Let $X_t=N_t^+-N_t^-$, where $N^+$ and $N^-$ are independent <Poisson processes> of rate $\lambda$. This <symmetric Poisson difference process> is centered, has jumps $\pm1$, and has variance rate $\sigma^2=2\lambda$. If $a,b$ are positive integers, then $X_T\in\{-a,b\}$ exactly. Optional sampling of the martingale $X_t$ gives
$$
\mathbb P(X_T=b)=\frac{a}{a+b},
\qquad
\mathbb P(X_T=-a)=\frac{b}{a+b}.
$$
Consequently
$$
\mathbb E[X_T^2]
=b^2\frac{a}{a+b}+a^2\frac{b}{a+b}
=ab.
$$
Part (d) now gives the <expected value>[mean exit time]
$$
\mathbb E T=\frac{ab}{2\lambda}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 201 / 6 / e / Solution

The difference of two independent rate-$\lambda$ <Poisson processes> is a continuous-time simple symmetric random walk with total jump rate $2\lambda$ and variance $2\lambda t$ at time $t$.

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 Incoming links: Symmetric Poisson difference process