Clark’s Formula for Brownian Functionals and Itô’s Lemma

Summary

The document asks how Itô’s lemma can help derive Clark’s representation for a smooth, bounded function of Brownian motion at a fixed horizon. The stated formula expresses the terminal function value as its expectation plus a stochastic integral whose integrand is the conditional expectation of the function’s derivative given the filtration at each time. The questioner considers conditioning on the current Brownian value and using the Gaussian distribution of the terminal value, then tries applying Itô’s lemma directly to the function along the path.

The material is primarily a mathematical question and does not include an answer completing the derivation. It does show why the conditional distribution of Brownian increments is relevant, but its attempted conditional-expectation step does not establish the displayed integrand for a general function. The topic can inform stochastic calculus and derivative-pricing theory, though the document itself offers no proof, application to a trading strategy, or empirical evidence.

Key ideas

• Clark’s formula represents a Brownian functional as its expectation plus a stochastic integral.
• The integral’s integrand is a conditional expectation of the terminal function’s derivative.
• Brownian motion’s conditional Gaussian distribution can help evaluate related conditional expectations.
• Applying Itô’s lemma directly to the function along the Brownian path does not by itself complete the requested derivation.

Tags

# Brownian function and Clark's formula


# 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 :

- First, observe that, conditional to $X_s$, $X_1$ is normally distributed with mean $X_s$ and variance $(1-s)$

- 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 $\mathbb{E}[g'(X_1)|F_s]=g'(X_s)e^{-\frac{1}{2}u^2(1-s)}$

- 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 !

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