Let (N t ) t≥0 be a Poisson process with parameter𝜆. (a) Find the quantity m(t)such that M t = (N t −𝜆t) 2 −m(t) is a martingale. (b) For fixed integer k>0, let T=min {t∶ N t =k} be the first-time k...

Let (N_t)_t≥0
be a Poisson process with parameter𝜆.

(a) Find the quantity m(t)such that M_t
= (N_t−𝜆t)²−m(t) is a martingale.

(b) For fixed integer k>0, let T=min {t∶ N_t=k} be the first-time k arrivals occur for a Poisson process. Show that T is a stopping time that satisfies the conditions of the optional stopping theorem.

(c) Use the optional stopping theorem to find the standard deviation of T.