Study the Stochastic Exponential Function 

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

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

Simulate the process.

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X data = RandomFunction[\[ScriptCapitalE]w /. \[Sigma] -> 1, {0, 10, 0.01}, 4];

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X ListLinePlot[data, PlotRange -> All]

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

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X {Mean[\[ScriptCapitalE]w[t]], Variance[\[ScriptCapitalE]w[t]]}

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Compare with the result using the corresponding SDE .

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X \[ScriptCapitalE]sde = ItoProcess[\[DifferentialD]u[t] == u[t] \[DifferentialD]b[t], u[t], {u, 1}, t, b \[Distributed] WienerProcess[0, \[Sigma]]];

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X {Mean[\[ScriptCapitalE]sde[t]], Variance[\[ScriptCapitalE]sde[t]]}

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

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X \[ScriptCapitalE]p = TransformedProcess[2^p[t] Exp[-\[Lambda] t], p \[Distributed] PoissonProcess[\[Lambda]], t];

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X ListLinePlot[ RandomFunction[\[ScriptCapitalE]p /. \[Lambda] -> 1, {0, 10, 0.01}, 4], PlotRange -> All]

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X {Mean[\[ScriptCapitalE]p[t]], Variance[\[ScriptCapitalE]p[t]]}

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