# Question about Stochastic Calculus,(change of measure)?

Can any one give some hint for this question?

Let $$\{S_t\}_{t=0}^\infty$$ be an asset price process defined on the probability space $$(\Omega,\mathcal{F},\mathbb{P})$$. Assume that the log-return of $$S_t$$ follows a discrete time stochastic process under the probability measure $$\mathbb{P}$$: $$\begin{split} y_t &\equiv \log S_t - \log S_{t-1} \\ y_t &= 0.5 y_{t-1} + \eta_t \sqrt{1 + y_{t-1}^2} \end{split}$$ where $$y_0=0,S_0=1$$ and $$y_i \sim \mathcal{N}(0,1)$$ are iid.

1. Find one $$\theta_t\in F_t$$ such that $$\frac{e^{y_t \theta_t}S_t}{M_t(\theta_t)}$$ is a martingale with respect to $$F_t$$ under the probability measure P, where $$M_t(z)$$ be the conditional moment-generating function of $$y_t$$, given $$F_{t-1}$$
2. Give a new probability measure $$Q$$ such that $$S_t$$ is a martingale with respect to $$F_t$$ under $$Q$$.

Many thanks!