This is a follow-up question on this question: Use of Ito's lemma in ItoProcess
My problem is to find some method how to use Ito's lemma in Mathematica. As an example:
How can I 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 $S$,
$$S_t=S_0\exp\left(\sigma B_t+ \left (\mu-\tfrac{\sigma^2}{2} \right )t\right).$$
It was suggested to do the following:
[...] you'll need load the package ItosLemma, by Mark Fisher, that can be downloaded here: http://library.wolfram.com/infocenter/MathSource/1170/
The ItosLemma.nb has a clear example on how to use it. Following his example, you only need to load the package
<< ItosLemma`
and call ItoMake to represent the SDE satisfied by xt
dx = ItoMake[x[t], μ-σ^2/2, σ]
By denoting y = f(x,t), you should use the function ItoD[y] that gives you the Ito's lemma application for any "well behaved" function f.
y = f[x[t], t] ItoD[y]
The problem that I have is that it assumes what I what to derive: basically the $-\tfrac{\sigma^2}{2}$ part.
The original SDE above was defined without it and it appeared in the derivation because of Ito's lemma!
My question
How can I use this package (or any other method) to arrive at the solution above with Mathematica?