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In the documentation for the ItoProcess it says:

Converting an ItoProcess to standard form automatically makes use of Ito's lemma.

It is unclear to me how this is done, also the example given for the standard form doesn't help.

How can I, for example, apply Ito's lemma on the following stochastic differential equation (SDE) $dS=S(σdB+μdt)$, with $B$ being Brownian motion. Applying Itō's lemma with $f(S)=log(S)$ gives

$$\begin{align} d\log(S) & = f^\prime(S)\,dS + \frac{1}{2}f^{\prime\prime} (S)S^2\sigma^2 \,dt \\ & = \frac{1}{S} \left( \sigma S\,dB + \mu S\,dt\right) - \frac{1}{2}\sigma^2\,dt \\ &= \sigma\,dB +\left (\mu-\tfrac{\sigma^2}{2} \right )\,dt. \end{align}$$

It follows that

$$\log (S_t) = \log (S_0) + \sigma B_t + \left (\mu-\tfrac{\sigma^2}{2} \right )t,$$

exponentiating gives the expression for $S$,

$$S_t=S_0\exp\left(\sigma B_t+ \left (\mu-\tfrac{\sigma^2}{2} \right )t\right).$$

How can I achieve that in Mathematica?

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