# Brownian function and Clark's formula

I was reading a paper (link) from Richard Bass about Brownian functionals, and came across the following passage :

Let $$X_t$$ be a Brownian motion, $$F_t$$ its filtration and $$g$$ a real-valued function. We first assume that $$g$$ is bounded, has compact support, and is in $$C^2$$. By Clark's formula applied to the functional $$g(X_1)$$, $$g(X_1)=\mathbb{E}g(X_1) + \int_0^1 \mathbb{E}[g'(X_1)|F_s]dX_s$$ (Another derivation of this representation is to use Ito’s lemma to take care of the case $$g(x) = e^{iux}$$ and then use linearity and a limiting process.)

My question is about the proposed alternative derivation of the given formula.

I tried to use Ito's lemma on $$Y_t^u = e^{iuX_t}$$ but didn't seem to be the right direction.

Can somebody give me some hints on how to proceed ?

Thanks

Here are the steps I tried :

1. First, observe that, conditional to $$X_s$$, $$X_1$$ is normally distributed with mean $$X_s$$ and variance $$(1-s)$$
2. Then, we know $$\mathbb{E}(e^{iuX_1}|F_s)$$, which is given by the characteristic of a gaussian : $$\mathbb{E}(e^{iuX_1}|F_s)=e^{iuX_s-\frac{1}{2}u^2(1-s)}$$. We then have $$\mathbb{E}[g'(X_1)|F_s]=g'(X_s)e^{-\frac{1}{2}u^2(1-s)}$$
3. Using Ito lemma, we have that $$g(X_1)=1+\int_0^1 g'(X_s)dX_s + \frac{1}{2}\int_0^1 g''(X_s)d\langle X,X\rangle_s$$

I'm still wondering if it's the right path to the proof !

• What do you mean by not the right direction? You can't prove it for that choice of $g$ or don't know what to do next? – LazyCat Sep 19 '20 at 16:56
• I meant for that particular choice of $g$. I guess that, since $g$ can be decomposed as a sum of Fourrier series, I should be able to arrive to the result. – Aguelmame Sep 21 '20 at 20:19
• I've searched around a bit, take a look at page 25 of the following notes: math.wisc.edu/~kurtz/NualartLectureNotes.pdf I think, one can extract the details from there. If not - let me know I can take another look. – LazyCat Sep 21 '20 at 21:14
• Hello, not sure to see the link ! The page goes through another route to prove Clark's formula, for general random variables. – Aguelmame Sep 22 '20 at 19:56