=+9. Let Nt denote the number of random points that occur by time t in a Poisson

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=+9. Let Nt denote the number of random points that occur by time t in a Poisson process on [0, ∞) with intensity λ. Show that the following stochastic processes Xt = Nt − λt Xt = (Nt − λt)

2 − λt Xt = e−θNt+λt(1−e−θ)

enjoy the martingale property E(Xt+s| Nr, r ∈ [0, t]) = Xt for s > 0.

(Hint: Nt+s − Nt is independent of Nt and distributed as Ns.)

266 10. Martingales

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