# Is this application of Ito's lemma correct?

Suppose that $S$ follows a geometric brownian motion

$$dS=S(\mu dt+\sigma dB).$$

It is well understood that

$$S_{T}=S_{0}exp((\mu-\dfrac{\sigma^{2}}{2})T+\sigma B_{T}).$$

## Method 1 (I have no problem with this)

Letting $f(S)=log(S)$ and doing a 2nd order Taylor expansion and noting that $(dB)^{2}=dt$. For example:

$$d(log(S)) = f'(S)dS+\dfrac{1}{2}f''(S)S^{2}\sigma^{2}dt=\dfrac{1}{S}(\mu dt+\sigma SdB)-\dfrac{1}{2}\sigma^{2}dt=(\mu-\dfrac{\sigma^{2}}{2})dt+\sigma dB.\quad (*)$$

It follows that

$$log(S_{T})=log(S_{0})+(\mu-\dfrac{\sigma^{2}}{2})T+\sigma B_{T}$$ and hence

$$S_{T}=S_{0}exp((\mu-\dfrac{\sigma^{2}}{2})T+\sigma B_{T}).$$

The above derivation is perfectly fine and can be found on Wikipedia for example.

## Method 2 (is this allowed?)

Let's use Ito's lemma

$$dF(t,X(t))=(F_{t}'+a(t)F_{x}'+\dfrac{1}{2}b(t)^{2}F''_{xx})dt+(b(t)F'_{x})dB$$ for a process $dX(t)=a(t)dt+b(t)dB(t)$ and for a function $F(t,X(t))$. Let $F(t,X(t))=log(X(t)).$ Let $dS=S\mu dt+S\sigma dB$, and let $a(t)=S\mu$ and $b(t)=S\sigma$. This is my question: $a(\cdot)$ is a function of $t$ and not $S$, so is this still OK to do?. Then

$$dF=(0+\mu+\dfrac{1}{2}\sigma^{2}S^{2}\times(\dfrac{-1}{S^{2}}))dt+\sigma\dfrac{S}{S}dB=(\mu-\dfrac{1}{2}\sigma^{2})dt+\sigma dB.$$ Here we arrive at (*) from Method 1 and hence the result follows.

Is this derivation technically correct? Just because it gives the right answer, it does not imply the method is sound.

Notice that Wikipedia's formulation of Ito's lemma is a bit misleading, as they write $$dX_t = \mu_t dt + \sigma_t dB_t$$ but, actually, the functions $\mu$ and $\sigma$ are allowed to depend on $X$, that is $$\mu_t = \mu(t,X_t) \qquad \text{and}\qquad \sigma_t = \sigma(t,X_t).$$ Hence your application of Ito's lemma is formally correct.
PS: See for instance here for a derivation of the lemma, where the fact that $\mu$ and $\sigma$ can depend on $X_t$ is made explicit after equation (1).
• Yes. But my question is that $a(\cdot)$ and $b(\cdot)$ are not functions of $S$ -- and whether this substitution should be allowed in the form of Ito's lemma that was given. (Indeed, Ito's lemma in general is nothing but Taylor expansion to 2nd order of $dt$ anyway. I get that... and that's fine. But using the lemma given in the original question, surely that's not fine?) – Antonius Gavin Mar 25 '15 at 15:30