Use of Ito's lemma in ItosLemma.m (or any other method in Mathematica)

Question

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?

Answer 1

Another possibility with built-in functions:

(* Define the process *)
proc = ItoProcess[ⅆs[t] == μ s[t] ⅆt +
 σ s[t] ⅆw[t], s[t], {s, s0}, {t, 0}, 
 w \[Distributed] WienerProcess[]]

Now the process can be used as:

Expectation[z, z \[Distributed] proc[t]]
(* E^(t μ) s0 *)

Simplify[Expectation[Log[z], z \[Distributed] proc[t]], 
  Assumptions -> {σ > 0}]
(* t (μ - σ^2/2) + Log[s0] *)

Answer 2

A comment from @Diogo brought me on the right track:

<<ItosLemma`
dS=ItoMake[S[t],μ *S,σ*S] 
(* dt S μ+S σ Subscript[dB, 1] *)

ItoD[Log[S[t]]]
(* dt (μ-σ^2/2)+σ Subscript[dB, 1] *)