝ^{d}}|y|^{2}∧1 dν(y)<∞]. Prove that φ is a chf by considering exp{∫_{|y|>ε}(e^{i⟨θ,y⟩}−1−1_{{|y|≤1}}i⟨θ, y⟩) dν(y)} and sending ε↓0.

2.3 Moments

Definition 2.8 A function g:ℝ^{d} →[0, ∞) is \underline{submultiplicative} if

g(x+y) ≤ a g(x)g(y) for x, y ∈ℝ^{d}, for some constant a>0.

Exercise 21 Show that products of submultiplicative functions are submultiplicative, as are g(x) = |x|∨1 and g(x) = e^{|x|^{b}} for b∈(0, 1].

Theorem 2.9 (Kruglov) For g locally bounded submultiplicative measurable, and X a Lévy process with Lévy measure ν, the following conditions are equivalent:

(1) E{g(X(t))} < ∞ for some t>0;

(2) E{g(X(t))} < ∞ for each t>0;

(3) ∫_{|x|>1} g(x) dν(x) < ∞.

Exercise 22 Show that, for a Lévy process X with Lévy measure ν, E{ |X(t)|} < ∞ for some (each) t>0 iff. ∫_{|x|>1} |x| dν(x) < ∞.

Corollary 2.10 An id Y with triplet (A, ν, γ) has well-defined expected value, which in that case coincides with the center γ_{1}, iff.

∫_{|x|>1} |x| dν(x) < ∞.

Proof. The existence issue is contained in Exercise 22. Assuming existence, expressing E\{e^{i⟨θ,Y ⟩}\} with the triplet (A, ν, γ_{1})_{1}, we readily get \frac{∂}{∂θ_{k}} E\{e^{iθ_{k}Y_{k}}\} |_{θ_{k}=0}= i(γ_{1})_{k}. □

Existence of an odd derivative at zero for a chf is necessary but not sufficient for existence of the corresponding odd moment. However, existence of an even derivative at zero for a chf is equivalent with existence of the corresponding even moment.

Lemma 2.11 g(x) ≤ be^{C|x|} for some constants b, C >0.

Proof. With b ≡ max{1/a, sup_{x∈[0,1]} g(x)}, we get, for |x|∈[n−1, n],

g(x) = g(∑_{k=1}^{n} \frac{x}{n}) ≤ a^{n−1}g(\frac{x}{n})^{n} ≤ a^{n−1}b^{n} ≤ b(ab)^{|x|} = be^{ln(ab)|x|}. □

Lemma 2.12 An ℝ-valued id rv with triplet (Â, ν̂, γ̂_{1})_{1} such that ν̂ has bounded support, has an entire chf.

Proof. Since |(e^{iy(θ+h)}−1−iy(θ+h))−(e^{iyθ}−1−iyθ)|/|h| = O(|θ|^{2}) as |θ|→0, uniformly for |h| small enough, the following chf is differentiable, by elementary arguments,

φ(θ) = exp\{−\frac{1}{2} Âθ^{2} + iγ̂_{1}θ + ∫_{ℝ}(e^{iyθ}−1−iyθ) dν̂(y)\}, θ ∈ℂ. □

Lemma 2.13 For an ℝ-valued rv Y with entire chf, E{e^{C|Y |}} < ∞ for C >0.

Proof. Existence of φ^{(2n)}(0) gives E{Y^{2n}}<∞ for n∈ℕ. Specifically,

φ(θ) = ∑_{n=0}^{∞} φ^{(n)}(0) θ^{n}/(n!) = ∑_{n=0}^{∞} i^{n} E{Y^{n}} θ^{n}/(n!) for θ ∈ℂ.

This power series is absolutely convergent on ℂ, so that

∑_{n=0}^{∞} |E{Y^{n}}| |θ|^{n}/(n!) < ∞ ⇒ ∑_{n=0}^{∞} E{Y^{2n}} |θ|^{2n}/((2n)!) < ∞ for θ ∈ℂ.

Clearly, it is enough to show that E{cosh(C|Y |)} < ∞. However,

E{cosh(C|Y |)} = E{∑_{n=0}^{∞}(CY)^{2n}/((2n)!)} = ∑_{n=0}^{∞} E{Y^{2n}} C^{2n}/((2n)!) < ∞. □

Proof of Kruglov’s Theorem. Let X have generating triplet (A, ν, γ). Denote ν_{0} ≡ ν1_{{|x|≤1}} and ν_{1} ≡ ν1_{{|x|>1}}. Let X^{0} and X^{1} be independent Lévy processes with generating triplets (A, ν_{0}, γ) and (0, ν_{1}, 0)_{0}, respectively, so that X =_{d} X^{0}+X^{1}.

(1) ⇒ (3) If E{g(X(t))}<∞, since X(t) =_{d} X^{0}(t)+X^{1}(t), we conclude that

∫_{ℝ^{d}×ℝ^{d}} g(x+y) dF_{X^{1}(t)}(y) dF_{X^{0}(t)}(x) < ∞ ⇒ ∫_{ℝ^{d}} g(x+y) dF_{X^{1}(t)}(y) < ∞

for some x∈ℝ^{d}. By Lemma 2.11, g(y) ≤ ag(−x)g(x+y) ≤ ab e^{C|x|} g(x+y), so that

∫_{ℝ^{d}} g(y) dF_{X^{1}(t)}(y) = E{g(X^{1}(t))} < ∞.

For ν_{1}(ℝ^{d})>0, since ν_{1}(ℝ^{d})<∞, by Exercise 18, X^{1} is a compound Pp. Thus

∞ > E\{g(X^{1}(t))\} = E\{g(S_{N(t)})\} = ∑_{n=0}^{∞} E\{g(S_{n})\}\frac{(ct)^{n}}{n!}e^{−ct}.

In particular, it follows that

∞ > E{g(S_{1})} = ∫_{ℝ^{d}} g(x) dσ(x) = c^{−1} ∫_{ℝ^{d}} g(x) dν_{1}(x) = c^{−1} ∫_{|x|>1} g(x) dν(x). Of course, for ν_{1}(ℝ^{d})=0, the right-hand side is trivially finite.

(3) ⇒ (2) When ∫_{ℝ^{d}} g(x) dν_{1}(x) < ∞, submultiplicativity gives ∫_{(ℝ^{d})^{n}} g(x_{1}+…+x_{n}) dν_{1}(x_{1}) … dν_{1}(x_{n}) ≤ a^{n−1}[∫_{ℝ^{d}} g(x) dν_{1}(x)]^{n}.

From this we immediately get (cf. above) E\{g(X^{1}(t))\} = ∑_{n=0}^{∞} E\{g(S_{n})\}\frac{(ct)^{n}}{n!} e^{−ct} ≤ ∑_{n=0}^{∞} a^{n−1}[∫_{ℝ^{d}} g(x) dν_{1}(x)]^{n} \frac{t^{n}}{n!} e^{−ct} < ∞. Thus we are done if E{e^{C|X^{0}(t)|}} < ∞ since, by submultiplicativity and Lemma 2.11,

E{g(X(t))} ≤ aE{g(X^{0}(t))g(X^{1}(t))} ≤ abE{e^{C|X^{0}(t)|}}E{g(X^{1}(t))}. However, by Hölder’s inequality together with Lemmas 2.12 and 2.13, we have

E{e^{C|X^{0}(t)|}} ≤ E{e^{C ∑_{k=1}^{d} |X_{k}^{0}(t)|}} ≤ ∏_{k=1}^{d} E{e^{C|X_{k}^{0}(t)|}} < ∞. □

Exercise 23 Explain the representation X =_{d} X^{0}+X^{1} in the above proof. Exercise 24 Explain why Lemmas 2.12 and 2.13 really apply to X_{k}^{0}(t).

2.4 Something I came to think of this time ...

16

3.1 Stable Random Variables and Processes

Definition 3.1 Y is \underline{stable} if, for each n∈ℕ, with Y_{1}, …, Y_{n} iid copies of Y,

Y_{1}+…+Y_{n} =_{d} bY + c for some constants b = b(n) > 0 and c = c(n) ∈ ℝ^{d}.

Y is \underline{strictly stable} if it is possible to take c(n)=0 for n∈ℕ.

Definition 3.1 may be rewritten Y^{⋆1/n} =_{d} \frac{1}{b}Y − \frac{c}{n}. And so stable rv's are id.

Definition 3.2 A Lévy process X with X(1) (strictly) stable is called a \underline{(strictly)} \underline{stable Lévy motion}.

Definition 3.3 An ℝ-valued stochastic process \{Z(t)\}_{t∈T} is \underline{stable} if, for each n∈ℕ, with Z_{1}, …, Z_{n} iid copies of Z,

Z_{1}+…+Z_{n} =_{d} b_{n} Z + c_{n} for some constant b_{n} >0 and function c_{n}:T →ℝ.

Z is \underline{strictly stable} if it is possible to take c_{n} ≡0 for n∈ℕ.

Sato uses the old-fashioned language to call stable Lévy motions stable processes. This was standard some decades ago, when stable processes in the sense of Definition 3.3 had not been studied. Now most authors have switched to the language we use.

Stable distributions are among the few most important id distributions. Two reasons are their stability under additon (Definition 3.1), and the explicitness of their chf (see below). Stable processes occupy a similar position among id processes. They have become fashionable since naturally having heavy tails and long range dependence.

Exercise 25 Show that stable Lévy motions are stable processes.

Definition 3.4 Y is \underline{trivial} (or \underline{degenerate}), if Y =_{d} c for some constant c∈ℝ^{d}.

Definition 3.5 A Lévy process X is \underline{trivial} if X(1) is trivial.

Exercise 26 Show that a Lévy process X is trivial iff. X =_{d} {ct}_{t≥0} for some constant c∈ℝ^{d}.

Theorem 3.6 For Y non-trivial stable, there exists a unique constant α ∈ (0, 2] such that

Y^{⋆t} =_{d} t^{1/α}Y + c for t>0, for some constant c = c(t) ∈ ℝ^{d}.

For Y non-trivial strictly stable, c(t)=0 for t>0.

The proof is not difficult, but a bit technical. It belongs to basic probability courses.

Definition 3.7 A stable Y is called \underline{α-stable}, α∈(0, 2], whenever

Y^{⋆t} =_{d} t^{1/α}Y + c for t>0, for some constant c = c(t) ∈ ℝ^{d}.

Y is called \underline{strictly α-stable} if c(t)=0 for t>0.

Exercise 27 For Y (strictly) α-stable, show that Y^{⋆t} is (strictly) α-stable for t>0. Show that values of (strictly) α-stable Lévy motions are (strictly) α-stable.

Definition 3.8 X is \underline{self-similar with index κ>0} if

{X(λt)}_{t≥0} =_{d} {λ^{κ}X(t)}_{t≥0} for λ>0.

Fact 3.9 A Lévy process is self-similar iff. it is a strictly stable Lévy motion.

Proof. ⇒ Self-similarity and Corollary 1.27 give X(1)^{⋆1/n} =_{d} X(\frac{1}{n}) =_{d} n^{−κ}X(1).

⇐ Take Y strictly stable and X a Lévy process with X(1)=_{d} Y. For Y trivial, self-similarity with κ = 1 follows from Exercise 26. For Y non-trivial, so that Y is strictly α-stable (Theorem 3.6), we get X(λt)−X(λs) =_{d} λ^{1/α}(X(t)−X(s)), since X(λ(t−s)) =_{d} X(1)^{⋆λ(t−s)} =_{d} (λ(t−s))^{1/α}X(1) =_{d} λ^{1/α}X(1)^{⋆(t−s)} =_{d} λ^{1/α}X(t−s). From this we conclude that, for 0<t_{1} <…<t_{n}, (X(λt_{1}), X(λ(t_{2}−t_{1})), …, X(λ(t_{n}−t_{n−1}))) =_{d} λ^{1/α}(X(t_{1}), X(t_{2}−t_{1}), …, X(t_{n}−t_{n−1})), which by a familiar argument is equivalent with sought after

(X(λt_{1}), …, X(λt_{n})) =_{d} (λ^{1/α}X(t_{1}), …, λ^{1/α}X(t_{n})). □

We now give the two most important representations of stable chf. Historically, the

literature in this area, articles as well as books, is heavily polluted by technical errors, see “Hall: A comedy of errors: the cannonical form for a stable characteristic function”. The book “Samorodnitsky & Taqqu: Stable Non-Gaussian Processes” from 1994 has put things in order. (Be careful with earlier literature!)

Theorem 3.10 Y is 2-stable iff. it is Gaussian. Y is α-stable, α∈(0, 2), iff.

E\{e^{i⟨θ,Y ⟩}\} = \begin{cases}exp\{− ∫_{S^{d}} |⟨θ, y⟩|^{α}(1−i tan(\frac{πα}{2})sign(⟨θ, y⟩)) dμ(x) + i⟨θ, τ⟩\}, α≠1 \\ exp\{− ∫_{S^{d}} |⟨θ, y⟩|(1+i\frac{2}{π} ln(|⟨θ, y⟩|)sign(⟨θ, y⟩)) dμ(x) + i⟨θ, τ⟩\}, α=1 \end{cases}

for θ ∈ℝ^{d}, where μ is a unique finite measure on \underline{S^{d}} ≡ \{x∈ℝ^{d}: |x|=1\}, and τ ∈ℝ^{d} a unique constant that is the drift γ_{0} for α∈(0, 1) and the center γ_{1} for α∈(1, 2).

Y is α-stable, α∈(0, 2), iff. it is id with triplet (0, ν, γ) such that

ν(B) = ∫_{x∈S^{d}} ∫_{x=0}^{x=∞} 1_{B}(rx) r^{−(α+1)} dr dλ(x) for B ∈B(ℝ^{d}),

where λ is a unique finite measure on S^{d}.

The proof is somewhat difficult, and will not be attempted here. The result is only marginally important for us, so there is really no reason to do it.

Exercise 28 In one direction Theorem 3.10 is trivial: Prove that part.

Corollary 3.11 An ℝ-valued id rv Y is α-stable, α ∈ (0, 2), iff. it has triplet (0, ν, γ) such that, for some unique constants c_{1}, c_{2} ≥0,

dν(x) = (c_{1}1_{(0,∞)}(x) + c_{2}1_{(−∞,0)}(x)) |x|^{−(α+1)} dx for x∈ℝ.

An ℝ-valued rv Y is α-stable, α∈(0, 2), iff.

E\{e^{iθY} \} = \begin{cases}exp\{−c|θ|^{α}(1−iβ tan(\frac{πα}{2})sign(θ)) + iθτ\}, α≠1 \\ exp\{−c|θ|(1+iβ \frac{2}{π} ln(|θ|)sign(θ)) + iθτ\}, α=1 \end{cases} for θ ∈ℝ,

with c≥0, β = \frac{c_{1}−c_{2}}{c_{1}+c_{2}} ∈ [−1, 1] and τ ∈ℝ constants that are unique for Y non-trivial. Here τ is the drift γ_{0} for α∈(0, 1) and the center (mean) γ_{1} for α∈(1, 2).

For Y with the chf in the second part of Corollary 3.11 we write \underline{Y ∼ S_{α}(c, β, τ)}.

Exercise 29 Derive Corollary 3.11 from Theorem 3.10.

Corollary 3.12 For an ℝ-valued α-stable rv, α∈(0, 2), E{|Y |^{p}}<∞ iff. p<α.

Exercise 30 Prove Corollary 3.12.

There are four univariate α-stable distributions that are known explicitely in closed form, one of which is the trivial distribution. The other three are given in Exer-

cises 31-33 below. Further, the distribution with α= \frac{3}{2} and β =0 is called Holtsmark distribution.

Exercise 31 Show that an rv with the density below is S_{1}(c, 0, τ)-distributed

f(y) = (c/π) ((y−τ)^{2}+c^{2})^{−1} for y ∈ℝ (Cauchy Distribution).

Exercise 32 Show that an rv with the density below is S_{1/2}(c, 1, τ)-distributed

f(y) = \sqrt{c/(2π)} (y−τ)^{−3/2} e^{−c/(2(y−τ))} for y >τ (Lévy Distribution).

Exercise 33 Show that Gaussian rv's are 2-stable. Are they strictly stable?

There exists a simple explicit formula to simulate univariate α-stable rv's in a

computer. Multivariate α-stable rv's are considerably more difficicult to simulate.

Theorem 3.13 For V and W independent rv's with uniform distribution over (−\frac{π}{2}, \frac{π}{2}) and standard exponential distribution, respectively, and for constants α ∈ (0, 2), c≥0, β ∈[−1, 1] and τ ∈ℝ, we have

c \frac{sin[αV +tan^{−1}(β tan(\frac{πα}{2}))] (cos[(1−α)V −tan^{−1}(β tan(\frac{πα}{2}))])^{1/α−1}}{(cos[tan^{−1}(β tan(\frac{πα}{2}))])^{1/α} (cos(V))^{1/α} W^{1/α−1}} + τ ∼ S_{α}(c, β, τ).

The proof is not difficult at all, but is of little interest from our point of view.

Exercise 34 Derive a formula for simulating Gaussian rv's by sending α↑2.

3.2 Something I came to think of this time ...

Vi väcker er inte då ni skall gå hem!

4.1 Continuity Properties of id Distributions

Definition 4.1 A measure ρ on ℝ^{d} is ∘ \underline{discrete} if ρ(ℝ^{d}∖C)=0 for some countable C ⊆ℝ^{d}; ∘ \underline{continuous} if ρ(\{x\})=0 for x∈ℝ^{d}; ∘ \underline{absolutely continuous} if ρ(B)=0 for null-events B ∈B(ℝ^{d}).

Lemma 4.2 For ρ = ρ_{1}⋆ρ_{2} with ρ_{1} and ρ_{2} non-zero finite measures on ℝ^{d},

(1) ρ is discrete iff. ρ_{1} and ρ_{2} are discrete;

(2) ρ is continuous iff. ρ_{1} or ρ_{2} is continuous;

(3) ρ is absolutely continuous iff. ρ_{1} or ρ_{2} is absolutely continuous.

Exercise 35 Prove Lemma 4.2.

Theorem 4.3 (Döblin) For a Lévy process X the following three conditions are equivalent

(1) L(X(t)) is continuous for each t>0;

(2) L(X(t)) is continuous for some t>0;

(3) X is of type B or type C.

Proof. ¬(3) ⇒ ¬(2) Let X be type A, i.e., A = 0 and ν(ℝ^{d}) < ∞. Then either ν(ℝ^{d}) = 0, so that X is trivial and L(X(t)) not continuous, or ν(ℝ^{d}) > 0, so that X(t)−γ_{0}t is compound Pp and has an atom at 0 with mass P{N(t)=0} = e^{−ν(ℝ^{d})t}.

(3) ⇒ (1) If A ≠ 0, then X has a non-trivial Gaussian component, which is continuously distributed (albeit not necessarily absolutely continuous). Hence (1) follows from Lemma 4.2.2. In the rest of the proof we may thus assume that ν(ℝ^{d})=∞. For \underline{ν discrete}, let \{x_{j}\}_{j=1}^{∞} be the atoms with ν(\{x_{j}\}) ≡ m_{j} > 0. Put m′_{j} =m_{j}∧1, so that ∑_{j=1}^{∞} m′_{j} =∞ (since ∑_{j=1}^{∞} m_{j} =∞). Let X =_{d} X_{n}^{0}+X_{n}^{1}, with X_{n}^{0} and X_{n}^{1} independent, and X_{n}^{0} compound Pp with Lévy measure ν_{n} ≡∑_{j=1}^{n} m′_{j}δ_{x_{j}}. Put D(Y) ≡ sup{P{Y =x}: x∈ℝ^{d}}. Here D(Y_{1}+Y_{2})≤D(Y_{1}) for Y_{1} and Y_{2} independent, since

P{Y_{1}+Y_{2} =x} = ∫_{ℝ^{d}} P{Y_{1} =x−y_{2}} dF_{Y_{2}}(y_{2}) ≤ D(Y_{1}) for x∈ℝ^{d}.

Hence D(X(t))≤D(X_{n}^{0}(t)). Let c_{n} =ν_{n}(ℝ^{d}) and σ_{n} =c_{n}^{−1}ν_{n}. Since D(σ_{n}⋆…⋆σ_{n}) ≤ D(σ_{n}) ≤ c_{n}^{−1}, by the construction of ν_{n}, we have, for any x∈ℝ^{d}, as n→∞,

P\{X_{n}^{0}(t)=x\} = e^{−c_{n}t} ∑_{k=0}^{∞} \frac{(c_{n}t)^{k}}{k!}(σ_{n}⋆…⋆σ_{n})(\{x\}) ≤ e^{−c_{n}t}+c_{n}^{−1} → 0.

For \underline{ν continuous}, let X_{n}^{0} be a compound Pp with Lévy measure ν_{n} = ν1_{\{|x|>1/n\}}. Since ν_{n}⋆…⋆ν_{n} is continuous (by Lemma 4.2.2), X_{n}^{0}(t) only has an atom at zero with mass e^{−c_{n}t} →0 as n→∞, so that again D(X(t))=0 [since ≤D(X_{n}^{0}(t))]. \underline{General case}. Since ν1_{\{|x|>1/n\}} is finite, ν = ν_{c}+ν_{d} with ν_{c} continuous and ν_{d} discrete, and one of them infinite. To that guy we apply one of the above arguments, giving continuity for that component, and thus for X(t) by Lemma 4.2.2. □

Exercise 36 Explain in detail what is going on in the last part of the proof.

Corollary 4.4 For a Lévy process X the following three conditions are equivalent (1) L(X(t)) is discrete for each t>0;

(2) L(X(t)) is discrete for some t>0;

(3) X is of type A with discrete Lévy measure ν.

Exercise 37 Derive Corollary 4.4 from Theorem 4.3 (and Lemma 4.2).

Lemma 4.5 Given n∈ℕ, Y has a density f_{Y} ∈ℂ^{n}(ℝ^{d}), with corresponding partial derivatives that all tend to zero at infinity, provided that

∫_{ℝ^{d}} |θ|^{n} |φ_{Y} (θ)| dθ < ∞.

Proof. By elementary theory of chf, Y has a continuous density

f_{Y} (y) = (2π)^{−d} ∫_{ℝ^{d}} e^{−i⟨θ,y⟩} φ_{Y} (θ) dθ for y ∈ℝ^{d}.

Using the assumption of the lemma together with dominated convergence, we readily get f_{Y} ∈ℂ^{n}(ℝ^{d}). The claim at infinity is the Riemann-Lebesgue Lemma. □

Theorem 4.6 (Orey) An ℝ-valued id Y with Lévy measure ν such that

lim inf_{r↓0} r^{α−2} ∫_{[−r,r]} x^{2} dν(x) > 0 for some α∈(0, 2),

has a density f_{Y} ∈ℂ^{∞}(ℝ), all derivatives of which tends to zero at infinity.

Proof. We check that the hypothesis of Lemma 4.5 holds for all n: Pick c_{1} > 0 such that ∫_{[−r,r]} x^{2} dν(x) ≥ c_{1}r^{2−α} for r > 0 small enough. Pick c_{2} > 0 such that 1−cos(u) ≥ c_{2}u^{2} for |u|≤1. By inspection of Lévy-Khintchin Formula, we get

|φ_{Y} (θ)| ≤ exp{∫_{ℝ}(cos(θx)−1) dν(x)} ≤ exp{−c_{2} ∫_{|x|≤1/|θ|} θ^{2}x^{2} dν(x)} ≤ e^{−c_{1}c_{2}|θ|^{α}}. □

Exercise 38 Show that non-trivial univariate α-stable rv's have ℂ^{∞}-densities.

4.2 Selfdecomposable Distributions

Definition 4.7 An id Y is \underline{selfdecomposable} (or of \underline{class L}), if there exists a function k:S^{d}×(0, ∞)→[0, ∞), that is measurable in its first argument and non-incre-asing in the second, such that, for some finite measure λ on S^{d}, Y has Lévy measure

ν(B) = ∫_{x∈S^{d}} ∫_{x=0}^{x=∞} 1_{B}(rx) k(x, r) dr/r dλ(x) for B ∈B(ℝ^{d}).

By Therem 3.10, univariate α-stable rv's are selfdecomposable with k(x, r)=r^{−α}.

Exercise 39 Show that exponential rv's are selfdecomposable but not stable.

Definition 4.8 A Lévy process X with X(1) selfdecomposable is called a \underline{selfde}- \underline{composable process}.

Definition 4.9 An ℝ-valued Y is \underline{unimodal with mode a∈ℝ}, if it has probability distribution dF_{Y} (y) = cδ_{a}(y) + f(y)dy, for some constant c ∈ [0, 1] and subprobability density f that is non-decreasing on (−∞, a) and non-increasing on (a, ∞).

Unimodality is an important property that simplifies or even “saves” many proofs.

Theorem 4.10 (Yamazato) Selfdecomposable distributions on ℝ are unimodal.

The proof is technical, and not really of interest to us. It is famous for having been preceeded by several faulty proofs, as well as disproofs (i.e., counterexamples).

Theorem 4.11 (Sato) Non-trivial selfdecomposable distributions on ℝ are absolutely continuous.

Againg the proof is technical, and not of interest to us.

Theorem 4.12 Y is selfdecomposable iff., for each b>1, φ_{Y} (θ) = φ_{Y} (θ/b)φ_{ρ_{b}}(θ) for θ ∈ℝ^{d}, for some probability measure ρ_{b} on ℝ^{d}.

Proof. ⇐ This is quite long and technical and cannot be done here. ⇒ For the Lévy measure ν of Y, we have, for b>1, ν(bB) = ∫_{S^{d}}∫_{0}^{∞} 1_{bB}(rx)k(x, r) dr/r dλ(x) = ∫_{S^{d}}∫_{0}^{∞} 1_{B}(r̂x)k(x, r̂b) dr̂/r̂dλ(x) ≤ ν(B). Write ν = ν_{0}+ν_{1}, where ν_{0}(B)=ν(bB) and ν_{1}(B) = ν(B)−ν_{0}(B) are both Lévy measures. Then Lévy-Khintchine Formula together with routine calculations give

φ_{Y} (θ) = φ_{Y} (θ/b) φ_{((1−b^{−2})A , ν_{1} , ...)}(θ). □

Exercise 40 Check the last formula in the proof of Theorem 4.12.

4.3 Subordination

Lemma 4.13 The chf of an ℝ-valued id Y cannot be zero.

Proof. Let {φ_{n}(θ)}_{n=1}^{∞} be univariate chf such that φ_{n}(θ)→1 as n→∞ for θ in an open interval around zero. For a_{1} =a_{3} =1, a_{2} =−2, t_{1} =0, t_{2} =t and t_{3} =2t, we get

0 ≤ ∑_{i,j=1}^{3} a_{i}a_{j}φ_{n}(t_{j} −t_{i}) = 6φ_{n}(0) − 4(φ_{n}(t)+φ_{n}(−t)) + (φ_{n}(2t)+φ_{n}(−2t)), by non-negative definiteness. Taking real parts and rearranging, this gives ℜφ_{n}(±2t) = \frac{1}{2}ℜ(φ_{n}(2t)+φ_{n}(−2t)) ≥ 2ℜ(φ_{n}(t)+φ_{n}(−t))−3φ_{n}(0) = 4ℜφ_{n}(±t)−3φ_{n}(0). By iteration we therefore readily conclude that φ_{n}(θ)→1 for θ ∈ℝ.

We have φ_{n}(θ) ≡ φ_{Y} (θ)^{1/n} → 1 as n→∞, for θ in an open interval around zero, by continuity of φ_{Y}. Thus φ_{Y}^{1/n} →1 on ℝ. This means that φ_{Y} (θ)≠0 for θ ∈ℝ. □

Fact 4.14 An ℝ-valued id Y is the weak limit of a sequence of compound Poison distributions.

Proof. By Lemma 4.13, φ_{Y} (θ)^{1/n} = 1+n^{−1} ln φ_{Y} (θ)+O(n^{−2}) → 1, so that φ_{Y} (θ) ∼ exp{n(φ_{Y} (θ)^{1/n}−1)} ∼ exp{n ∫_{ℝ}(e^{iθx}−1) dF_{Y^{⋆1/n}}(x)} as n→∞. □

Exercise 41 Can Fact 4.14 be useful to simulate Y in a computer?

Corollary 4.15 A non-negative ℝ-valued id Y is the weak limit of a sequence of non-negative compound Poison distributions.

Exercise 42 Derive Corollary 4.15 from Theorem 4.14.

Definition 4.16 An ℝ-valued Lévy process X is a \underline{subordinator} if it is non-de-creasing a.s.

Exercise 43 Show that an ℝ-valued Lévy process X is a subordinator iff. its generating id distribution X(1) is non-negative a.s.

The following theorem is of fundamental importance.

Theorem 4.17 An ℝ-valued Lévy process X is a \underline{subordinator} iff. it is of type A or B, with Lévy measure supported on (0, ∞) and non-negative drift γ_{0}, i.e., iff.

E{e^{iθX(1)}} = exp{iθγ_{0} + ∫_{(0,∞)}(e^{iθx}−1) dν(x)}  with  γ_{0} ≥0.

Proof. ⇐ Let the right statement in the theorem hold. Clearly, it is enough to show that the triplet (0, ν, 0)_{0} (with γ_{0} removed) corresponds to a non-negative rv. For ν(ℝ)=0 this is trivial. For ν(ℝ)≠0 we put ν_{n} ≡ν1_{{|x|>1/n}}. Then, as n→∞,

∫_{(0,∞)}(e^{iθx}−1) dν_{n}(x) = ∫_{(0,∞)}(e^{iθx}−1)1_{{|x|>1/n}} dν(x) → ∫_{(0,∞)}(e^{iθx}−1) dν(x), (⋆)

i.e., (0, ν_{n}, 0)_{0} →_{d} (0, ν, 0)_{0}: Recall that X is type A or B when A = 0 and ∫_{|x|≤1} |x| dν(x) < ∞. Thus ∫_{ℝ} |x|∧1 dν(x) < ∞, which gives dominated convergence in (⋆).

Now (0, ν, 0)_{0} is a weak limit of non-negative compound Poisson rv's, since ν_{n} is supported on (0, ∞). Therefore (0, ν, 0)_{0} is non-negative.

⇒ Clearly, for the generating triplet (A, ν, γ) of X, we cannot have A ≠ 0, see Exercise 44 below. Hence, with γ_{η} ≡ γ − ∫_{η<|x|≤1} x dν(x), Ψ(−iθ) is given by

∫_{|x|≤η}(e^{iθx}−1−iθx) dν(x)+∫_{|x|>η}(e^{iθx}−1) dν(x)+iθγ_{η} ≡ Ψ_{1}(−iθ)+Ψ_{2}(−iθ)+iθγ_{η} (⋆⋆)

for η ∈ (0, 1]. Let Y_{1} and Y_{2} be id with cgf's Ψ_{1} and Ψ_{2}, respectively. Since Y_{2} is compound Poisson, ν1_{{|x|>η}} must be supported on (0, ∞) for η > 0 (Exercise 45 below), i.e., ν is supported on (0, ∞). Since E{Y_{1}} = 0 (Exercise 46), similar arguing give that γ_{η} ≥ 0 for η > 0 (Exercise 47). Hence ∫_{|x|≤1} |x| dν(x) < ∞, since otherwise γ_{η} →−∞ as η ↓0. Therefore Y_{1} →_{d} 0 as η ↓0, by dominated convergence. And so X(1) is of type A or B, with triplet (0, ν, γ_{0})_{0}, since we must have γ_{η} →γ_{0} (by uniqueness of triplets). Consequently, γ_{0} ≥0. □

Exercise 44 Why is the Gaussian component zero for a non-negative ℝ-valued

id rv?

Exercise 45 In (⋆⋆), why has ν1_{{|x|>η}} support (0, ∞) for X(1) non-negative?

Exercise 46 In (⋆⋆), why is E{Y_{1}}=0?

Exercise 47 In (⋆⋆), for X(1) non-negative, why is γ_{η} ≥0 for η >0?

Exercise 48 Show that an α-stable Lévy motion is a subordinator iff. α < 1, β =1 and τ =γ_{0} ≥0.

Exercise 49 In Exercise 48, why is τ =γ_{0}?

Theorem 4.18 Let X_{1} be a Lévy process with generating triplet (A, ν, γ), and X_{2} an independent subordinator with generating triplet (0, ρ, β)_{0}. Then X ≡ X_{1}∘X_{2} is a Lévy process with generating triplet (Â, ν̂, γ̂) given by

\begin{cases}Â & = & βA \\ ν̂(B) & = & βν(B) + ∫_{(0,∞)} P\{X_{1}(1)^{⋆s} ∈B\} dρ(s) \\ γ̂ & = & βγ + ∫_{(0,∞)} ∫_{|x|≤1} x dF_{X_{1}(1)^{⋆s}}(x) dρ(s) \end{cases} .

Exercise 50 Explain why X is a Lévy process in Theorem 4.18.

Exercise 51 Try to derive the triplet (Â, ν̂, γ̂) in Theorem 4.18.

Exercise 52 Show that X(t) really is a rv for t≥0 in Theorem 4.18.

Exercise 53 In Theorem 4.18, show that X is a subordinator if X_{1} is.

Definition 4.19 In Theorem 4.18, the transformation of X_{1} to X is called \underline{sub}- \underline{ordination by X_{2}}. Further, any Lévy process with the law of X, for a suitable X_{2}, is called \underline{subordinate to X_{1}}.

4.4 Something I came to think of this time ...

Vi väcker er då ni skall gå hem! OBS: Limmet är hälsovådligt att indandas.

5.1 Ornstein-Uhlenbeck Processes

Here we give an introductory coverage of id Ornstein-Uhlenbeck processes.

Let {L(t)}_{t∈ℝ} be an ℝ^{d}-valued Lévy process. This means that L satisfies an obvious version of Definition 1.8 [including L(0)=_{d} 0], extended to the whole real line.

For a constant c>0, the ℝ^{d}-valued \underline{Ornstein-Uhlenbeck (OU) process} is given by

Z(t) ≡ ∫_{−∞}^{t} e^{−c(t−s)} dL(s) for t∈ℝ.

The stochastic integral can be defined as a limit of Riemann sums (see Lecture 6). If Z is well-defined, it will be a \underline{stationary process}, since by homogeneity of L,

{Z(t+h)}_{t∈ℝ} = {∫_{−∞}^{t+h} e^{−c(t−(s−h))} dL(s)}_{t∈ℝ} = {∫_{−∞}^{t} e^{−c(t−s)} dL(s+h)}_{t∈ℝ} =_{d} {Z(t)}_{t∈ℝ}.

By Riemman sums and id of L (Definition 1.34, Exercise 14), we see that \underline{Z is id}. Further, Z is a (continuous time) \underline{Markov [AR(1)-] process}, because (trivially)

Z(t) = e^{−c(t−t_{0})}Z(t_{0}) + ∫_{t_{0}}^{t} e^{−c(t−s)} dL(s) for t≥t_{0},

and is a strong weak solution to the \underline{Langevin stochastic differential equation} (sde)

dZ(t) = −cZ(t) dt + dL(t) for t>0, Z(0) = ∫_{−∞}^{0} e^{cs} dL(s),

because the corresponding integrated equation holds, simply by insertion of Z,

Z(0) − c ∫_{0}^{t} Z(s) ds + ∫_{0}^{t} dL(s)

= ∫_{−∞}^{0} e^{cs} dL(s) − c ∫_{s=0}^{s=t} ∫_{r=−∞}^{r=s} e^{c(r−s)} dL(r)ds + ∫_{0}^{t} dL(s) = ∫_{−∞}^{0} e^{cs} dL(s) − c ∫_{r=−∞}^{r=0} ∫_{s=0}^{s=t} e^{c(r−s)} dsdL(r) − c ∫_{r=0}^{r=t} ∫_{s=r}^{s=t} e^{c(r−s)} dsdL(r) + ∫_{0}^{t} dL(s) = ∫_{−∞}^{0} e^{cs} dL(s) − ∫_{−∞}^{0}(e^{cr}−e^{c(r−t)}) dL(r) − ∫_{0}^{t}(1−e^{c(r−t)}) dL(r) + ∫_{0}^{t} dL(s) = ∫_{−∞}^{t} e^{c(r−t)} dL(r) = Z(t).

Since Ψ_{L(s)} =sΨ_{L(1)}, the cgf of Z(t) (independent of t by stationarity) is

Ψ_{Z(t)}(−iθ) = ln E{e^{i⟨θ,∫_{−∞}^{0}e^{cs}dL(s)⟩}} = lim ∑ Ψ_{L(s_{n+1}−s_{n})}(ie^{cs_{n}}θ) = ∫_{−∞}^{0}Ψ_{L(1)}(−ie^{cs}θ) ds.

Using Theorem 4.12, it follows that \underline{Z(t) is selfdecomposable}, since, for b>1,

φ_{Z(t)}(θ)/φ_{Z(t)}(θ/b) = e^{∫_{−∞}^{0}Ψ_{L(1)}(−ie^{cs}θ) ds−∫_{−∞}^{0}Ψ_{L(1)}(−ie^{cs}θ/b) ds} = e^{∫_{− ln(b)/c}^{0}Ψ_{L(1)}(−ie^{cs}θ) ds}.

Here the right-hand side is a chf (that defines ρ_{b} in Theorem 4.12), because (continuous at zero and) the limit of a product of chf's by Riemann sum approximation.

To check when Z is well-defined, let L have generating triplet (A, ν, γ), so that

∫_{−∞}^{0} Ψ_{L(1)}(−ie^{cs}θ) ds = −\frac{1}{2} ∫_{−∞}^{0}⟨e^{cs}θ, Ae^{cs}θ⟩ ds + i ∫_{−∞}^{0}⟨e^{cs}θ, γ⟩ ds + ∫_{−∞}^{0}[∫_{ℝ^{d}} i⟨e^{cs}θ, y⟩(1_{{|e^{cs}y|≤1}} − 1_{{|y|≤1}}) dν(y)] ds + ∫_{−∞}^{0}[∫_{ℝ^{d}}(e^{i⟨e^{cs}θ,y⟩}−1−i⟨e^{cs}θ, y⟩) 1_{{|e^{cs}y|≤1}} dν(y)] ds

+ ∫_{−∞}^{0}[∫_{ℝ^{d}}(e^{i⟨e^{cs}θ,y⟩}−1) 1_{{|e^{cs}y|>1}} dν(y)] ds.

Here the two first integrals on the right-hand side are trivially well-defined. The third integral is well-defined, since it is “dominated” by the integral

|θ| ∫_{ℝ^{d}}[∫_{−∞}^{−(ln(|y|)/c)^{+}}e^{cs}|y| ds] 1_{{|y|>1}} dν(y) = |θ| ν({y ∈ℝ^{d}: |y|>1}) < ∞,

while, for some constant C >0, the fourth is dominated by

C|θ|^{2}∫_{ℝ^{d}}[∫_{−∞}^{−(ln(|y|)/c)^{+}}e^{2cs}|y|^{2} ds] dν(y) = C|θ|^{2}∫_{ℝ^{d}} 1∧|y|^{2} dν(y) < ∞.

As for the fifth integral on the right, it is dominated by

2 ∫_{ℝ^{d}}[∫_{−(ln(|y|)/c)^{+}}^{0} ds] dν(y) = (2/c) ∫_{|y|>1} ln(|y|) dν(y).

Thus the requirement ensuring that the OU process is well-defined becomes

∫_{{y∈ℝ^{d}: |y|>1}} ln(|y|) dν(y) < ∞.

5.2 Weak Convergence of ID Distributions

Theorem 5.1 For id {Y_{n}}_{n=1}^{∞} with triplets (A_{n}, ν_{n}, γ_{n}) we have Y_{n} →_{d} Y for some Y, iff. Y is id with triplet (A, ν, γ) such that the following conditions hold;

(1) lim_{n→∞} ∫_{|y|>ε} f(y) dν_{n}(y) = ∫_{|y|>ε} f(y) dν(y) for bounded f ∈ℂ(ℝ^{d}) and ε>0;

(2) lim_{ε↓0} lim sup_{n→∞} |⟨θ, (A_{n}−A)θ⟩ + ∫_{|y|≤ε}⟨θ, y⟩^{2} dν_{n}(y)| = 0 for θ ∈ℝ^{d};

(3) lim_{n→∞} γ_{n} = γ.

Exercise 54 In one direction Theorem 5.1 is immediate: Prove that part.

5.3 Lévy-Itô Decomposition

The Lévy-Itô decomposition expresses the sample function of an additive process as a sum of a continuous part and an independent compensated sum of independent jumps. It is very important! The proof is long and difficult and cannot be given here. We state the result for Lévy processes only. The general formulation with additive processes is only notationally more complicated.

On http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Ito.html we found

31

Definition 5.2 Let (S, 𝔖, σ) be a σ-finite measure space. An \overline{ℕ}-valued stochastic process \{N(B)\}_{B∈𝔖} is a \underline{Poisson random measure on S with intensity measure σ}, if the following conditions hold;

(1) N(B) is Po(σ(B))-distributed for B ∈𝔖;

(2) {N(B_{i})}_{i=1}^{n} are independent for disjoint {B_{i}}_{i=1}^{n} ⊆𝔖;

(3) N(·, ω) is a measure on S for ω ∈Ω.

Theorem 5.3 (Lévy-Itô Decomposition) Let X be a Lévy process with generating triplet (A, ν, γ). Define

N(B) ≡ #{s : (s, X(s)−X(s^{−})) ∈ B} for Borel B ⊆ (0, ∞)×ℝ^{d}.

The following assertions hold:

(1) N is a Poisson random measure with intensity measure σ given by

σ((0, t]×B) ≡ tν(B) for t>0 and B ∈B(ℝ^{d}).

(2) With a.s. locally uniform convergence in t, the following limit X_{1} is a Lévy process with generating triplet (0, ν, 0),

X_{1}(t) = lim_{ε↓0} ∫_{(0,t]×{ε<|x|≤1}}[x dN(s, x) − x dσ(s, x)] + ∫_{(0,t]×{|x|>1}} x dN(s, x).

(3) X_{2} ≡X−X_{1} is a Lévy process with generating triplet (A, 0, γ). It is a.s. continuous and independent of X_{1}.

Theorem 5.4 (Lévy-Itô Decomposition) Let X be a Lévy process with generating triplet (A, ν, γ_{0})_{0}. With the notation of Theorem 5.3, the following assertions hold:

(1) The following process X_{3} is a Lévy process with generating triplet (0, ν, 0)_{0},

X_{3}(t) = ∫_{(0,t]×ℝ^{d}} x dN(s, x).

(2) X_{4} ≡ X −X_{3} is a Lévy process with generating triplet (A, 0, γ_{0}). It is a.s. continuous and independent of X_{3}.

5.4 Sample Function Behaviour

Theorem 5.5 A Lévy process is a.s. continuous iff. its Lévy measure is zero.

Proof. By Theorem 5.3, J ≡ #{s∈(0, t] : |X(s)−X(s^{−})|>ε} satisfies E{J} = E{N((0, t]×{x∈ℝ^{d}:|x|>ε})} = t ∫_{|x|>ε} dν(x). □

Definition 5.6 A function f:[0, ∞)→ℝ^{d} is \underline{piecewise constant} if there exist 0=t_{0} <t_{1} <… with lim_{n→∞} t_{n} =∞ and f constant on the intervals {[t_{n−1}, t_{n})}_{n=1}^{∞}.

Theorem 5.7 A Lévy process is a.s. piecewise constant iff. it is type A with γ_{0} =0.

Proof. ⇐ For A=0, ν(ℝ^{d})<∞ and γ_{0} =0, by Theorem 5.4, the number of jumps of X =_{d} X_{3} in (0, t] has a Po(tν(ℝ^{d}))-law, and thus is finite a.s.

⇒ Since #{s ∈ (0, t] : |X(s)−X(s^{−})| > ε} has a Po(tν({x ∈ ℝ^{d}: |x| > ε}))-law, by Theorem 5.3, ν(ℝ^{d})<∞, since otherwise the number of jumps in (0, t] is infinite a.s., by sending ε ↓ 0. For ν(ℝ^{d}) < ∞, X_{3} is piecewise constant by ⇐ . Now Theorem 5.4 gives A=0 and γ_{0} =0, since X_{4} is not piecewise constant, if non-zero. □

Theorem 5.8 (1) A Lévy process with infinite Lévy measure has jumping times that a.s. are countable and dense in [0, ∞).

(2) A Lévy process with finite Lévy measure ν has jumping times that a.s. are countable in increasing order, with the time to the first jump exp(ν(ℝ^{d}))-distributed.

Proof. Jumps are countable for càdlàg sample paths. Let T_{ε} ≡ inf{s > 0 : |X(s)− X(s^{−})|>ε}. By Theorem 5.3, unless zero or infinite, T_{ε} is exp-distributed with

P{T_{ε} ≤t} = P{N((0, t]×{x∈ℝ^{d}:|x|>ε})≥1} = 1 − exp{−t ∫_{|x|>ε} dν(x)}. For ν(ℝ^{d})=∞, lim_{ε↓0} P{T_{ε} ≤t} = 1 for t>0, giving assertion (1) (see Exercise 55 below). For ν(ℝ^{d})<∞, we get an exp(ν(ℝ^{d}))-law as ε↓0, and Theorem 5.3 gives the claimed countability, since the number of jumps in (0, t] has a Po(tν(ℝ^{d}))-law. □

Exercise 55 Explain the argument in the proof of Theorem 5.8 for ν(ℝ^{d})=∞.

Definition 5.9 The \underline{variation v(t, f)} over (0, t] of f:[0, ∞)→ℝ^{d} is defined

v(t, f) ≡ sup{∑_{k=1}^{n} |f(t_{k})−f(t_{k−1})| : 0=t_{0} <t_{1} <…<t_{n} =t, n∈ℕ}.

The following important result is again proved making crucial use of the Lévy-Itô Decomposition. But now arguments are quite long and difficult.

Theorem 5.10 (1) A Lévy process X of type A or B with generating triplet (0, ν, γ_{0})_{0} has a.s. locally finite variation V (t)≡v(t, X), that is a subordinator with generating triplet (0, ρ, |γ_{0}|)_{0}, where

ρ(B) = ν({x∈ℝ^{d}: |x|∈B}) for B ∈B(ℝ).

(2) A Lévy process of type C has a.s. locally infinite variation.

Exercise 56 Exercise E 22.1 in Sato’s book.

Exercise 57 Exercise E 22.5 in Sato’s book.

5.5 Recurrence and Transience

There are many results on recurrence and transience for Lévy processes. We only display a single result, to give some flavour of the topic (see Sato’s book and tutorials on more information). An application is given to stable Lévy motions.

Definition 5.11 A Lévy process X is \underline{reccurent} if

lim inf_{t→∞} |X_{t}| = 0 a.s.,

and \underline{transient} if

lim_{t→∞} |X_{t}| = ∞ a.s.

Theorem 5.12 Given an a>0, a Lévy process X is reccurent iff.

∫_{0}^{∞} P{|X(t)|<a} dt = ∞ a.s.,

and transient iff.

∫_{0}^{∞} P{|X(t)|<a} dt < ∞ a.s.

Exercise 58 Show that an ℝ-valued non-trivial α-stable Lévy motion X, α≠1, may be written X(t) = L(t)+τt, where L is strictly stable and τ ∈ ℝ a constant. Conclude that X is strictly α-stable iff. τ =0.

Example 2 Let X(t)=L(t)+τt be an ℝ-valued non-trivial α-stable, α∈(1, 2], Lévy motion, with L strictly stable and τ ∈ ℝ a constant (cf. Exercise 58). Recall that L(1) has a density function f_{L(1)}, by Exercise 38. Assume that f_{L(1)}(0)>0.

By self-similarity of L (this is Fact 3.9), we have ∫_{0}^{∞} P{|L(t)|<a} dt = ∫_{0}^{∞} P{|L(1)|<t^{−1/α}a} dt ≥ ∫_{t_{0}}^{∞} f_{L(1)}(0) t^{−1/α}a dt = ∞

(for some constant t_{0} > 0). Hence L is reccurent, by Theorem 5.12. I.e., strictly α-stable Lévy motions with α∈(1, 2] are reccurent.

If X is not strictly α-stable, so that τ ≠0, then we get P{|X(t)|<a} = P{|t^{1/α}L(1)+τt|<a} = P{−t^{−1/α}a−τt^{1−1/α} <L(1)<t^{−1/α}a−τt^{1−1/α}}, i.e., L(1) belongs to an interval of width 2t^{−1/α}a located “around” −τt^{1−1/α}. By unimodality of α-stable distributions (recall Section 4.2), this is at most

2t^{−1/α}a f_{L(1)}(±t^{−1/α}a−τt^{1−1/α}) ≤ 2t^{−1/α}a f_{L(1)}(−\frac{1}{2}τt^{1−1/α})

for t large enough. Hence X is transient, by Theorem 5.12, since ∫_{0}^{∞} t^{−1/α}f_{L(1)}(−\frac{1}{2}τt^{1−1/α}) dt = ∫_{0}^{∞} 2/(τ(1−1/α)) f_{L(1)}(−t̂) dt̂< ∞. #

Exercise 59 Show that a non-zero α-stable Lévy motion, α∈(0, 1), is transient.

5.6 Something I came to think of this time ...

Go ahead, make my day punk!

In Lectures 6 and 7, as preparation for sde, we study stochastic integrals of nonrandom functions wrt. additive processes, following the standard text “Rajput & Rosiński: Spectral Representation of Infinitely Divisible Processes”. The treatment has been slightly

“watered down”, since it is estimated that most participants do not benifit from the full truth.

6.1 Outline of Stochastic Integration wrt. Semi-martingales

Let G be an ℝ-valued Gaussian additive process, where G(t) has triplet (A(t), 0, γ(t)), with γ of finite variation. For measurable x:[0, ∞)→ℝ the stochastic integral

∫_{0}^{t} x dG exists ⇔ ∥1_{[0,t]}x∥_{(A,0,γ)} ≡ [∫_{0}^{t} A(s)x(s)^{2} ds]^{1/2} + ∫_{0}^{t} |x(s)| d|γ|(s) < ∞.

We may integrate a predictable (essentially meaning left-continuous and adapted to the filtration associated with G) ℝ-valued process X wrt. G,

∫_{0}^{t} X dG exists  ⇔  P{∥1_{[0,t]}X∥_{(A,0,γ)} <∞} = 1.

An continuous semi-martingale S is a time-changed version S = G ∘ T of G, where {T (t)}_{t≥0} is an non-decreasing, predictable and continuous stochastic process. Formally, it is an additive process with time dependent stochastic triplet (A∘T, 0, γ∘T). We may integrate a predictable process X wrt. S,

∫_{0}^{t} X dS exists (for t≥0) ⇔ P{∥1_{[0,t]}X∥_{(A∘T,0,γ∘T)} <∞} = 1 (for t≥0).

The stochastic integral process {∫_{0}^{t} X dS}_{t≥0} is again a continuous semi-martingale.

For an ℝ-valued additive process L such that L(t) has triplet (A(t), ν(t), γ(t)) [recall that L(t) is id by Fact 1.33], and measurable x:[0, ∞)→ℝ, the integral

∫_{0}^{t} x dL exists  ⇔  ∥1_{[0,t]}x∥_{(A,ν,γ)} < ∞,

where ∥·∥_{(A,ν,γ)} is a F -pseudo-norm on a Musielak-Orlicz space (see Section 6.3 below), that reduces to ∥·∥_{(A,0,γ)} for L Gaussian ν(t)=0. We may integrate a predictable (wrt. the filtration associated with L) process X wrt. L,

∫_{0}^{t} X dL exists  ⇔  P{∥1_{[0,t]}X∥_{(A,ν,γ)} <∞} = 1.

A not necessarily continuous semi-martingale S formally is an additive process with (time dependent) stochastic triplet (Â(t), ν̂(t), γ̂(t)). For a predictable X,

∫_{0}^{t} X dS exists (for t≥0)  ⇔  P{∥1_{[0,t]}X∥_{(Â,ν̂,γ̂),t} <∞} = 1 (for t≥0).

The process {∫_{0}^{t} X dS}_{t≥0} is again a semi-martingale. (This is the general theory!)

6.2 Representation of Additive Processes

Fact 6.1 For X additive in law, X(t)−X(s) is id for 0≤s≤t.

Proof. For s≥0, X′ ≡X(· +s)−X(s) is an additive process in law, because

\begin{matrix}(X(s), X′(t_{1}), X′(t_{2})−X′(t_{1}), …, X′(t_{n})−X′(t_{n−1})) \\ =_{d} (X(s), X(t_{1}+s)−X(s), X(t_{2}+s)−X(t_{1}+s), …, X(t_{n}+s)−X(t_{n−1}+s)) \end{matrix}

are independent for 0≤t_{1} ≤…≤t_{n}. The assertion thus follows from Fact 1.33. □

Fact 6.2 Given a family of probability densities {μ_{s,t}}_{0≤s≤t<∞} on ℝ^{d}, there exists X that is additive in law with L(X(t)−X(s)) = μ_{s,t} for 0≤s≤t, iff. the following conditions hold;

(1) μ_{r,s}⋆μ_{s,t} = μ_{r,t} for 0≤r ≤s ≤t;

(2) μ_{s,t} →_{d} δ_{0} as s↑t>0 and as t↓s≥0 [in particular, μ_{t,t} = δ_{0} for t≥0].

Exercise 60 Prove ⇒ in Fact 6.2.

Exercise 61 Prove ⇐ in Fact 6.2.

Definition 6.3 An X that is additive in law such that X(t) has triplet (A_{t}, ν_{t}, γ_{t}) for t≥0, has \underline{system of triplets \{(A_{t}, ν_{t}, γ_{t})\}_{t≥0}}.

Corollary 6.4 An additive process in law is uniqely determined in law by its system of triplets.

Proof. Let X and X′ be additive with common systems of triplets. With μ_{s,t} ≡ L(X(t)−X(s)) and μ′_{s,t} ≡ L(X′(t)−X′(s)) for 0 ≤ s ≤ t, we have μ_{0,t} = μ′_{0,t} for t≥0, by assumption. This gives

μ_{0,s}⋆μ′_{s,t} = μ′_{0,s}⋆μ′_{s,t} = μ′_{0,t} = L(X′(t)) = L(X(t)) = μ_{0,t} = μ_{0,s}⋆μ_{s,t} for 0≤s≤t.

Hence μ′_{s,t} = μ_{s,t}, by using chf [since the chf of μ_{0,s} is non-zero, by Lemma 4.13, since X(s) is id, by Fact 6.1]. It follows that X =_{d} X′ (see Exercise 62 below). □

Exercise 62 Explain why μ_{s,t} = μ′_{s,t} for 0≤s≤t, gives the X =_{d} X′ requested in the proof of Corollary 6.4.

Theorem 6.5 The triplets {(A_{t}, ν_{t}, γ_{t})}_{t≥0} is system of triplets for an additive process in law, iff. the following conditions hold;

(1) A_{0} =0, ν_{0} =0 and γ_{0} =0;

(2) for 0≤s≤t, ⟨θ,A_{s}θ⟩ ≤ ⟨θ,A_{t}θ⟩ for θ ∈ℝ^{d}, and ν_{s}(B)≤ν_{t}(B) for B ∈B(ℝ^{d});

(3) as s→t≥0, A_{s}→A_{t}, ν_{s}(B)→ν_{t}(B) for B ∈B(ℝ^{d}) with 0∉clos(B), and γ_{s}→γ_{t}.

Proof. ⇐ For μ_{s,t} the id law with triplet (A_{t}−A_{s}, ν_{t}−ν_{s}, γ_{t}−γ_{s}), for 0 ≤ s ≤ t, conditions (1) and (2) of Fact 6.2 hold.

⇒ Property (1) expresses that X(0) =_{d} 0. Further, L(X(t) − X(s)) is id, by Fact 6.1, with triplet (A_{t}−A_{s}, ν_{t}−ν_{s}, γ_{t}−γ_{s}), since φ_{X(t)} = φ_{X(s)}φ_{X(t)−X(s)}, so that φ_{X(t)−X(s)} = φ_{X(t)}/φ_{X(s)}. This gives claim (2). As s→t, X(s)→_{d} X(t), by stochastic continuity. Hence, by Theorem 5.1.3, γ_{s} →γ_{t}, and by Theorem 5.1.2,

lim_{ε↓0} lim sup_{s→t} |⟨θ, (A_{s}−A_{t})θ⟩ + ∫_{|y|≤ε}⟨θ, y⟩^{2} dν_{s}(y)| = 0 for θ ∈ℝ^{d}.

Thus A_{s} →A_{t}, since taking t̂>t, the property (2) and Dominated Convergence give

lim_{ε↓0} lim sup_{s→t} ∫_{|y|≤ε}⟨θ, y⟩^{2} dν_{s}(y) ≤ lim_{ε↓0} ∫_{|y|≤ε}⟨θ, y⟩^{2} dνt̂(y) ≤ lim_{ε↓0} ∫_{|y|≤ε} |θ|^{2}|y|^{2} dνt̂(y) = 0.

Moreover, ν_{s} ≤ ν_{t̂} gives dν_{s}(x) = g_{s}(x)dνt̂(x) for some measurable g_{s}: ℝ^{d} → ℝ, for s≤t̂, by Radon-Nikodym, with g_{s} ↑g_{s′} and g_{s} ↓g_{s′} a.e. (νt̂), as s↑s′ ≤t̂ and s↓s′ <t̂, by Exercise 64 below. Hence Monotone Convergence gives, as s↑t, for ε>0,

ν_{s}(1_{{|x|≥ε}}B) = ∫_{|y|>ε} 1_{B}(x)g_{s}(x) νt̂(x) → ∫_{|y|>ε} 1_{B}(x)g_{t}(x) νt̂(x) = ν_{t}(1_{{|x|≥ε}}B).

Similarly, ν_{s}(1_{{|x|≥ε}}B) ↓ ν_{t}(1_{{|x|≥ε}}B) as s↓t. This finishes the proof of (3). □

Exercise 63 Explain why (3) [together with a bit of (2)] gives stochastic continuity of X in the proof of ⇐ in Theorem 6.5.

Exercise 64 Explain why g_{s} ↑g_{s′} and g_{s} ↓g_{s′} a.e. (νt̂) as s↑s′ ≤t̂ and s↓s′ <t̂, respectively, in the proof of ⇒ in Theorem 6.5.

Corollary 6.6 A family of triplets {(0, ν_{t}, 0)}_{t≥0} is system of triplets for an additive process in law, iff.

ν([0, t]×B) ≡ ν_{t}(B) for t≥0 and B ∈B(ℝ^{d})

defines a measure on [0, ∞)×ℝ^{d}, such that

ν({t}×ℝ^{d}) = 0  and  ∫_{[0,t]×ℝ^{d}} 1∧|x|^{2} dν(s, x) < ∞  for t≥0.

Exercise 65 Prove ⇒ in Corollary 6.6.

Exercise 66 Prove ⇐ in Corollary 6.6.

6.3 Some Facts from Measure Theory and Functional Analysis

Definition 6.7 A family S of subsets of a set S is a \underline{semi-ring} if the following conditions hold;

(1) ∅ ∈ S;

(2) A, B ∈ S ⇒ A∩B ∈ S;

(3) A, B ∈ S ⇒ B∖A = ⋃_{k=1}^{n} E_{k} for some disjoint {E_{k}}_{k=1}^{n} ⊆ S.

Definition 6.8 A family R of subsets of a set S is a \underline{ring} if the following conditions hold;

(1) A, B ∈ R ⇒ A∩B ∈ R;

(2) A, B ∈ R ⇒ AΔB ≡ (A∖B)∪(B∖A) ∈ R.

Exercise 67 Show that rings are semi-rings. Show that there exists a smallest ring, the \underline{generated ring}, that contains any given semi-ring.

Definition 6.9 A family ℜ of subsets of a set S is a \underline{δ-ring} if it is a ring such that

{A_{k}}_{k=1}^{∞} ⊆ ℜ ⇒ ⋂_{k=1}^{∞} A_{k} ∈ ℜ.

Definition 6.10 A family ℜ of subsets of a set S is a \underline{σ-ring} if it is a ring such that

{A_{k}}_{k=1}^{∞} ⊆ ℜ ⇒ ⋃_{k=1}^{∞} A_{k} ∈ ℜ.

Definition 6.11 A famliy S of subsets of a set S is \underline{σ-R} if

S = ⋃_{k=1}^{∞} A_{k} for some {A_{k}}_{k=1}^{∞} ⊆ S.

Example 3 A σ-ring ℜ is a δ-ring, since for {A_{k}}_{k=1}^{∞} ⊆ℜ, with A=⋃_{k=1}^{∞} A_{k},

ℜ ∋ AΔ ⋃_{k=1}^{∞}(AΔA_{k}) = A ∩ [⋃_{k=1}^{∞}(A ∩ A_{k}^{c})]^{c} = A ∩ ⋂_{k=1}^{∞}(A^{c} ∪ A_{k}) = ⋂_{k=1}^{∞} A_{k}.

A “typical” δ-ring (that is not a σ-ring) is {B ∈B(ℝ^{d}) :∫_{B} dx<∞}.

σ-finite integration theory can be developed by considering measures (σ-additive set functions) over the σ-R δ-ring of sets with well-defined and finite measure (set function values). (See below for notation!) In advanced integration theory and in stochastic integration, this is the standard. #

Definition 6.12 A family A of subsets of a set S is an \underline{algebra} if it is a ring such that S ∈A.

Exercise 68 Show that a ring R is an algebra iff. it is closed under taking complements. Show that there exists a smallest algebra, the \underline{generated algebra}, that contains R.

Definition 6.13 A family 𝔄 of subsets of a set S is a \underline{σ-algebra} if it is a σ-ring such that S ∈𝔄. A \underline{measurable space} (S, 𝔄) is a set S together with a σ-algebra 𝔄 of subsets S.

Exercise 69 For a ring R, show that there exists a smallest δ-ring, the \underline{generated δ-ring}, a smallest σ-ring, the \underline{generated σ-ring}, that is a σ-algebra if R is σ-R, and a smallest σ-algebra, the \underline{generated σ-algebra σ(R)}, that contains R. What happens if R is a semi-ring instead?

Definition 6.14 A function μ : S → \overline{ℝ} ≡ [−∞, ∞] on a family S of subsets of a set S is an \underline{additive set function} if the following conditions hold;

(1) μ(∅) = 0;

(2) μ(⋃_{k=1}^{n} A_{k}) = ∑_{k=1}^{n} μ(A_{k}) for disjoint {A_{k}}_{k=1}^{n} ⊆ S with ⋃_{k=1}^{n} A_{k} ∈ S.

Exercise 70 Show that (2) ⇒ (1), in Definition 6.14, if μ(A) ∈ ℝ for some A ∈ S. Why cannot an additive set function on a ring take both the values −∞ and ∞?

Definition 6.15 An additive set function μ on a family S of subsets of a set S is \underline{σ-additive} if

μ(⋃_{k=1}^{∞} A_{k}) = ∑_{k=1}^{∞} μ(A_{k}) for disjoint {A_{k}}_{k=1}^{∞} ⊆ S with ⋃_{k=1}^{∞} A_{k} ∈ S.

Definition 6.16 A \underline{measure} is a non-negative σ-additive set function. A \underline{measure space} (S, 𝔄, μ) is a set S together with a measure μ on a σ-algebra 𝔄 of subsets of S.

Definition 6.17 A function μ:S →\overline{ℝ} on a family S of subsets of a set S is \underline{σ-finite} if

B ⊆ ⋃_{k=1}^{∞} A_{k} for some {A_{k}}_{k=1}^{∞} ⊆ S with μ(A_{k})∈ℝ, for each B ∈S.

Exercise 71 Which of Definitions 6.14-6.17 apply to μ(A)≡#{t: t∈A} for A⊆S?

Theorem 6.18 An additive set function μ on a semi-ring S of subsets of a set S that is finite, or non-negative (non-positive), has a unique extension to the generated ring (see Exercise 67). If μ is σ-additive, then so is the extension.

Theorem 6.19 A σ-additive set function on a ring R of subsets of a set S that is finite, or nonnegative (non-positive) and σ-finite, has a unique extension to the generated δ-ring (see Exercise 69) that is finite, or non-negative (non-positive) and σ-finite, respectively.

Theorem 6.20 A measure on a ring R of subsets of a set S has an extension to the generated σ-ring (see Exercise 69). The extension is unique and σ-finite if the mesure is σ-finite.

Definition 6.21 A function ρ: T ×T → [0, ∞) is a \underline{pseudo-metric} on the set T if the following conditions hold;

(0) ρ(t, t) = 0 for t∈T;

(1) ρ(r, t) ≤ ρ(r, s) + ρ(s, t)  for r, s, t∈T  (Δ-inequality);
(2) ρ(s, t) = ρ(t, s)  for s, t∈T  (symmetry).

Exercise 72 Explain why, on the space of rv's with well-defined mean, ρ(Y_{1}, Y_{2}) ≡ E{|Y_{1}− Y_{2}|} is a pseudo-metric, but not a metric.

Definition 6.22 A pseudo-metric ρ on a set T is a \underline{metric} if

ρ(s, t) = 0 ⇒ s = t for s, t∈T.

Virtually all results for metric spaces have versions for pseudo-metric spaces. In fact, often results extends immediately, without any changes at all.

Definition 6.23 A function ∥·∥ : T → [0, ∞) is a \underline{pseudo-norm} (also called \underline{semi-norm}) on a linear space T (over ℝ say), if the following conditions hold;

(0) ∥0∥ = 0;

(1) ∥s+t∥ ≤ ∥s∥ + ∥t∥  for s, t∈T  (Δ-inequality);
(2) ∥λt∥ = |λ|∥t∥  for λ∈ℝ and t∈T  (homogeneity).

Definition 6.24 A pseudo-norm ∥·∥ on T is a \underline{norm} if

∥t∥ = 0 ⇒ t = 0 for t∈T.

Exercise 73 Show that for ∥·∥ a pseudo-norm (norm), ∥· − ·∥ is a pseudo-metric (metric).

Definition 6.25 A function ∥·∥: T → [0, ∞) is a \underline{F -pseudo-norm} (also called \underline{F -semi-norm}) on a linear space T (over ℝ say), if the following conditions hold;

(0) ∥0∥ = 0;

(1) ∥s+t∥ ≤ ∥s∥ + ∥t∥ for s, t∈T;

(2) ∥λ_{k}t_{k}−λt∥ → 0 whenever ℝ ∋ λ_{k} → λ ∈ ℝ and t_{1}, t_{2}, …, t∈T with ∥t_{k}−t∥ → 0.

Definition 6.26 An F -pseudo-norm ∥·∥ is an \underline{F -norm} if

∥t∥ = 0 ⇒ t = 0 for t∈T.

Exercise 74 Show that a pseudo-norm (norm) is an F -pseudo-norm (F -norm).

Definition 6.27 A \underline{pseudo-modular of moderate growth ( pmmg)} on a linear space T (over ℝ say), is a function Φ:T →[0, ∞) such that the following conditions hold;

(0) Φ(0) = 0;

(1) ℝ ∋ λ → Φ(λt) is continuous, even, non-incresing on (−∞, 0], and non-decreasing on [0, ∞), for t∈T;

(2) Φ(s+t) ≤ C(Φ(s) + Φ(t)) for s, t∈T, for some constant C >0.

Exercise 75 Define “modular of moderate growth”. Verify that Φ(Y) ≡ E{|Y |^{2}} is a pmmg, but not a pseudo-norm, on the space of Y that satisfy E{|Y |^{2}}<∞.

Theorem 6.28 A pmmg Φ on T induces a pseudo-metrizable topology on T with open subbasis

{{t∈T : Φ(t−t_{1})<r_{1}, …, Φ(t−t_{n})<r_{n}}}_{t_{1},...,t_{n}∈T, r_{1},...,r_{n}>0, n∈ℕ}.

Further, addition and scalar multiplication are continuous (i.e., T is a topological vector space).

Theorem 6.29 (Musielak-Orlicz) Let (S, 𝔄, μ) be a complete σ-finite measure space, with 𝔄 the Borel σ-algebra of a separable topological space. Let ψ: S×ℝ → [0, ∞) be a function such that the following conditions hold;

(1) ψ(s, 0)=0 for s∈S;

(2) ψ(s, ·) is continuous, even, non-incresing on (−∞, 0], and non-decreasing on [0, ∞), for s∈S;

(3) ψ(·, t) ∈ 𝕃^{0}(S, 𝔄) for t∈ℝ;

(4) ψ(s, C_{1}t) ≤ C_{2}ψ(s, t) for s∈S and t∈ℝ, for some constants C_{1}, C_{2} >1. Then ψ(·, f(·)) ∈ 𝕃^{0}(S, 𝔄) for f ∈𝕃^{0}(S, 𝔄), and

Φ(f) ≡ ∫_{S} ψ(s, f(s)) dμ(s)  is a pmmg on  𝕃^{ψ}(S, 𝔄, μ) ≡ {f ∈𝕃^{0}(S, 𝔄) : Φ(f)<∞}.

The induced topology is complete pseudo-metric, with addition and scalar multiplication continuous. If in addition Φ(1)<∞, then 𝕃^{ψ}(S, 𝔄, μ) is separable, with simple functions as a dense subset.

Definition 6.30 The space \underline{𝕃^{ψ}(S, 𝔄, μ)} in Theorem 6.29 is a \underline{Musielak-Orlicz space}. If ψ(s, ·)= ψ(·) does not depend on s∈S, then it is an \underline{Orlicz space}.

Exercise 76 Show that the space of rv's with the modular Φ(Y) ≡ E{|Y |∧1} is an Orlicz space, and that the convergence Y_{n} →Y ⇔_{def} Φ(Y_{n}−Y)→0 is convergence in probability.

Exercise 77 Many important Musielak-Orlicz spaces do not satsify Φ(1)<∞. Give an example of this. However, the criterion Φ(1)<∞ for denseness of simple functions can often be used anyway, by a σ-finiteness type of argument. Explain how!

6.5 Presentation av Examensarbete, Fredag 27 September kl. 11.15, S1

44

7.1 Stochastic Integrals wrt. ID Random Measures

In Lecture 7, \underline{𝔖} is a σ-R δ-ring of subsets of a set \underline{S}. (Simply view 𝔖 as a σ-algebra!)

Definition 7.1 An \underline{independently scattered random measure ( isrm) on (S, 𝔖)} is an ℝ-valued stochastic process {Λ(A)}_{A∈𝔖}, such that the following conditions hold;

(1) {Λ(A_{k})}_{k=1}^{n} are independent for disjoint {A_{k}}_{k=1}^{n} ⊆𝔖;

(2) Λ(⋃_{k=1}^{∞} A_{k}) =_{a.s.} ∑_{k=1}^{∞} Λ(A_{k}) for disjoint {A_{k}}_{k=1}^{∞} ⊆𝔖 with ⋃_{k=1}^{∞} A_{k} ∈𝔖.

Definition 7.2 An \underline{id isrm ( idisrm)} is an isrm with id process values.

Theorem 7.3 A function Λ:𝔖→𝕃^{0}(Ω, 𝔉) is an idisrm on (S, 𝔖) iff.

E\{e^{iθΛ(A)}\} = exp\{iθν_{0}(A) − \frac{1}{2}θ^{2}ν_{1}(A) + ∫_{ℝ}(e^{iθx}−1−iθx1_{\{|x|≤1\}}) dF_{A}(x)\}

for θ ∈ℝ and A∈𝔖, for some (necessarily unique) signed finite measure ν_{0} on 𝔖, finite measure ν_{1} on 𝔖, Lévy measures {F_{A}}_{A∈𝔖} on ℝ with {F_{(·)}(B)}_{B∈B(ℝ),0∉clos(B)} finite measures on 𝔖.

Conversely, given such ν_{0}, ν_{1} and F_{A}, there exists an idisrm Λ with the chf above.

Exercise 78 Show that an idisrm on (S, 𝔖) is an id stochastic process. Explain why an isrm on ℝ or ℝ^{+} must be an idisrm.

Exercise 79 Motivate (or prove) Theorem 7.3.

Definition 7.4 For an idisrm on (S, 𝔖) with chf as in Theorem 7.3, the \underline{con}- \underline{trol measure λ} is the unique extension to σ(𝔖) of the measure on 𝔖

λ = |ν_{0}| + ν_{1} + ∫_{ℝ} 1∧x^{2} dF_{(·)}(x).

Corollary 7.5 For an idisrm Λ on (S, 𝔖) with control measure λ, we have

λ(A_{n})→0 for {A_{n}}_{n=1}^{∞} ⊆𝔖 ⇔ Λ(A′_{n})→0 whenever 𝔖∋A′_{n} ⊆A_{n} for n∈ℕ.

Exercise 80 Prove Corollary 7.5.

Definition 7.6 For an idisrm on (S, 𝔖) with chf as in Theorem 7.3 and control measure λ, we define the \underline{characteristic (σ^{2}, ρ, b)} by

σ^{2} = dν_{1}/dλ, ρ(·, B) = dF_{(·)}(B)/dλ and b = dν_{0}/dλ.

Fact 7.7 For the characteristic (σ^{2}, ρ, b) of an idisrm with control measure λ,

|b(s)| + σ^{2}(s) + ∫_{ℝ} 1∧x^{2} ρ(s, dx) = 1 a.e. (λ).

Proof. For A∈𝔖, we have

∫_{A} [|b|+σ^{2}+∫_{ℝ}1∧x^{2} ρ(·, dx)] dλ = |ν_{0}(A)|+ν_{1}(A)+∫_{ℝ}1∧x^{2} dF_{A}(x) = λ(A) = ∫_{A} dλ. □

Fact 7.8 For an idisrm on (S, 𝔖) with control measure λ and characteristic (σ^{2}, ρ, b), we have, for θ ∈ℝ and A∈𝔖,

E\{e^{iθΛ(A)}\} = exp\{∫_{s∈A} [iθb(s) − \frac{1}{2}θ^{2}σ^{2}(s) + ∫_{ℝ}(e^{iθx}−1−iθx1_{\{|x|≤1\}}) ρ(s, dx)] dλ(s)\}.

Exercise 81 Prove Fact 7.8.

Definition 7.9 For an idisrm Λ on (S, 𝔖), and a 𝔖-simple function f = ∑_{k=1}^{n} a_{k}1_{A_{k}}, with \{A_{k}\}_{k=1}^{n} ⊆ 𝔖 disjoint, the \underline{stochastic integral ∫ f dΛ} is defined

∫ f dΛ = ∑_{k=1}^{n} a_{k} Λ(A_{k}).

Fact 7.10 For an idisrm Λ on (S, 𝔖) with characteristic (σ^{2}, ρ, b) and control measure λ, we have for 𝔖-simple f:S →ℝ,

E\{e^{iθ∫f dΛ}\} = exp\{∫_{S} [iθfb − \frac{1}{2}θ^{2}f^{2}σ^{2} + ∫_{ℝ}(e^{iθfx}−1−iθfx1_{\{|x|≤1\}}) ρ(·, dx)] dλ\}.

Hence ∫ f dΛ is id with triplet (A_{f}, ν_{f}, γ_{f}) given by

\begin{cases}A_{f} = ∫_{S} f^{2}σ^{2} dλ \\ ν_{f} = ∫_{S} ∫_{ℝ} 1_{(·)}(f(s)x) ρ(s, dx)dλ(s) \\ γ_{f} = ∫_{S} [fb + ∫_{ℝ} fx(1_{\{|fx|≤1\}}−1_{\{|x|≤1\}}) ρ(·, dx)] dλ \end{cases} .

Proof. For f = ∑_{k=1}^{n} a_{k}1_{A_{k}}, Fact 7.8 shows that E{e^{iθ∫f dΛ}} is

∏_{k=1}^{n} exp\{∫_{A_{k}}[iθa_{k}b − \frac{1}{2}θ^{2}a_{k}^{2}σ^{2} + ∫_{ℝ}(e^{iθa_{k}x}−1−iθa_{k}x1_{\{|x|≤1\}}) ρ(s, dx)] dλ(s)\}. □

Exercise 82 Verify the claim regarding the triplet (A_{f}, ν_{f}, γ_{f}) in Fact 7.10.

Theorem 7.11 For an idisrm Λ on (S, 𝔖) with characteristic (σ^{2}, ρ, b) and control measure λ, the following function ψ satisfies the hypothesis of Theorem 6.29,

ψ(s, t) ≡ [ sup_{|τ|≤|t|}|τ||b(s)+∫_{ℝ} x(1_{{|τx|≤1}}−1_{{|x|≤1}}) ρ(s, dx)|]+t^{2}σ^{2}(s)+ ∫_{ℝ} 1∧|tx|^{2} ρ(s, dx).

The Musielak-Orlicz space 𝕃^{ψ}(S, 𝔄, λ) is metrized by the complete F -pseudo-norm

∥f∥_{ψ} ≡ inf{c>0 : Φ(f/c)≤c}  where  Φ(f) = ∫_{S} ψ(s, f(s)) dλ(s).

We have f ∈𝕃^{ψ}(S, 𝔄, λ) iff. ∥f∥_{ψ} <∞ iff. Φ(f)<∞ iff.

∫_{S}[|f||b +∫_{ℝ} x(1_{{|fx|≤1}}−1_{{|x|≤1}}) ρ(·, dx)| + f^{2}σ^{2} + ∫_{ℝ} 1∧|fx|^{2} ρ(·, dx)] dλ < ∞.

Definition 7.12 With the notation of Theorem 7.11, we write \underline{𝕃^{Λ}} for 𝕃^{ψ}(S, 𝔄, λ).

Fact 7.13 For an idisrm Λ on (S, 𝔖), with the notation of Theorem 7.11, ∥ ∫ f dΛ−∫ g dΛ∥_{ψ} ≡ ∥f−g∥_{ψ} is an F -pseudo-norm on I^{Λ} ≡ {∫ f dΛ : simple f:S →ℝ}. The completion of I^{Λ} can be topologically identified with 𝕃^{Λ}.

Exercise 83 Provide arguments for Fact 7.13.

Lemma 7.14 For an idisrm Λ on (S, 𝔖), if {f_{n}}_{n=1}^{∞} and {f′_{n}}_{n=1}^{∞} are 𝔖-simple sequences with ∥f_{n}−f∥_{ψ}, ∥f′_{n}−f∥_{ψ} →0 for f ∈𝕃^{Λ}, then ∫ f_{n} dΛ − ∫ f′_{n} dΛ →_{P} 0.

Proof. By Fact 7.10, ∫ f_{n} dΛ − ∫ f′_{n} dΛ is id with triplet (A_{n}, ν_{n}, γ_{n}) given by

\begin{cases}A_{n} = ∫_{S}(f_{n}−f′_{n})^{2}σ^{2} dλ \\ ν_{n} = ∫_{S} ∫_{ℝ} 1_{(·)}(f_{n}(s)x−f′_{n}(s)x) ρ(s, dx)dλ(s) \\ γ_{n} = ∫_{S} [(f_{n}−f′_{n})b + ∫_{ℝ}(f_{n}−f′_{n})x(1_{\{|(f_{n}−f′_{n})x|≤1\}}−1_{\{|x|≤1\}}) ρ(·, dx)] dλ \end{cases} .

It follows from Theorem 5.1, and the definition of Φ, that ∫ f_{n} dΛ − ∫ f′_{n} dΛ →_{d} 0 when Φ(f_{n}−f′_{n})→0. This in turn holds by Theorem 7.11, since ∥f_{n}−f′_{n}∥_{ψ} →0. □

Exercise 84 Elaborate on the details in the proof of Lemma 7.14.

Definition 7.15 For an idisrm Λ on (S, 𝔖), with the notation of Theorem 7.11, the \underline{stochastic integral ∫ f dΛ} of f ∈ 𝕃^{Λ} is defined as lim_{n→∞} ∫ f_{n} dΛ, for any 𝔖simple sequence {f_{n}}_{n=1}^{∞} such that ∥f_{n}−f∥_{ψ} →0.

Corollary 7.16 For an idisrm Λ on (S, 𝔖), ∫ f dΛ is id with triplet (A_{f}, ν_{f}, γ_{f}) as in Fact 7.10, and ∫ f_{n} dΛ →_{P} ∫ f dΛ for some 𝔖-simple {f_{n}}_{n=1}^{∞}, for f ∈𝕃^{Λ}.

For an idisrm Λ on ℝ, the following process L is an additive process on ℝ,

L(t) = Λ((0, t]) for t≥0 and L(t) = Λ((t, 0]) for t<0.

Conversely, for ℝ-valued additive process {L(t)}_{t∈ℝ}, there exists an idisrm Λ on ℝ with {Λ((s, t])}_{ℝ∋s<t∈ℝ} =_{d} {L(t)−L(s)}_{ℝ∋s<t∈ℝ}. The integral is denoted ∫ f dL.

For an ℝ-valued Lévy process {L(t)}_{t∈ℝ}, the characteristic of dL coincide with the generating triplet, and the control measure a multiple of Lebesgue measure.

Exercise 85 Verify the claims above for additive processes and Lévy processes.

Remark 7.17 In the sequel, we will neither emphasize nor make crucial use of the mathematical aspects of stochastic integration brought up in Lectures 6 and 7. Rather, they are there to give a complementary view on things. #

7.2 Stochastic Integrals wrt. α-stable Lévy Motion

Let {L(t)}_{t∈ℝ} be α-stable Lévy motion, α∈(0, 1)∪(1, 2), with L(1) ∼ S_{α}(c, β, 0). For α<1, since γ_{0} =0 (see Corollary 3.11), the last integral in Theorm 7.11 is

∫_{ℝ}|∫_{ℝ} f(s)x1_{{|f(s)x|≤1}} dν(x)| ds + ∫_{ℝ}∫_{ℝ} 1∧|f(s)x|^{2} dν(x)ds.

Exercise 86 For α<1 and c>0, show that 𝕃^{dL} =𝕃^{α}(ℝ).

For α>1, since γ_{1} =0 (see Corollary 3.11), the last integral in Theorm 7.11 is

∫_{ℝ}|∫_{ℝ} f(s)x1_{{|f(s)x|>1}} dν(x)| ds + ∫_{ℝ}∫_{ℝ} 1∧|f(s)x|^{2} dν(x)ds.

Exercise 87 For α>1 and c>0, show that 𝕃^{dL} =𝕃^{α}(ℝ).

Exercise 88 Determine 𝕃^{dB} for {B(t)}_{t∈ℝ} ℝ-valued Bm.

7.3 Några tankar om forskningens psykopatografi ...

ur “Sigmund Freud (1910): Leonardo da Vinci - ett barndomsminne”:

“Betänker vi Leonardos kombination av övermäktig forskardrift och förtvinat sexualliv, inskränkt till så kallad ideell homosexualitet, är vi benägna att använda honom som mönsterfall för vår tredje typ. Det skulle vara kärnan och hemligheten i hans natur, att han - efter att infantilt ha använt vetgirigheten för sexuella intressen - lyckades sublimera största delen av sin libido till forskarsträvan. Men att presentera bevis för denna uppfattning är förvisso inte lätt.”

“Låt oss uttryckligen betona att vi aldrig räknat Leonardo till neurotikerna, eller de ”nervsjuka” som den klumpiga termen lyder. Den som beklagar att vi över huvud taget vågar ha patografiska aspekter på honom, den klamrar vid fördomar som vi i dag med rätta har övergett. Vi tror inte längre att hälsa och sjukdom, normal och nervöst, klart kan skiljas åt, och att neurotiska drag måste uppfattas som bevis för allmänn mindervärdighet. Vi vet i dag att neurotiska symptom är ett substitut för vissa bortträngningar som vi måste göra under vår utveckling från barn till kulturmänniska. Att vi alla utför sådana substitututbildningar och att det bara är dessa substituts antal, intensitet och fördelning som motiverar det praktiska begreppet sjukdom och slutsatsen om konstitutionell mindervärdighet. Utifrån små symptom i Leonardos personlighet kan vi nu föra honom till den neurotikertyp som vi kallar ”tvångstypen”. Och kan jämföra hans forskande med neurotikerns ”grubbeltvång” och hans hämningar med dessas så kallade abuli (viljesvaghet, ö.a.).”

“I blomman av sin ungdom verkar Leonardo till en början arbeta utan hämningar. När han i sin livsföring utåt gör fadern till förebild, så upplever han i Milano en tid av manlig skaparkraft och konstnärlig produktivitet. Där erbjuds han också av en slump ett fadersubstitut i form av hertig Lodovico Moro. Men snart gör hans erfarenhet sig påmind: hans nästan totala undertryckande av egentligt sexualliv är inte bästa förutsättningen för att hans sexualitet ska kunna sublimeras i annan aktivitet. Sexuallivet gör sig gällande som något positivt, driftigheten och förmågan till snabba beslut börjar förlamas, tendensen att fundera och tveka uppträder redan vid Nattvarden som störande och avgör tillsammans med tekniken detta storslagna verks öde. Inom honom pågår nu långsamt en process som man bara kan jämföra med regressionerna hos neurotiker. Hans pubertetsutveckling till konstnär överflyglas av hans tidiga infantilt betingade utveckling till forskare. Och den andra sublimeringen av hans erotiska drifter ger nu vika för den ursprungliga sublimeringen, förberedd för den första bortträngningen. Han blir forskare, i början fortfarande i sin konsts tjänst, senare oberoende av den och på väg bort från den. Sedan han förlorat den faderssubstituerande skyddspatronen och livet blivit alltmer dystert så utbreder sig denna regressiva substituering mer och mer. Han blir ”mycket irriterad av att måla”, som en brevskrivare till markgrevinnan Isabella d’Este berättar, som ovillkorligen ville ha en målning av honom. Hans förflutna som barn har tagit makten över honom. Men forskandet, som nu ersätter hans konstnärliga skapande, tycks ha några drag som är typiska för hur omedvetna drifter aktiveras: omättligheten, den otyglade halsstarrigheten, oförmågan att anpassa sig till realiteter.”

8 Lectures 8-9 4-8/10 - 02: Sections I.1-I.5 in Protter

The remaining lectures cover Chapters I-III of “Protter: Stochastic Integration and Differential Equations”. Since Protter’s book is more readable than Sato’s, we mainly indicate selection of material. We have added explanations when we found that we were in need of such ourselves, or thought we had better arguments.

We do list many exercises. There are no exercises in Protter’s book, labeled as such. But Protter leaves many verifications of minor details to the reader, and to check these do in general make very useful exercises. Many of our exercises have this origin.

Our initial ambition is to cover Protter’s whole book, with few omissions.

See e.g., Chapter 1 in “Chung: Lectures from Markov processes to Brownian Motion” for more on stopping times, and Chapter 6 in “Kallenberg: Foundations of Modern Probability” on martingales.

8.1 Basic Definitions and Notation: Section I.1 in Protter

Definition of \underline{filtration}. (See also Definition 1.14.) Notice that Protter includes 𝔉_{∞} in the filtration. Filtrations with other parametersets than [0, ∞] are defined similarly.

Definition of \underline{usual conditions}.

Convention: We always assume the usual conditions! But notice that the usual

conditions are not assumed for the natural filtrations introduced below.

Definition of \underline{stopping time}. In the literature, stopping times are called Markov times, optional times, predictable times, and strong stopping times, strong ... . (Without the usual conditions, {T <t}∈𝔉_{t} is not the same as {T ≤t}∈𝔉_{t}, and names are needed for for both these properties.)

Theorem 1, with proof. Proof: {T <t} = ⋃_{u<t}{T ≤u} and {T ≤t} = ⋂_{u>t}{T <u}. □

Definition of \underline{modification}. (See also Definition 1.11.)

Definition of \underline{indistinguishable}. Recall our Remark 1.12 on this language.

Definition of \underline{hitting time}.

Theorems 3-4, with proof of Theorem 3. The analogue result for any Borel set is mentioned.

Theorem 5.

Exercise 89 Prove Theorem 5.

Definition of \underline{the stopping time σ-algebra 𝔉_{T}}. 𝔉_{T} has many properties one intuitively expects (see e.g., Exercise 90 below), but they do in general require a non-trivial proof.

Exercise 90 Show that 𝔉_{T} is a σ-algebra for T a stopping time.

Theorem 6, with proof of ⊆ . (An “adapted” is missing at one place in the hypothesis.)

Exercise 91 Elaborate on the details in the proof of ⊇ in Theorem 6.

Exercise 92 For S and T stopping times, show that 𝔉_{S} ⊆ 𝔉_{T} if S ≤ T a.s., and that 𝔉_{S} ∩𝔉_{T} = 𝔉_{S∧T}.

Because of the usual conditions, S ≤T a.s. is as good as S ≤T for all ω ∈Ω, e.g., when checking memberships in filtrations. (Readers should make sure to take this in!)

Notation \underline{ΔX}. Notice that \underline{ΔX} is not cádlág, except if it is the zero process.

Exercise 93 Show that ΔX is adapted when X is.

Theorem 7 (possibly with proof).

Corollary to Theorem 7. (That this really is a corollary is trivial.)

8.2 Martingales: Section I.2 in Protter

Definition of \underline{martingale}, \underline{supermartingale} and \underline{submartingale}. Many results are proved e.g., for supermartingales, and thus hold automatically for martingales. They often follow also for submartingales, by Exercise 94 below. Martingales with other parametersets are defined similarly.

Exercise 94 Show that −X is a supermartingale when X is a submartingale.

Definition of \underline{closed martingale}. This is much more important than one may initially belive.

Theorem 9. Notice that the final statement “Such a modification is unique.” in the theorem means that such a modification is indistinguishable from any other such modification. (This is a direct consequence of the fact that modifications of cádlág processes are indistinguishable.)

Exercise 95 Show that modifications of martingales, supermartingales and submartingales are still martingales, supermartingales and submartingales, respectively.

Corollary 1 to Theorem 9. (That this is a corollary is of course trivial.)

Convention: We always assume that martingales are right-continuous!

Theorem 10. (The “Moreover ......”-part of the result may be skipped, at least for now.)

Exercise 96 Why are (right-continuous) martingales cádlág?

Notation \underline{⋁} for σ-algebras (filtrations).

Definition of \underline{uniform integrability (ui)}.

Exercise 97 Show that {Y_{α}}_{α∈𝔄} is ui if sup_{α∈𝔄} E{|Y_{α}|^{p}} < ∞ for some p>1.

Theorem 13. (Theorem 12 is included in Theorem 13.)

Theorem 16. This is really very important!

Theorem 17. For martingales, this is a version of Theorem 16 with other technical conditions.

Exercise 98 Why is T∧t a stopping time for T a stopping time and t∈[0, ∞]?

Definition of \underline{stopped process}.

Lemma 8.1 For S and T stopping times, {S ≤T }, {S <T }, {S =T } ∈ 𝔉_{S∧T}.

Proof: By Exercise 99 below, it suffices to check that {S <T } ∈ 𝔉_{S}, 𝔉_{T}. Here {S <T } ∈ 𝔉_{T}, since

{S <T } ∩ {T ≤t} = ⋂_{ℚ∋s∈(t,t̂]} ⋃_{ℚ∋r<s}{S <r} ∩ {r <T <s} ∈ 𝔉t̂ ↓ 𝔉_{t} as t̂↓t.

To show that also {S <T } ∈ 𝔉_{S}, it is enough to check that {S ≥T } ∈ 𝔉_{S}, which we get from

\{T ≤S\} ∩ \{S ≤t\} = ⋂_{n=1}^{∞} ⋂_{ℚ∋s∈(t,t̂]} ⋃_{ℚ∋r<s}\{T + \frac{1}{n} <r\} ∩ \{r <S<s\} ∈ 𝔉t̂ ↓ 𝔉_{t} as t̂↓t. □

Theorem 18, with proof. Proof: By Theorem 13, it is sufficient to prove that X_{t∧T} = E{X(T)|𝔉_{t}}. This we get (recalling that 𝔉_{t∧T} ⊆𝔉_{t} by Exercise 92), provided that

X(T)1_{{T <t}} is 𝔉_{t∧T} -measurable  and  E{X(T)1_{{T ≥t}}|𝔉_{t∧T} } = E{X(T)1_{{T ≥t}}|𝔉_{t}},

from the fact that, by Theorem 16,

X_{t∧T} = E{X(T)|𝔉_{t∧T} } = X(T)1_{{T <t}} + E{X(T)1_{{T ≥t}}|𝔉_{t}} = E{X(T)|𝔉_{t}}.

The rv X(T)1_{{T <t}} is 𝔉_{t∧T} -measurable, since

{X(T)∈B} ∩ {T <t} ∩ {t∧T ≤s} = {X(t∧T)∈B} ∩ {t∧T <t} ∩ {t∧T ≤s}

= \begin{cases}\{X(t∧T)∈B\} ∩ \{t∧T <t\} for s≥t \\ & ∈ 𝔉_{s} for B ∈B(ℝ), \\ \{X(t∧T)∈B\} ∩ \{t∧T ≤s\} for s<t \end{cases}

by definition of stopping σ-algebras, since X(t∧T) is 𝔉_{t∧T} -measurable by Theorem 6 (where t∧T is a stopping time by Exercise 98), and since {t∧T <t} = ⋃_{ℚ∋u<t}{t∧T ≤u} ∈ 𝔉_{t} ⊆ 𝔉_{s} for s≥t.

Since 1_{{T ≥t}} is 𝔉_{t∧T} -measurable, by Lemma 8.1, E{X(T)1_{{T ≥t}}|𝔉_{t∧T}} = E{X(T)|𝔉_{t∧T}}1_{{T ≥t}}. For H ∈𝔉_{t}, H ∩ {T ≥t} ∈ 𝔉_{t∧T} = 𝔉_{t} ∩ 𝔉_{T} (cf. Exercise 92), since trivially in 𝔉_{t}, and in 𝔉_{T} by

H ∩ \{T ≥t\} ∩ \{T ≤s\} = \begin{cases}\{H ∩ \{T <t\}^{c} ∩ \{T ≤s\} ∈ 𝔉_{s} for s≥t \\ & ∈ 𝔉_{s}. \\ ∅ for s<t \end{cases}

Hence we get E{X(T)1_{{T ≥t}}|𝔉_{t∧T} } = E{X(T)1_{{T ≥t}}|𝔉_{t}}, using Lemma 8.1, since for H ∈𝔉_{t},

E{E{X(T)1_{{T ≥t}}|𝔉_{t∧T} }1_{H}} = E{E{X(T)|𝔉_{t∧T} }1_{{T ≥t}}1_{H}} = E{E{X(T)1_{{T ≥t}}1_{H}|𝔉_{t∧T} }}
= E{X(T)1_{{T ≥t}}1_{H}}
= E{E{X(T)1_{{T ≥t}}|𝔉_{t}}1_{H}}. □

Exercise 99 Why is it enough to check that {S <T } ∈ 𝔉_{S}, 𝔉_{T} in Lemma 8.1?

Corollary to Theorem 18. Proof: By Lemma 8.1, 1_{{T ≤S}} is 𝔉_{S∧T} -measurable, and thus 𝔉_{T} -measurable, so that E{Y 1_{{T ≤S}}|𝔉_{T} } is 𝔉_{S∧T} -measurable, since

E{Y 1_{{T ≤S}}|𝔉_{T} }1_{{S∧T ≤t}} = E{Y |𝔉_{T} }1_{{T ≤S}}1_{{T ≤t}}

is 𝔉_{t}-measurable. This gives

E{E{Y 1_{{T ≤S}}|𝔉_{T} }|𝔉_{S}} = E{E{Y 1_{{T ≤S}}|𝔉_{T} }|𝔉_{S∧T} } = E{Y 1_{{T ≤S}}|𝔉_{S∧T} }.

Similarly, E{E{Y 1_{{T >S}}|𝔉_{T} }|𝔉_{S}} is 𝔉_{S∧T} -measurable, and thus 𝔉_{S}-measurable, since

E{E{Y |𝔉_{T} }|𝔉_{S}}1_{{T >S}}1_{{S∧T ≤t}} = E{E{Y |𝔉_{T} }|𝔉_{S}}1_{{T >S}}1_{{S≤t}}

is 𝔉_{t}-measurable (by Lemma 8.1), which gives

E{E{Y 1_{{T >S}}|𝔉_{T} }|𝔉_{S}} = E{E{Y 1_{{T >S}}|𝔉_{T} }|𝔉_{S∧T} } = E{Y 1_{{T >S}}|𝔉_{S∧T} }.

Adding things up, we conclude that E{E{Y |𝔉_{T} }|𝔉_{S}} = E{Y |𝔉_{S∧T} }. □

Theorem 19. Corollary 1 to Theorem 19. Exercise 100 Prove Corollary 1 to Theorem 19. Corollary 2 to Theorem 19. Exercise 101 Prove Corollary 2 to Theorem 19.

Theorem 20. This is really very important!

Theorem 21, with proof.

8.3 Poisson Process, Brownian Motion: Section I.3 in Protter

Exercise 102 Show that {X(t)−E{X(1)}·t}_{t≥0} and {[X(t)−E{X(1)}·t]^{2}−Var {X(1)} · t}_{t≥0} are martingales for a suitably integrable ℝ-valued Lévy process X.

Theorem 24. This follows directly from Exercise 102.

Definition of \underline{natural filtration}.

Definition of \underline{n-dimensional Brownian motion (Bm) and standard Bm}. Stochastic calculus uses a different definition of BM than is usual: The news are that BM is an adpated process, with independent increments defined relative the σ-algebra of “the past”, that nothing is said about the value at zero, and that the variance matrix (at 1 say), may have non-zero off-diagonal elements.

Theorem 27. This follows directly from Exercise 102.

Theorem 28. This is well-known from basic stochastic calculus.

Theorem 29. This is well-known from basic stochastic calculus.

8.4 Levy Processes: Section I.4 in Protter

Definition of \underline{Lévy process}. The definition corresponds to Definition 1.6 of Lévy process in law. The stochastic calculus Lévy process definition differs from the usual one in that the process is adapted, with independent increments defined relative the σ-algebra of the past. A stochastic calculus Lévy process (in law) is always a “ususal” Lévy process in law (see Exercise 103 below).

Exercise 103 Why is a stochastic calculus Lévy process a Lévy process in law?

Convention: We always assume that Lévy processes are cádlág!

Theorem 31 (possibly with an idea of the proof).

Exercise 104 Show that {e^{iθX(t)}/E{e^{iθX(t)}}}_{t≥0} is a ℂ-valued martingale, and has a cádlág modification, for X a Lévy process and θ ∈ℝ.

Theorem 32, with proof. The proof exemplifies that Optional Sampling often cannot be used directly, since requiring closedness or bounded stopping times, so that one has to proceed e.g., by limits of bounded stopping times, or by other technical tricks.

Corollary to Theorem 33. This is well-known from basic stochastic calculus.

Theorem 34.

Exercise 105 Give an alternative proof of Theorem 34.

Theorem 40.

Exercise 106 Give an alternative proof of Theorem 40.

8.5 Local Martingales: Section I.5 in Protter

Definition of \underline{local martingale}.

Exercise 107 Show that martingales are local martingales.

Definition of \underline{reducing} stopping time.

Theorem 44, with proof. Help to Proof: In (a), by Theorem 17 for the ui martingale M(T∧·), first with stopping times S∧t ≤ t, and then with S∧s ≤ t, Corollary to Theorem 18 gives

E{M(S∧t)|𝔉_{s}} = E{E{M(T ∧t)|𝔉_{S∧t}}|𝔉_{s}} = E{M(T ∧t)|𝔉_{S∧s}} = M(S∧s) for s<t.

Hence M(S∧ ·) is a martingale, which is ui by Theorems 13 and 16 (see Exercise 108 below).

In (c), skip the details with M_{0}, to get an equally good proof, that is immediate from (a). □

Exercise 108 Show that M(S∧ ·) is ui in the above proof.

Corollary to Theorem 44. This is immediate from Theorem 44.b.

Definition of \underline{local} property. Theorem 45. Exercise 109 Prove Theorem 45. Theorem 46. Exercise 110 Prove Theorem 46.

Theorem 47, with proof. [It is not terribly difficult to prove that a local martingale X is a martingale iff. it is of Dirichlet class DL, i.e., iff. {X(T) : T ≤t stopping time} is ui for each t>0.]

   On http://www.orie.cornell.edu/~protter/ we found

8.8 Några tidiga tankar om detta med Bioinformatik (likn.) ...

ur “Falstaff Fakir (1895): Ny och nyttig lärobok i zoologi. Förtjänstfull avhandling för

att erhålla en bra plats, med åtskilliga illustrationer efter iaktagelser i naturen.”:

“Då ...... en bra plats såsom direktör för zoologien nu är ledig, gör jag vad jag länge tänkt. Jag ger ut

mitt vetande om djuren,

och gör det med rätta.

* Djuren

indelar jag naturligtvis i två slag:

I. Djur. II. Odjur.

Som de sistnämda äro äldst, bjuder mig redan vördnaden för ålderdomen att först beskriva dem.

II. Odjur.

Odjuren skilja sig från djuren dels genom sitt o, dels genom sin skapnad. De flesta av odjuren äro hur som helst, medan de mesta av djuren följa vissa reglor i avseende på sitt skelett, sin färg, sin själ, sitt kynne, sina födoämnen m.m. dylikt.

Det äldsta och vackraste odjuret är

Leviathan.

Job omtalar detta odjur med en bestämd hätskhet, och andra förf. i samma ämne äro även genomträngda av en viss aversion ...... Numera har den spelat ut sin roll och förekommer endast sporadiskt vid läsaremöten i ...... Afrika, Stockholm m.fl. svårtillgängliga trakter.

Fågeln Phoenix

är ett hundraårsodjur. När detta vackra flygfä uppnått 100 års ålder, ställer det på sin födelsedag till en mycket besynnerlig fest, genom att tända eld på sig själv, ......

Vad dessa fåglar göra under mellantiderna, har alltjämt varit höljt i dunkel. Till och med rockemottagaren på Hôtel Phoenix i Stockholm vet int därom. Frid över hans tioöring i alla fall.

Att sagde rockavdragare skulle hava goda kunskaper om

Fågeln Rock

skulle man gärna antaga. Men så är icke fallet. Sagda fågel levde dock säkert förr i världen, ...... Darwin påstår helt rått, att det var en ko med en rock på, men detta torde vara ett av de många obevisade hugskott, som på sin tid så ofta förmörkade den eljest ganska klarsynte vetenskapsgubbens själ.

Gripen står i heraldiskt hänseende mycket nära de s.k. statsdjuren, om vilka mera ......

Draken ...... levde mest av jungfrur samt av kristet blod. Syresättningen i dess eget blod var så intensiv, att dess andedräkt var eld. ......

Endast en drakhona är med visshet känd, den s.k. drakan i Babel, ......

Bibeln har sin Behemot att uppvisa, och Jämtland sitt unika Storsjö-odjur. ......

Helhästen är, såsom namnet angiver, en hel häst, men dessutom ett spöke, ......

Fen-ris-ulven och Cerberus likna varandra däruti, att de båda äro utdöda, men äro varandra olika genom antalet huvuden. ......

Sfinxen levde helst i Egypten, vanligen i sällskap med ......

Om vi se närmare på Fågel Blå, finna vi densamma ofta antaga formen av en bok, ......

Stadsodjuren.

Av dessa finnes ett i varje respektabel stat; det underhålles på allmänn bekostnad och dyrkas ......

Schweiziska huvudstadens Berns statsodjur är en björn, och även dess läte fruktansvärt, såsom du nog ofta själv erfarit, och ännu oftare skall erfara, så vitt jag misstänker dig rätt. Låt denna delikata vink förbättra ditt sinne, så kan du med ljuvare samvetsfrid höra, vad jag i det följande skall sia för dig och de dina om de ännu levande Djuren.

I. Djur.

Om det är svårt, ja, rent av omöjligt uppställa någon indelning av de resp. odjuren emellan, ställer sig detta dock jämförelsevis lätt när det gäller de för oss alla mer än välkända djuren. Härvidlag måste jag dock bestämt protestera och avvika mot och från gamla, slentrianmässiga uppdelningen i däggdjur, fåglar, fiskar, krypdjur, kräk, o.s.v. - av skäl, som jag strax skall blotta.

Förhållandet är nämligen, att det är på hög tid att även inom zoologien införa ett s.k.,

Naturligt system

såsom ju redan fallet är i den vetenskapliga växtvärlden eller botanismen. Då du alltså ser på en krokodil med högra öga och på en luktviol med ditt vänstra öga, observerar du genast, att växten sitter tyst och stilla, medan djuret kvittrar, kväker, morrar m.m. - med andra ord: rör sig.

Denna huvudskillnad mellan djuren och växter tvingar oss uppställa den såsom huvudprincip vid de förras indelning i klasser och stånd. Då emellertid all rörelse i allmännhet försigår medelst fötterna, uppdela vi djuren i två bjärt från varandra skilda grupper:

Djur med fötter och Djur utan fötter.

Sedan jag sålunda fotat mitt system på fötterna, vill jag endast i förbigående nämna, att en man med pedantiska böjelser lätt ur den allmänna principen kan uppställa underklasser, såsom djur med en fot (vakant), d:o med två, d:o med tre fötter, med trettiotre fötter etc.

A. Djur med fötter.

Det högsta och ädlaste djuret i zoologisk mening är naturligtvis tusenfotingen, alldenstund han uteslutande består av fötter, såsom framgår av den fotogravyr jag i ett obevakat ögonblick tog av honom. Dessa fötter äro jämt 1000, vilket glädjande förmått mången entusiastisk djurolog att glädjestrålande skrika: ”Det var tusan!”

Över denna tusenfotings framstående ställning i systemet känna sig dock de andra djuren icke avundsjuka, härvid följande den gyllene och egyptiska vishetsregeln:

O b s a l v e r a mycket noga: med din lott, du får, dig foga!

Av dylika vishetsregler och vetenskapsdefinitioner vimlar i själva verket den egyptiska vitterheten; och jag skall i det följande draga fram några av dem ur deras dunkel, i den mån de kunna upp- och belysa vetenskapen. Över en pharaonisk munkskänks sakrofag i Mempispyramiden står tex. inristat:

A n j o f i s k e n i sin ask passar präktigt efter gask.

(Förstår du en egyptisk pik, träffade läsare? Rodna gärna! Bättra dig och bliv såsom bältan: B ä l t a n utan skryt och skrävlan låter världen ha sin ävlan.)

Ehuru tusenfotingen står högst hos vetenskapsmännen, är dock lejonet sedan gammalt djurens konung, men åtnjuter intet regelbundet apanage. Det är lika gult som grymt, lever helst i Afrika och av niggrar. Dessa senare bära kring fotknölarna amuletter av koppar, vilka lejonet aldrig förtär, varuti man vill spåra en gärd åt religionen.

A propos lejon ......

B. Djuren utan fötter

vilka äro de förra underlägsna i de flesta hänseenden. Så sakna de t.ex. fötter och tydlig själ. Somliga av dem kunna nöja sig med att helt simpelt kräla, andra åter måste för sin existens på ett eller annat sätt hava lärt sig simma. Därför är t.ex. den fisk, som icke är simkunnig, fullkomligt värnlös, om han faller i vattnet.

Åter andra varken simma eller kräla, utan flyta menlöst omkring på Oceanen - s.k. urdjur - eller sitta still på ett eller annat värdelöst föremål. T.ex. korallen, ......

10 Lecture 10 15/10 - 02: Sections I.6-7 and II.1-3 in Protter 10.1 Stieltjes Integration: Section I.6 in Protter

Definition of \underline{increasing process} and \underline{finite variation process (FV)}. Notice that cádlág is baked into the definition. Remark. In my view, unlike Protter, it is best to also bake “adapted” into the definition of increasing (FV) processes, since this is what we meet later, and the additional generality is not needed.

Definition of \underline{total variation process \{|·|_{t}\}_{t≥0}.} Exercise 111 Show that a process has finite variation iff. it is the difference between two increasing processes.

Definition 10.1 A stochastic process \{X(t)\}_{t≥0} = \{X(ω; t)\}_{t≥0} is \underline{measurable} if

X:Ω×[0, ∞)→ℝ is F ×B([0, ∞))-measurable.

This is what Protter calls “jointly measurable”.

Exercise 112 Show that a process is measurable if right- or left-continuous. Notation \underline{F · A}. Exercise 113 Calculate N · N for a Pp N.

Theorem 48, with proof. The only non-immediate thing in the proof is the technicalities of the measurability issue. These may well be skipped. Corollary to Theorem 48, with proof. I prefer d|A|_{t} to Protter’s notation |dA_{t}| (which is inconsistent, at best).

Theorem 49. Exercise 114 Outline a proof of Theorem 49. Theorem 50, with proof. Corollary to Theorem 50. Exercise 115 Prove Corollary to Theorem 50.

10.2 Naive Stochastic Integration: Section I.7 in Protter

Naive Stochastic Integration is Impossible!

10.3 Introduction to Semimartingales: Section II.1 in Protter

Definition of \underline{simple predictable process}. Notice their left-continuity.

Notation \underline{𝕊} and \underline{𝕊_{u}}.

Notation \underline{𝕃^{0}}. Exercise 76 characterizes this topology. Notation \underline{I_{X}(H)}. Please, look at this carefully!

Definition of \underline{total semimartingale}. Definition of \underline{semimartingale}.

10.4 Properties of Semimartingales: Section II.2 in Protter

Theorem 1, with proof. This result is immediate (in the meaning “immediate”). Theorem 2, with proof. This result is immediate.

Exercise 116 Give the full details of the continuity argument in proof of The-

orem 2 (i.e., that convergence in Q-probability follows from that in P-probability). Theorem 3, with proof. This result is immediate. Theorem 4, with proof. This result is immediate. Theorem 5. This result is simple, but slightly technical to prove anyway.

Corollary to Theorem 5, with proof. Theorem 6, with proof.

Corollary to Theorem 6. This is immediate from Theorem 6.

10.5 Examples of Semimartingales: Section II.3 in Protter

Theorem 7, with proof. In the theorem, “finite total variation” means finite variation over [0, ∞). In the proof, ∥·∥_{u} is the sup- (uniform) norm over Ω×[0, ∞). Theorem 10.2 (Optional Sampling) Let X be a submartingale and S and T stopping times. If T is bounded, or if X is ui, then

E{X(T)|𝔉_{S}} ≥ X(S∧T).

Theorem 8, with proof. In the theorem, “square-integrable” means sup_{t≥0} E{X(t)^{2}} < ∞. Proof: Pick a t≥0. Since H_{i} is 𝔉_{T_{i}}-adapted, the Optional Sampling Theorem 10.1 (which applies by Theorem I.13, since X is ui by Exercise 97), give

E{(X(T_{i+1}∧t)−X(T_{i}∧t))^{2}|𝔉_{T_{i}}} = E{X(T_{i+1}∧t)^{2}−X(T_{i}∧t)^{2}|𝔉_{T_{i}}}−2E{X(T_{i+1}∧t)−X(T_{i}∧t)|𝔉_{T_{i}}}X(T_{i}∧t),

with the second term on the right zero. Hence it follows that

E{H_{i}^{2}(X(T_{i+1}∧t)−X(T_{i}∧t))^{2}} ≤ ∥H_{i}∥_{u}^{2}E{X(T_{i+1}∧t)^{2}−X(T_{i}∧t)^{2}}.

In a similar fashion, we see that

E{H_{i}H_{j}(X(T_{i+1}∧t)−X(T_{i}∧t))(X(T_{j+1}∧t)−X(T_{j} ∧t))}
= E{E{X(T_{j+1}∧t)−X(T_{j} ∧t)|𝔉_{T_{j}}} H_{j}H_{i}(X(T_{i+1}∧t)−X(T_{i}∧t))}
= E{[X(T_{j} ∧t)−X(T_{j} ∧t)] H_{j}H_{i}[X(T_{i+1}∧t)−X(T_{i}∧t)]} = 0 for j ≥i.

Furthermore, X^{2} is a submartingale (by Corollary 1 to Theorem I.19), so that

E{X(T ∧t)^{2}} ≤ E{E{X(t)^{2}|𝔉_{T} }} = E{X(t)^{2}}

for any stopping time T, by the Optional Sampling Theorem 10.1. We may assume that X(0) = 0, since X is a semimartingale iff. X −X(0) is. Putting things together, we get 𝕃^{2}-continuity

E{I_{X(t∧·)}(H)^{2}} = E{[∑_{i=0}^{n−1} H_{i}(X(T_{i+1}∧t)−X(T_{i}∧t))]^{2}}
= E{∑_{i=0}^{n−1} H_{i}^{2}(X(T_{i+1}∧t)−X(T_{i}∧t))^{2}}
≤ ∥H∥_{u}^{2} ∑_{i=0}^{n−1} E{X(T_{i+1}∧t)^{2}−X(T_{i}∧t)^{2}}
≤ ∥H∥_{u}^{2}E{X(T_{n}∧t)^{2}} ≤ ∥H∥_{u}^{2}E{X(t)^{2}} → 0 as H →_{𝕊_{u}} 0. □

Corollary 1 to Theorem 8, with proof. The proof also uses Theorem I.44.a.

Corollary 2 to Theorem 8. Protter’s proof is not good in my opinion.

Exercise 117 Give a better proof of Corollary 2 to Theorem 8.

Definition of \underline{decomposable process}.

Theorem 9, with proof.

Corollary to Theorem 9.

Exercise 118 Prove Corollary to Theorem 9.

10.6 \underline{Vi} tror i alla fall på “the General Theory” ...

61

11 Lecture 11 18/10 - 02: Sections II.4-6 in Protter

11.1 Stochastic Integrals: Section II.4 in Protter

Definition of \underline{𝔻}, \underline{𝕃} and \underline{b𝕃}. Definition of convergence \underline{uniformly on compacts in probability} ucp. Notation \underline{X_{t}^{⋆}}.

Notation \underline{𝔻_{ucp}}, \underline{𝕃_{ucp}} and \underline{b𝕃_{ucp}}. Notice that 𝔻_{ucp} is complete metrizable.

Exercise 119 Show that ucp-limits of cádlág processes are cádlág. Exercise 120 Show that 𝔻_{ucp} is complete. Is it an Orlicz space? Theorem 10. Definition of \underline{J_{X}:𝕊_{ucp} →𝔻_{ucp}}. Notation \underline{J_{X}(H)}, \underline{∫ H dX} and \underline{H · X}. Theorem 11, with proof.

Definition of \underline{J_{X}:𝕃_{ucp} →𝔻_{ucp}}. The continuity argument for extension used here is standard.

Exercise 121 Show in detail how continuity follows from that at 0 for I_{X} and J_{X}.

11.2 Properties of Stochastic Integrals: Section II.5 in Protter

Theorem 12. This is simple for H simple predictable.

Exercise 122 Prove Theorem 12 in general, from the simple predictable case.

Theorem 13. This is simple for H simple predictable.

Exercise 123 Prove Theorem 13 in general, from the simple predictable case.

Protter’s Notation \underline{H_{P} · X} [which I find a bit unusual (isn’t (H · X)_{P} better?)]. Theorem 14. This is trivial for H simple predictable.

Exercise 124 Prove Theorem 14 in general, from the simple predictable case. Corollary to Theorem 15.

Exercise 125 Prove Corollary to Theorem 15 [e.g., using the probability (Q+P)/2].

Theorem 16. This is trivial for H 𝔉- and 𝔊-simple predictable.

Exercise 126 Prove Theorem 16 in general, from the simple predictable case.

Theorem 17, with proof. This is trivial for H simple predictable. Theorem 18, with proof. This follows from Theorem 14, and from Theorems 14 and 17, respectively. Corollary to Theorem 18, with proof. (One should not try to think too deep here, but instead buy the proof as cheaply as possible.) Theorem 19. Proof. It is a useful exercise to show associativity I_{H·X(t∧·)}(G) = J_{X}(GH)(t) for G∈𝕊 and H ∈𝕃. Picking t>0, H ∈𝕃, and G_{n} →_{𝕊_{u}} 0, we have G_{n}H →_{𝕃ucp} 0, so that

I_{H·X(t∧·)}(G_{n}) =_{associativity} I_{X(t∧·)}(G_{n}H) = J_{X}(G_{n}H)(t) →_{P} 0  since  J_{X}(G_{n}H) →_{𝔻ucp} 0. □

Exercise 127 Prove the above claimed associativity for G∈𝕊 and H ∈𝕃. Exercise 128 Prove associativity in general, from the case G∈𝕊 and H ∈𝕃. Exercise 129 Why is the difficult middle part of Protter’s proof needed? (Beets me!). Theorem 20. Exercise 130 Elaborate on the details in the proof of Theorem 20. (The proof

of Theorem 8 may be useful as guidance.)

Definition of \underline{random partition} σ = \{T_{0}, …, T_{k}\}. Definition of sequence of random partitions \underline{tending to the identity}. Notation Y^{σ} for Y ∈ 𝔻 ∪ 𝕃 and σ a random partition..

Theorem 21. This should be regarded as immediate from the continuity of J_{X}.

11.3 Det är viktigt att börja tidigt med “the General Theory”!

Därför och på allmänn begäran förevisar jag nu för första gången någonsin tre privata fotografiska bilder av mig själv, och det i tre olika stadier av teoretisk genialitet:

Jag vill se bevis av precis allting, varenda litet δpsilon, jag menar ϵpsilon. Det är det som är det svåra och viktiga, och som håller i längden. Annars blir allt ett luftslott utan någon ordentlig grund!

Det bästa är att bevisa utvalda resultat, och låta bli bevisa andra. Då kommer man någon vart, allt känns roligare och mera menigsfullt. Chansen ökar att man får verklig nytta av det man lärt!

Bevis är inte alls särskilt viktiga (tänker du försöka hitta fel i dem, eller vad?), utan det är själva tolkningen och ev. tillämpbarhet som kräver eftertanke. Det är ju faktiskt en verklig värld vi lever i!

12 Lecture 12 22/10 - 02: Sections II.6-8 in Protter

12.1 Quadratic Variation of Semimartingales: Section II.6 in Protter

Definition of \underline{quadratic variation process}. Definition of \underline{quadratic covariation (bracket) process}. Exercise 131 Prove the polarization identity in Protter and [X, Y] = \frac{1}{4}([X + Y, X +Y] − [X −Y, X −Y]). Theorem 22. Corollary 1 to Theorem 22, with proof. Corollary 2 to Theorem 22, with proof. Theorem 23. Corollary. Exercise 132 Prove Corollary to Theorem 25. Definition of \underline{[X, X]^{c}}. Definition of \underline{quadratic pure jump process}. Theorem 26, with proof. Exercise 133 The idea in the proof of Theorem 26 is to show that [X, X] = ΔX · X. Why does this give the theorem? Theorem 27. Corollary 1 to Theorem 27, with proof. Corollary 2 to Theorem 27, with proof. Corollary 3 to Theorem 27. Corollary 4 to Theorem 27. Exercise 134 For a local martingale M with E{[M, M](∞)} < ∞, show that sup_{t≥0} E{M(t)^{2}} = E{[M, M](∞)}. Corollary 5 to Theorem 27. Theorem 28, with proof.

Exercise 135 Why does [X,X]^{c} =0 ⇒ [X,Y]^{c} =0 in the proof of Theorem 28? Theorem 29.

Theorem 30, with proof. (It is really something to be able to prove this so easy!)

12.2 Itô’s Formula: Section II.7 in Protter Theorem 32. Theorem 31 (regarded as a Corollary to Theorem 32), with proof. Exercise 136 Explain how Theorem 31 follows from Theorem 32 for f ∈ℂ^{2}. Corollary to Theorem 32, with proof. This is “the usual” Itô’s Formula. Exercise 137 Show that for Bm (B · B)(t) = \frac{1}{2}B^{2} − \frac{1}{2}t. Exercise 138 Prove Corollary to Theorem 32. Theorem 33. Definition of \underline{(Fisk-) Stratonovich integral ∫ Y ∘ dX}. Theorem 34.

12.3 Application of Itô’s Formula: Section II.8 in Protter

Theorem 36. Obviously, this is extremly important!

Definition of \underline{stochastic (Doléans-Dade) exponential}. Special cases of continuous semimartingale, and of Bm (called geometric Bm). Theorem 37, with proof. Corollary to Theorem 37, with proof. Exercise 139 In the proof of Corollary to Theorem 37, why is [X, X] + [X, −X] + [X, [X, X]] = 0?

Theorems 38-39 and 41. These are fundamental in Bm stochastic calculus, and proofs belong there, albeit simple. Theorem 41 is the famous Dambis-Dubins-Schwarz Theorem.

12.4 Är Göteborgsvitsen död? Döm själv ...

mha. det utvalt roligaste (vitsigaste?)^{dödsroligaste? Sättarens anm.} från GP en fredag och

konsekutiv lördag. OBS: Inslaget skall ej uppfattas som uttryckande åsikt, smak, eller saknad av endera eller bådadera hos Lecture Notes utsände.^{Hos GP’s skribenter då? Korrekturläsarens anm.}

13 Lecture 13 24/10 - 02: Sections III.1-3 in Protter

This chapter develops technology needed to extend the stochastic integral J_{X}:𝕃→𝔻 from 𝕃 to the larger class of \underline{predictable processes} (see Section III.8).

13.1 Introduction: Section III.1 in Protter (Recall) Definition of \underline{decomposable process}. Definition of \underline{classical semimartingale}.

Now follows a list of the main results of Chapter III: Theorem 1. This is the main result! Definition of \underline{(locally) natural process}.

The natural processes are closely related to predictable processes. Exercise 140 Explain all details of the argument that A(T ∧ · ) is natural for A natural and T a stopping time. Theorem 2, with proof. Exercise 141 Why have continuous FV processes locally integrable variation? Theorem 3.

13.2 The Doob-Meyer Decompositions: Section III.2 in Protter

Definition of \underline{potential}. Exercise 142 Why exist lim_{t→∞} E{X(t)} for non-negative supermartingales?

Theorem 4, with proof. This Doob Decomposition is the discrete analogue of the continuous time Doob-Meyer Decomposition, that in turn is crucial for us.

Exercise 143 Give all details of the existence part of proof of Theorem 4.

Theorem 5, with proof. Notice: Protter has forgotten to say that Z is UI in the theorem (which is needed to prove Corollary to Theorem 6 below).

Exercise 144 Explain why Z is non-negative in the proof of Theorem 5. Theorem 6.

Corollary to Theorem 6, with proof. Theorem 7.

13.3 Quasimartingales: Section III.3 in Protter Definition of \underline{partition of [0, ∞]}. Definition of \underline{variation of X along a partition}. Definition of \underline{variation of X}. Definition of \underline{quasimartingale}.

Exercise 145 Which quasimartingales are martingales? Theorem 8. Exercise 146 In one direction Theorem 8 is easy: Prove that part. Theorem 9, with proof. Exercise 147 Explain how uniqueness in Theorem 9 follows from Lemma on page 92 in Protter’s book (or in some other way). Exercise 148 Why are locally integrable variation processes local quasimartingales? Definition of \underline{compensator}. Fact. Compensators of increasing processes with (locally) integrable variation are increasing with (locally) integrable variation. With proof! Exercise 149 Why are decreasing processes with adequate integrability properties (local) supermartingales?

13.4 Theoretical Background for Chapters III-V in Protter’s Book

Following the introductory Chapters I-II, the main treatment of stochastic integrals and differential equations is in Chapters III-V. To take in that material properly, a bit more is needed in terms of theoretical background, than for the introduction. Most participants have much of that background already. However, there will certainly be topics to “fresh up” for everyone, so that not too much time is spent later on things of background character. For convenience, I have assembled a list of what one ideally should know, to do well with Chapters III-V. Please notice that the listed material is fundamental also in many other areas of applied probability, so that it is always well spent time to work with it, if not well-known previously.

Section III.2. The Doob-Meyer Decompositions. K.M. Rao’s proof is much more simple and intiutive than previous ones, and uses basic potential theory. See e.g.,

Doob, J.L. (1983). Classical Potential Theory and Its Probabilistic Counterpart. Springer. Part 1.

Chapter IV. General Stochastic Integration. Here the Musielak-Orlicz space machinery outlined in Lectures 6-7 comes into play. See e.g.,

Kwapień, S. and Woyczyński, W. (1992). Random Series and Stochastic Integrals. Birkhäuser. Chapters 0 and 7-9.

Musielak, J. (1983). Orlicz Spaces and Modular Spaces. Springer. Chapters I-II.

Rao, M.M. and Ren, Z.D. (1991). Theory of Orlicz Spaces. Dekker.

Chapter V. SDE. In general, this chapter requires distribution theory together with elliptic PDE, and variational calculus, deterministic and stochastic (i.e., Malliavin calculus). Standard texts are e.g.,

Federer, H. (1969). Geometric Measure Theory. Springer. (This one takes time, but is worth it!) Hörmander, L. (1983-4). The Analysis of Linear Partial Differential Operators. Springer. Part

I and Part III Chapters XVII-XIX.

Nualart, D. (1995). The Malliavin calculus and Related Topics. Springer.

Section V.2. H^{p}-norms for Semimartingales. This requires Hardy spaces. See e.g., Douglas, R.G. (1972). Banach Algebra Techniques in Operator Theory. Academic. Chapter 6.

Section V.6. Markov Nature of Solutions. This requires Markov Processes. (Riktiga såna!) See e.g.,

Blumenthal, R.M. and Getoor, R.K. (1968). Markov Processes and Potential Theory. Academic. Chapters 0-2.

Sections V.7-8. Flows of SDE and flows as Diffeomorhisms. This requires basic knowledge of infinite dimensional differential geometry (Hilbert space generality), and of stochastic flows. See e.g.,

Kunita, H. (1990). Stochastic flows and stochastic differential equations. Cambridge. Chapters 4-5. Okubo, T. (1987). Differential geometry. Dekker.

Chapters III-V. Rigorous treatment of stochastic differentials requires stochastic nonstandard analysis. Elementary aspects of complex analysis in several variables is used on multidimensional complex martingales, as is functors (Banach operator ideals), and probability in Banach spaces. See e.g.,

Hörmander, L. (1988). Complex Analysis in Several Variables. North-Holland. Chapter 2.

Ledoux, M. and Talagrand, M. (1980). Probability in Banach Spaces. Springer.

Michnor, P.W. (1978). Functors and Categories of Banach Spaces. Springer.

Stoyan, K.D. and Bayod, J.M. (1986). Infinitesimal Stochastic Analysis. North-Holland.

14 Lecture 14 6/11 - 02 8 am: Sections III.4-5 in Protter 14.1 The Fundamental Theorem: Section III.4 in Protter

Definition of \underline{predictable} stopping time. Exercise 150 Show that non-random times are predictable. Definition of \underline{announcing} a predictable stopping time. Exercise 151 Explain why T_{n} = inf\{t > 0 : |X(t)| ≥ c− \frac{1}{n}\} (n large enough) announces T = inf{t > 0 : |X(t)| ≥ c} for c > 0 a constant and X an adapted continuous process with X(0)=0. Definition of \underline{accessible} stopping time. Exercise 152 Explain why the time to the first jump for a Poisson process is not accessible. Definition of an \underline{envelop} of an accessible stopping time. Exercise 153 Find an envelop of a stopping time whose set of possible values is countable. Definition of \underline{totally inaccessible} stopping time. Exercise 154 Is the time to the first jump for a Poisson process totally inaccessible? Notation T_{Λ} for Λ∈𝔉_{T} and T stopping time. Exercise 155 Prove that T_{Λ} is a stopping time. Theorem 10, with proof. Theorem 12. Exercise 156 Motivate Theorem 12. Theorem 13. Corollary to Theorem 13.

14.2 Classical Semimartingales: Section III.5 in Protter

Theorem 14, with proof. Exercise 157 Show that classical semimartingales are decomposable. Exercise 158 Show that, for a classical semimartingale X, we really have X =M +A, as in the proof of Theorem 14. Corollary to Theorem 14, with proof. Theorem 15, with proof. Theorem 16, with proof. Exercise 159 In the proof of Theorem 16, why is X(t∧ ·) a supermartingale? Exercise 160 Show that a submartingale is a semimartingale. Theorem 17, with proof (albeit a bit longish). Exercise 161 Explain the details of the inequalities E{∫_{0}^{t∧T} d|A|} ≤ m + β + E{|(ΔM)(T)|} < ∞, in the proof of Theorem 17. Exercise 162 In the last computation in Theorem 17, motivate thorougly (not just repeating Protter’s words) the exchange of order of limits E{lim ...} = lim E{...}. Exercise 163 In the last computation in Theorem 17, prove the last equality. Definition of \underline{special semimartingale}. Theorem 18. Exercise 164 Check the proof of Theorem 18. Definition of \underline{canonical decomposition}. Theorem 19, with proof. Exercise 165 Explain how Theorem 13 gives X = X(0)+N +B, in the proof of Theorem 19. Exercise 166 Explain, using e.g., Exercise 148, why B is a local quasimartingale, in the proof of Theorem 19.

Corollary to Theorem 19. (The proof of the Corollary is harder than that of the theorem!)

Exercise 167 Show that bounded jump Lévy processes are special semimartingales (there it came!), and that Lévy processes with ∫_{1}^{∞} x dν(x) = ∞ are not.

14.3 Inte ens Föreläsaren är Perfekt ...

73

15 Lecture 15 8/11 - 02: Sections III.6-8 in Protter

15.1 Girsanov’s Theorem: Section III.6 in Protter Definition of \underline{equivalent} probability measures. Notation P ∼Q. Lemma before Theorem 20. Exercise 168 prove Lemma. Theorem 20, with proof. Exercise 169 Verify that Z^{−1} = E_{Q}\{\frac{dP}{dQ}|𝔉_{t}\} in the proof of Theorem 20.

Theorem 21, with proof. (This is “usual” Girsanov’s Theorem.)

15.2 Bichteler-Dellacherie Theorem: Section III.7 in Protter

Theorem 22.

15.3 Natural and Predictable Processes: Section III.8 in Protter

Definition of \underline{predictable} σ-algebra. Theorem 23. Exercise 170 Check the details of the proof of Theorem 23.

Corollary 1 to Theorem 23, with proof. Notice: The conclusion should be that A is \underline{locally} natural (i.e., Protter has forgotten “locally”).

Corollary 2 to Theorem 23, possibly with proof. Theorem 26. Theorem 27.

15.4 Lite Julpyssel.

Först en vinpalör som ger tillräckligt kunnande i standardårgångar för typviner, för

att lura nästan varje sommelier [men kräver att man har med den stora portmonän (med ett undantag)]: Para alltså ihop vin med år. (Minnesluckor bör kunna hjälpas upp mha. www.)

Annus mirabilis, förutom Douro  1921, 1985
Annus mirabilis i Douro  1931
Cheval Blanc  1945, 1963, 1994
Cru Beaujolais  1947, (1982?)
DRC  1961, (1982?)
Haut Brion  1961, 1990
Hermitage la Chapelle  1967, 1983, 1988-1990
Latour  1971, 1982, 1998
Leoville las Cases, Pichon Comtesse de Lalande  1975
Marquis Alexandre de Lur-Saluce  1976, 1983, 1988-1990
la Mission haut Brion  1978, 1985, 1988-1990
Montrachet  1982
Montrose  1982
Mouton  1982, 1986
MSR  1982, 1985, 1988-1990
Palmer  1983
Petrus  1989
Le Pin  1990
Quinta do Noval Nacional  1990
Romanee Conti  1991
Syndicate Grande Marque  1994
Taylor  1995

Dessutom något om regioner, för att testa din sommelier. (Tappa ej initiativet!)

Baron Lacoste Lynchad av Comtessa efter Malmcitat  Region ...... ?
Gevre Charmade LatRik Kamrer  Region ...... ?
Petrus Evangelium Pinade Gay Pastor  Region ...... ?

Goda lösningar till dessa grundläggande vinövningar räknas som en “Exercise”, i syfte att uppnå rimlig salongsfäighet. (Detta kan även doktorander ha nytta utav.) Dessutom, den som lämnar den bästa lösningen vinner en låda 2620 Oltina La Revedere.

A Appendix. Solutions to Difficult Exercises

Exercise 54 The easy part of Theorem 5.1 is that for id {Y_{n}}_{n=1}^{∞} and Y with triplets (A_{n}, ν_{n}, γ_{n}) and (A, ν, γ), respectively, Y_{n} →_{d} Y if

(1) lim_{n→∞} ∫_{|y|>ε} f(y) dν_{n}(y) = ∫_{|y|>ε} f(y) dν(y) for bounded f ∈ℂ(ℝ^{d}) and ε>0,

(2) lim_{ε↓0} lim sup_{n→∞} |⟨θ, (A_{n}−A)θ⟩ + ∫_{|y|≤ε}⟨θ, y⟩^{2} dν_{n}(y)| = 0 for θ ∈ℝ^{d}, and

(3) lim_{n→∞} γ_{n} = γ.

Proof. Define measures ρ_{n} on ℝ^{d} by dρ_{n}(y) = 1∧⟨θ, y⟩^{2} dν_{n}(y). By (2), for θ ∈ℝ^{d},

∫_{|y|≤ε} dρ_{n}(y) ≤ ∫_{|y|≤ε}⟨θ, y⟩^{2} dν_{n}(y) ≤ ∫_{|y|≤ε}⟨θ, y⟩^{2} dν_{n}(y) + ⟨θ, A_{n}θ⟩ < K_{1}

for n≥n̂, for some constants K_{1} >0 and n̂∈ℕ. Thus this holds for all n∈ℕ, possibly with a greater value for K_{1}. Hence Dominated Convergence gives

lim_{ε↓0} lim sup_{n→∞} ∫_{|y|≤ε}[e^{i⟨θ,y⟩}−1−i⟨θ, y⟩+ \frac{1}{2}⟨θ, y⟩^{2}] dν_{n}(y) = 0,

since |e^{i⟨θ,y⟩}−1−i⟨θ, y⟩+\frac{1}{2}⟨θ, y⟩^{2}| ≤ K⟨θ, y⟩^{2} for a constant K >0. Further, (1) gives

lim_{n→∞} ∫_{|y|>ε}e^{i⟨θ,y⟩}−1−i⟨θ,y⟩ dν_{n}(y) = ∫_{|y|>ε}e^{i⟨θ,y⟩}−1−i⟨θ,y⟩ dν(y) → ∫_{ℝ^{d}}e^{i⟨θ,y⟩}−1−i⟨θ,y⟩ dν(y)

as ε↓0, using Dominated Convergence and that ν is a Lévy measure at the end. Now we get just by inspection the desired conclusion that, as n→∞,

i⟨θ,γ_{n}⟩−\frac{1}{2}⟨θ,A_{n}θ⟩+ ∫_{ℝ^{d}}e^{i⟨θ,y⟩}−1−i⟨θ,y⟩ dν_{n}(y) → i⟨θ,γ⟩−\frac{1}{2}⟨θ,Aθ⟩+ ∫_{ℝ^{d}}e^{i⟨θ,y⟩}−1−i⟨θ,y⟩ dν(y).

Albeit correct in idea, there is an error in the proof in Sato’s book (in the direction carried out above), and his definition of ρ_{n} has to be changed to the above one. #

Exercise 91 Protter’s proof is not good in my opinion, and makes the problem harder than it is. We give a quicker and more transparent argument below. First it should be clerified that, if P{T = ∞} > 0, then it is only processes X with a welldefined (measurable) value X(∞) that feature in the theorem.

It suffices check that {X(T)∈G} ∩ {T ≤t} ∈ 𝔉_{t} for open G⊆ℝ, because then

{X(T)∈G} = {X(∞)∈G} ∩ {T =∞} ∪ ⋃_{n=1}^{∞}{X(T)∈G}∩{T ≤n} ∈ 𝔉.

But the required membership follows from cádláguity and adaptedness of X, since

1_{\{X(T)∈G\}}1_{\{T ≤t\}} = lim_{n→∞} ∑_{k=0}^{2^{n}−1}1_{\{X( \frac{k+1}{2^{n}} t)∈G\}}1_{\{ \frac{k}{2^{n}} t≤T < \frac{k+1}{2^{n}} t\}} + 1_{\{X(t)∈G\}}1_{\{T =t\}} #