Solution: Centering of an inhomogeneous Poisson process

With \(t \geq s\) and \(M(t) = N(t) - \int_0^t \lambda(u)du\) we get that \begin{align*}E[M(t) |\mathcal{F}_s] &= E\left[N(t) - \int_0^t \lambda(u)du {\huge |} \mathcal{F}_s\right] \\ &= E\left[N(t) - N(s) - \int_s^t \lambda(u)du + N(s) - \int_0^s \lambda(u)du {\huge |} \mathcal{F}_s\right]\\ &= E\left[N(t) - N(s) - \int_s^t \lambda(u)du{\huge |} \mathcal{F}_s\right] + E\left[N(s) - \int_0^s \lambda(u)du {\huge |} \mathcal{F}_s\right]\\ &= E\left[N(t) - N(s) | \mathcal{F}_s\right] - \int_s^t \lambda(u)du + E\left[N(s) |\mathcal{F}_s\right]- \int_0^s \lambda(u)du\\ &= 0 + N(s) - \int_0^s \lambda(u)du,\end{align*} and thus that \[\underline{\underline{E[M(t) |\mathcal{F}_s] = M(s).}}\]

Denote the cumulative density \(\Lambda(t) = \int_0^t \lambda(u)du\) and note that \(\Lambda(t) - \Lambda(s) = \int_s^t \lambda(u)du\) and \(M(t) = N(t) - \Lambda(t)\). We now use that \(M^2(t) = N^2(t)−2\Lambda (t) N(t)+(\Lambda (t))^2\) and the properties of the Poisson process \(N(t)\), in particular that \(E[N(t)-N(s)| \mathcal{F}_s] = Var[N(t)-N(s)| \mathcal{F}_s] = \Lambda (t) - \Lambda (s)\). Further we need to use that \begin{align*}E[N(t)| \mathcal{F}_s] &= E[N(t)-N(s) + N(s)| \mathcal{F}_s] \\ &= E[N(t)-N(s)| \mathcal{F}_s] + E[N(s)| \mathcal{F}_s] \\ &= (\Lambda (t) - \Lambda (s)) + N(s) \\ &= (N(s) - \Lambda (s)) + \Lambda (t),\end{align*} and that \begin{align*}E[N^2(t)| \mathcal{F}_s] &= Var[N(t)| \mathcal{F}_s] + E[N(t)| \mathcal{F}_s]^2,\end{align*} where \begin{align*} Var[N(t)| \mathcal{F}_s] &= Var[N(t)-N(s) + N(s)| \mathcal{F}_s] \\ &= Var[N(t)-N(s)| \mathcal{F}_s] + Var[N(s)| \mathcal{F}_s] \\ &= (\Lambda (t) - \Lambda (s)) + 0.\end{align*} Now we can calculate \begin{align*} E[M^2 (t)−\Lambda (t) |\mathcal{F}_s] &= E[N^2(t)−2\Lambda (t) N(t)+(\Lambda (t))^2 -\Lambda (t)|\mathcal{F}_s] \\ &= E[N^2(t)|\mathcal{F}_s] −2\Lambda (t) E[N(t)|\mathcal{F}_s] +(\Lambda (t))^2 -\Lambda (t) \\ &= Var[N(t)| \mathcal{F}_s] + E[N(t)| \mathcal{F}_s]^2−2\Lambda (t) \left[(N(s) - \Lambda (s)) + \Lambda (t)\right] +(\Lambda (t))^2 -\Lambda (t)\\ &= \Lambda (t) - \Lambda (s) + [(N(s) - \Lambda (s)) + \Lambda (t)]^2 −2\Lambda (t) [N(s) - \Lambda (s)] - 2(\Lambda (t))^2 + (\Lambda (t))^2 -\Lambda (t)\\ &= \Lambda (t) - \Lambda (s) + [N(s) - \Lambda (s)]^2 + 2\Lambda (t) [N(s) - \Lambda (s)] + (\Lambda (t))^2 −2\Lambda (t) [N(s) - \Lambda (s)] - 2(\Lambda (t))^2 + (\Lambda (t))^2 -\Lambda (t) \\ &= [N(s) - \Lambda (s)]^2-\Lambda (s) \\ &= M^2(s) - \Lambda (s).\end{align*} Thus we have shown that \(M^2 (t)−\Lambda (t)\) is a martingale.