Stochastic Integral of the exponential of a process to it self

Question

This is probably very easy, but i couldn't solve it so far.

Let $Y_{t}=\alpha dW_{t}$ for a constant $\alpha \in\mathbb{R}$ and a Brownian Motion $W$. Is the local Martingale $e^{Y_t}dY_t$ a true martingale ?

Answer 1

As discussed in the comments, to show $\int_0^t e^{Y_s}dY_s$ is a true martingale, it is sufficient to show $\mathbb{E}[\int_0^t (e^{Y_s})^2 ds] < \infty$. We compute

\begin{align*} \mathbb{E}[(e^{Y_s})^2] &= \mathbb{E}[e^{2 \alpha W_s}] \\ &= e^{2 \alpha^2 t} \end{align*} by using the known expression for the moment generating function of a $N(0,s)$ random variable. Thus, by Tonelli's theorem, \begin{align*} \mathbb{E}[\int_0^t (e^{Y_s})^2 ds] &= \int_0^t \mathbb{E}[(e^{Y_s})^2] ds \\ &= \int_0^t e^{2 \alpha^2 s} ds <\infty, \end{align*} so we conclude $\int_0^t e^{Y_s}dY_s$ is in fact a true martingale.