positive martingale process

Question

I would to like to prove that the process:

$$e^{\int_{0}^{T}\theta _{s}\,dW_{s}-\frac{1}{2}\int_{0}^{T}\theta _{s}^2\,ds}$$ is a martingale which is positive and has a mean=1, where $\theta_s$ is continuous deterministic function.

Thank you

Answer 1

Set $$X_{T}=\int_{0}^{T}\theta_{s}dW_{s}-\frac{1}{2}\int_{0}^{T}\theta_{s}^2ds$$ then $$dX_{T}=\theta_{T}dW_{T}-\frac{1}{2}\theta_{T}^2dT$$ and using Itô's Lemma: $$d(e^{X_{T}})=e^{X_{T}}dX_{T}+\frac{1}{2}e^{X_T}d\langle X\rangle_{T}$$

then ?? I don't know how to continue