distributed with parameters A and u, respectively. Then Z=X+y is Poisson distributed with parameter A+u. Proof. Use the characteristic function Definition. A continuous time stochastic process is a function X(t)where for each fixed t>0, X(t) is a random variable Definition. Let( Q, F, P, F) be a filtered probability space. A stochastic process IN(t,w); t20 is said to be a(P, E)-Poisson process with intensity A if ·Nisf- adapted. The trajectories of N are(with probability one)right continuous and piecewise continuous N(0)=0 ·△N(t)=0or1( with probability one) where f)=N(t)-N(t-) · For all s≤t,N(t)-N(s) is independent of F s N()-N(s) is Poisson distributed with parameter A(t-s), that is P(N(t)-N(s)=k|)=P(N()-N()=k)=c-A(-2(t-s Proposition. The time between jumps isis exponentially distributed. More pre- cisely, let T be defned via T=infN(t)>0)distributed with parameters λ and µ, respectively. Then Z = X + Y is Poisson distributed with parameter λ + µ. Proof. Use the characteristic function. Definition. A continuous time stochastic process is a function X(t) where for each fixed t ≥ 0, X(t) is a random variable. Definition. Let (Ω, F, P, F) be a filtered probability space. A stochastic process {N(t, ω);t ≥ 0} is said to be a (P, F)–Poisson process with intensity λ if • N is F–adapted. • The trajectories of N are (with probability one) right continuous and piecewise continuous. • N(0) = 0. • ∆N(t) = 0 or 1 (with probability one) where ∆N(t) = N(t) − N(t−). • For all s ≤ t, N(t) − N(s) is independent of Fs. • N(t) − N(s) is Poisson distributed with parameter λ(t − s), that is P (N(t) − N(s) = k | Fs)) = P (N(t) − N(s) = k) = e −λ(t−s)λ k (t − s) k k! . Proposition. The time between jumps is is exponentially distributed. More pre￾cisely, let τ be defined via τ = inf t≥0 {N(t) > 0}. 20