expectation of a product of a brownian motion and a poisson process

Question

I am trying to find out the expectation $\mathbb{E}[X_t|\mathcal{F}_s]$ for
$$X_t = (W_tN_t)^n$$ where $n\geq 1$, $N_t$ is a homogenous poisson process with intensity $\lambda$ and $\mathcal{F}_s$ represents the information generated by both the brownian motion and the jump process up to time s. I thought I could proceed using Ito's lemma for jump processes, however I am not sure how to apply it in this particular case.

Answer 1

Suppose that $t\geq s$ \begin{align} \mathbb{E}(X_t|\mathcal{F}_s) &= \mathbb{E}\left[(W_t-W_s+W_s)^n (N_t - N_s+N_s)^n\right|\mathcal{F}_s] \\ &= \mathbb{E}\left[ \sum_{k=0}^n\binom{n}{j}(W_t - W_s)^{n-k}W_s^k\ \sum_{j=0}^n\binom{n}{k}(N_t - N_s)^{n-j}N_s^j \right]\\ &= \sum_{k=0}^n\sum_{j=0}^n \binom{n}{k} \binom{n}{j} W_s^k N_s^j \mathbb{E}\left[ \left( W_t-W_s\right)^{n-k} \left( N_t - N_s\right)^{n-j} \right] \end{align} Furthermore, if the Poisson process is independent of the Brownian motion the last expectation becomes a product of two expectations of random variables whose distributions are known.