Statistical inference for Geometrical BM (cf .  Exercise 10.9)   dX = mu X dt + X dB ;  
    X (t) = Exp {(mu - 1/2) t + B (t)}

        ---------------------------------------------------------------------------------------------------------

<< Statistics`DescriptiveStatistics`; << 
      Statistics`ContinuousDistributions`;

The transition density function for the OU process   pBM[mu, 
          x, y, t]  =  
      P {BM (t) + (mu - 1/2) t ∈ dy | B (0) = 
            x} dy:

Clear[pBM]; 
    pBM[mu_, x_, y_, t_] := 
      PDF[NormalDistribution[x + (mu - 1/2)*t, Sqrt[t]], y];

Simulation of Geometrical BM is straightforward by the Euler method :

mu0 = 5; StepsPerTimeUnit = 100; T = 100; GBMData = {1}; 
    For[i = 2, i ≤ StepsPerTimeUnit*T, i++, 
      AppendTo[GBMData, 
        GBMData[[i - 1]] + mu0*GBMData[[i - 1]]/StepsPerTimeUnit + 
          GBMData[[i - 1]]*
            Random[NormalDistribution[0, 
                1/Sqrt[StepsPerTimeUnit]]]]];

ListPlot[GBMData, PlotJoined -> True];

ListPlot[Log[GBMData], PlotJoined -> True];

Method 1  :  
      Estimation of   mu   by Likelihood Ratio according to Equation (10.42)
-------------------------------------------------------------------------

mu1  =  
      Sum[(1/GBMData[[i - 1]])*(GBMData[[i]] - 
                GBMData[[i - 1]]), {i, 2, StepsPerTimeUnit*T}]/
        T

Method 2 :  
      Estimation of   mu   by Maximum Likelihood method based on    pBM[mu, x, y, t]
-------------------------------------------------------------------------------

SampleDistance = 50; NData = StepsPerTimeUnit*T/SampleDistance; 
    NewGBMData = Table[GBMData[[i*SampleDistance]], {i, 1, NData}]; 
    mu2 = mu /. NMaximize[{Sum[
                Log[pBM[mu, Log[NewGBMData[[i - 1]]], 
                    Log[NewGBMData[[i]]], T/NData]], {i, 2, NData}], 
              mu > 0}, {mu}][[2]]