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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!

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