show that the process is a martingale

Question

maybe you will have an idea how to show that : the process $(exp(X_t-\frac{1}{2}Y_t))$ is a martingale? Where $h \in L^2([0,T])$, $T< \infty$, $X_t=\int_0^th(s)dW_s$ and $Y_t=\int_0^th^2(s)ds$ for $t \le T$

Answer 1

If "exponential martingale" does not ring a bell that you may want to try the following approach. Note that your process $S_t := \exp(X_t - \frac{1}{2}Y_t)$ is a Geometric Brownian motion with a non-constant volatility parameter $h(t)$, that is, is satisfies the following SDE $$\mathrm{d} S_t = S_t h(t)\mathrm{d}W_t $$ which reads $$ S_t = 1+ \int_0^t S_s h(s) \mathrm{d}W_s.$$ You can use Ito's lemma to show it. Then for $\int_0^t S_s h(s) \mathrm{d}W_s$ apply the following

Theorem. The stochastic integral $\int_0^t Z_s(\omega) \mathrm{d}W_s(\omega) $ is a martingale for any $Z_t(\omega) \in L^2([0,T])$.

The proof of this theorem can be found in most textbooks, and possibly it was provided in your lecture. Let me know if you cannot find it.

You can also have a look here (point 13.), there is a direct proof of a special case when $h$ is constant, that is, $h(t) = \sigma$.