For a Poisson process prove that (a) $N (t) -λt $ and (b) $e^{(\log(1-u) N (t) + uλt)}$, are martingales

Question

For a Brownian motion ${z (t)}$ and for any $β ∈ R$, be

$V (t) = \exp\{ βz (t) - (t β ^ 2) / 2\}, t≥0 $

Show that ${V (t)}$ is a martingale with respect to a Brownian filtration. Also ${N (t)}$ be a Poisson process with $λ> 0$ intensity. Prove the following martingales regarding $\{F_t ^ {N}\}$, where $F_t^N = σ (N (s), s≤t$; (a) $N (t) -λt $ and (b) $e^{(\log(1-u) N (t) + uλt)}, 0 <u <1$.

Could could you give me some tips to solve it?, please. Or some bibliographic references with similar exercises?

Answer 1

Hints

1. Brownian motion: Apply Itô's formula to $f(t,x) := \exp(\beta x - t \beta^2/2).$
2. Poisson process: Use that $(N_t)_{t \geq 0}$ has independent increments, i.e. $N_t-N_s$ is independent of $F_s^N$. Start with the easier one: $$\mathbb{E}(N_t-\lambda t \mid F_s^N) = \mathbb{E}((N_t-N_s)+N_s \mid F_s^N) - \lambda t = \ldots$$