# Study the Stochastic Exponential Function

Define the stochastic exponential of the Wiener process as a transformed Wiener process.

 In[1]:= X\[ScriptCapitalE]w = TransformedProcess[E^(w[t] - (\[Sigma]^2 t)/2), w \[Distributed] WienerProcess[0, \[Sigma]], t];

Simulate the process.

 In[2]:= Xdata = RandomFunction[\[ScriptCapitalE]w /. \[Sigma] -> 1, {0, 10, 0.01}, 4];
 In[3]:= XListLinePlot[data, PlotRange -> All]
 Out[3]=

Compute the mean and variance for a time slice of the process.

 In[4]:= X{Mean[\[ScriptCapitalE]w[t]], Variance[\[ScriptCapitalE]w[t]]}
 Out[4]=

Compare with the result using the corresponding SDE .

 In[5]:= X\[ScriptCapitalE]sde = ItoProcess[\[DifferentialD]u[t] == u[t] \[DifferentialD]b[t], u[t], {u, 1}, t, b \[Distributed] WienerProcess[0, \[Sigma]]];
 In[6]:= X{Mean[\[ScriptCapitalE]sde[t]], Variance[\[ScriptCapitalE]sde[t]]}
 Out[6]=

Define the stochastic exponential of the compensated Poisson process as a transformed Poisson process.

 In[7]:= X\[ScriptCapitalE]p = TransformedProcess[2^p[t] Exp[-\[Lambda] t], p \[Distributed] PoissonProcess[\[Lambda]], t];
 In[8]:= XListLinePlot[ RandomFunction[\[ScriptCapitalE]p /. \[Lambda] -> 1, {0, 10, 0.01}, 4], PlotRange -> All]
 Out[8]=
 In[9]:= X{Mean[\[ScriptCapitalE]p[t]], Variance[\[ScriptCapitalE]p[t]]}
 Out[9]=

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