stochastic process: multiply two poisson process

Question

Consider the process $\{X(t),t\ge0\}$ defined by $$X(t)=N(t)-\lambda t$$ where $N$ is a Poisson process with rate $\lambda \gt 0$ I have multiple questions about this one:

1. Compute, for $t_1, t_2, n_1, n_2 \gt 0$, the second-order probability mass of $N, G(t_1,t_2;n_1,n_2)$.
2. Compute $\mathbb E[X(t_1)X(t_2)]$, for $0 \lt t_1 \le t_2$.

For the problem 2, I don't know if there is correlation between $N(t_1)\cdot N(t_2)$ to compute its expectation.

Answer 1

The Poisson process has a kind of Markov property: the distribution of $N(s_2) - N(s_1)$, and hence the distribution of $X(s_2) - X(s_1)$, depends only on the length of the interval and is in particular independent of $N(s_1)$ or $X(s_1)$. (This property comes from the definition of the Poisson process.)

To exploit that here, use the identity $X(t_1) X(t_2) = X(t_1) \left[( X(t_2) - X(t_1)) + X(t_1)\right].$

Answer 2

No way I'm doing parts (1)-(5) of a problem, but you did ask one actual question about the problem so I'll give a hint for that. We can write $$ N(t_1)N(t_2) = N(t_1)^2 + N(t_1)(N(t_2)-N(t_1)) = N(t_1)^2 + (N(t_1)-N(0))(N(t_2)-N(t_1))$$ and you know (from the definition of a Poisson process) a great deal about the joint distribution of the factors in the second term.