Exercise 8.1 Consider a model for the stock market where the short rate of interest $r$ is a deterministic constant. We focus on a particular stock with price process $S$. Under the objective probability measure $P$ we have the following dynamics for the price process. $$ d S(t)=\alpha S(t) d t+\sigma S(t) d W(t)+\delta S(t-) d N(t) . $$ Here $W$ is a standard Wiener process whereas $N$ is a Poisson process with intensity $\lambda$. We assume that $\alpha, \sigma, \delta$ and $\lambda$ are known to us. The $d N$ term is to be interpreted in the following way: - Between the jump times of the Poisson process $N$, the $S$-process behaves just like ordinary geometric Brownian motion. - If $N$ has a jump at time $t$ this induces $S$ to have a jump at time $t$. The size of the $S$-jump is given by $$ S(t)-S(t-)=\delta \cdot S(t-) . $$ Discuss the following questions. (b) Is the model complete