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Notebook[{
Cell[BoxData[
    \(Fit\ of\ midday\ Nord\ Pool\ eletricity\ prices\ \ year\ 2000\ to\ CKLS\
\ equation\ \ \ \ \ dX \((t)\)\  = \ 
      mu\ X \((t)\)\ dt + sigma\ X \((t)\)^gamma\ dB \((t)\)\)], "Input"],

Cell[BoxData[
    \(<< Statistics`DescriptiveStatistics`; << 
      Statistics`ContinuousDistributions`; $RecursionLimit = \
\($IterationLimit = Infinity\);\)], "Input"],

Cell[BoxData[{
    \(\(Clear[FEmp];\)\), "\n", 
    \(\(FEmp[data_, 
          x_] := {lt = Length[data], 
            Sum[If[data[\([i]\)] \[LessEqual] x, 1, 0], {i, 1, lt}]/
              lt}[\([2]\)];\)\)}], "Input"],

Cell[BoxData[
    \(Clear[CDFStationary]; 
    CDFStationary[mu_, sigma_, gamma_, x_] := 
      Gamma[\(1 - 2\ gamma\)\/\(2 - 2\ gamma\)]/
          Gamma[\(1 - 2\ gamma\)\/\(2\ gamma - 2\)] - \ \((Gamma[\(3 - 4\ \
gamma\)\/\(2 - 2\ gamma\)]/\((2\ gamma - 
                    1)\)\  - \((gamma\ Gamma[\(1 - 2\ gamma\)\/\(2 - 2\ gamma\
\)] + \((1 - 
                          2\ gamma)\)\ Gamma[\(1 - 2\ gamma\)\/\(2 - 2\ gamma\
\), \(mu\ x\^\(2 - 2\ gamma\)\)\/\(\((gamma - 1)\)\ sigma\^2\)])\)/2)\)/
          Gamma[1\/\(2 - 2\ gamma\)];\)], "Input"],

Cell[BoxData[
    \(Clear[KSDistance]; 
    KSDistance[data_, mu_, sigma_, 
        gamma_] := {sdata = Sort[data], lt = Length[data], 
          Max[Table[
              Max[{Abs[
                    CDFStationary[mu, sigma, gamma, 
                        sdata[\([i]\)]] - \((i - 1)\)/lt], 
                  Abs[CDFStationary[mu, sigma, gamma, sdata[\([i]\)]] - 
                      i/lt]}], {i, 1, lt}]]}[\([3]\)]\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(NP = 
      Part[Transpose[
          Import["\</users/math/palbin/user/courses/StokAnal/NP.csv\>", \
"\<CSV\>"]], 5]; lt = Length[NP]; 
    ListPlot[NP, PlotJoined \[Rule] True, 
      PlotLabel -> "\<NP prices 2000-2004\>"]; 
    NP2000 = Table[NP[\([i]\)], {i, 1, 350}]; 
    ListPlot[NP2000, PlotJoined \[Rule] True, 
      PlotLabel -> "\<NP prices 2000\>"];\)], "Input"],

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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mfstroke
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s
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s
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s
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Mistroke
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Cell[BoxData[
    \(Simulation\ of\ CKLS\ model\ is\ a\ research\ level\ problem, \ 
    and\ cannot\ be\ discussed\ here\ in\ \(\(detail\)\(.\)\)\)], "Input"],

Cell[BoxData[
    \(It\ turns\ out\ that\ it\ is\ suitable\ to\ simulate\ the\ transformed\ \
process\ \ Y \((t)\) = X \((t)\)^\((1 - gamma0)\), \ 
    rather\ that\ \ X \((t)\)\ itself, \ \[IndentingNewLine]where, \ 
    by\ the\ Ito\ \ formula, \ 
    dY \((t)\)\  = \ 
      phi \((Y \((t)\))\)\ dt\  + \ \((1 - gamma0)\)\ sigma0\ dB \((t)\), \ 
    with\ \ phi \((t)\)\ \ as\ given\ \(\(below\)\(.\)\)\)], "Input"],

Cell[BoxData[
    \(We\ use\ an\ implicit\ Euler\ scheme\ to\ simulate\ \ Y \((t)\), \ 
    and\ then\ get\ \ X \((t)\)\ \ as\ \ Y \(\(\((t)\)^\((1/\((1 - 
                  gamma0)\))\)\)\(.\)\)\)], "Input"],

Cell[BoxData[
    \(Clear[phi]; 
    phi[y_] := \((1 - gamma0)\)*mu0*y + 
        gamma0*\((gamma0 - 1)\)*sigma0^2/\((2*y)\); step = 200; days = 350; 
    xi = Random[UniformDistribution[0, 1]]; 
    Y = {\((y /. 
              FindRoot[
                CDFStationary[mu0, sigma0, gamma0, y] \[Equal] xi, {y, 
                  80}])\)^\((1 - gamma0)\)}; 
    For[i = 1, i \[LessEqual] step*days, \(i++\), 
      AppendTo[Y, 
        y /. FindRoot[
            Y[\([i]\)] + 
                phi[y]/step + \((1 - gamma0)\)*sigma0*
                  Random[NormalDistribution[0, Sqrt[1/step]]] \[Equal] y, {y, 
              SimNP[\([i]\)]}]]]; 
    SimNP = \((Table[
            Y[\([1 + \((i - 1)\)*step]\)], {i, 1, 
              days}])\)^\((1/\((1 - gamma0)\))\);\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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        PlotLabel -> "\<Simulated CKLS Model Fitted to NP Prices\>"];\)\)], \
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