Transcription of Chapter 1 Poisson Processes - NYU Courant
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Chapter 1 Poisson Processes - NYU Courant
Chapter 1 Poisson The Basic Poisson ProcessThe Poisson Process is basically a counting processs. A Poisson Process onthe interval [0, ) counts the number of times some primitive event hasoccurred during the time interval [0, t]. The following assumptions are madeabout the Process N(t).(i). The distribution ofN(t+h) N(t) is the same for eachh >0, isindependent oft.(ii). The random variablesN(t j) N(tj) are mutually independent if theintervals [tj, t j] are nonoverlapping.(iii).N(0) = 0,N(t) is integer valued, right continuous and nondecreasingint, with Probability 1.(iv).P[N(t+h) N(t) 2 ] =P[N(h) 2 ] =o(h) ash the above assumptions, the processN( )has the fol-lowing additional properties.(1). With probability1,N(t)is a step function that increases in steps ofsize1.]
Chapter 2 Continuous time Markov Processes 2.1 Jump Markov Processes. If we have a Markov Chain {Xn} on a state space X, with transition probabil-ities Π(x,dy), and a Poisson Process N(t) with intensity λ, we can combine the two to define a continuous time Markov process x(t) with X as state space by the formula x(t) = XN(t)
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