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I am trying to simulate the following process:

h(t)=B(t)+e[P1(t)-P2(t)]

in which B(t) is a Brownian motion and P1, P2 are Poisson process with lambda=5000 and t<-1:10000 and I choose e=0.5. I am able to simulate B(t) wiht the following code:

x <- rnorm(n = length(t) - 1, sd = sqrt(0.01)) 
B <- c(0, cumsum(x))

and one Poisson process 

P1<-cumsum(rpois(10000,5000/10000))

P2 is also poisson process with the same rate so that the difference between P1 and P2 will be a process that jumps around 0 as follow: ...0,lambda,-2*lambda,lambda,0.... I created P2 the same way I did for P1 and plot the difference P1-P2 but the result is not as I expected

I wonder if my algorithm to create P1 and P2 is correct? Can someone please tell me the step that I went wrong?

Many thanks in advance

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  • $\begingroup$ Why do you think P1 - P2 will behave like that? Your sample looks deterministic, but P1 - P2 will be random. $\endgroup$
    – Cam.Davidson.Pilon
    Commented Dec 8, 2014 at 4:58
  • $\begingroup$ sorry, this is my clarification for how I expected the difference P1-P2 behave: since the jump of Poisson process is 1, the difference of 2 Poisson processes should evolve around 0 with the jump of either 1 or -1. Therefore, h(t) will have the jump of 0.5 or -0.5 when the difference is not 0 $\endgroup$
    – Ocean
    Commented Dec 8, 2014 at 19:20

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