# Expectation of stochastic integral

Let us consider a diffusion process defined as $$dX_t = g(X_t,t) \, dt + \sigma \, dW_t$$ which induces a path measure $$Q$$ in the time interval $$[0,T]$$.

Is the following expectation $$\left\langle \int^T_0 \frac{f(X_t)-g(X_t,t)}{\sigma^2} \, dW_t \right\rangle_Q,$$ constant or zero? Here $$f$$ is a bounded function.

• do you want to specify $f$ ? Dec 23, 2022 at 17:45
• @CarloBeenakker It is a bounded function, nothing specific. Dec 23, 2022 at 19:39

A sufficient condition for the integral $$\int_0^t f(\omega, s)\, dB_s$$ to be a martingale on $$[0,T]$$ is that
1. $$f(\omega,s)$$ is adapted, measurable in s, and
2. $$\mathbb{E}\left(\int_0^T f^2(\omega,s)\,ds\right) < \infty$$. In this case, indeed, $$\mathsf{E} \left(\int_0^T f(\omega,s)\, dB_s\right)=0$$.
So if those conditions are satisfied for the given $$f,g$$ (eg. the integrability condition), then yes.
If we don't have the integrability, that integral will not even be well-defined/finite and so we cannot compute its expectation. Also, by taking $$f=x^{p}$$ for some $$0, we then deal with rough-integrals.