Exercise $3(10$ pts$).$ Let $P_0$ be the probability under which $=0$ and $P_1$ be the probability under which $N(0,1).$ Suppose that conditionally on $,\{x_i,i1\}$ is a sequence of iid $N(,1)$ random variables under both $P_0$ and $P_1.$ Let $f_j(x_1,$dots,$x_t)$ be the joint pdf of $(x_1,$dots,$x_t)$ under $P_j,j=0,1,$ and

${}_t=\frac{f_1(x_1,\text{d}\text{o}\text{t}\text{s}\text{,}{x}_t)}{f_0(x_1,\text{d}\text{o}\text{t}\text{s}\text{,}{x}_t)},t=1,2,$dots

Verify that $\{{}_1,$dots,${}_{10}\}$ is a martingale under $P_0$ with natural filtration.