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Generating a trajectory of a CP - gamma process
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A compound poisson process is a stochastic process{Xt, t>=0} that can be represented as the random sum X(t)=sum Yi (ps i=1 to N(t)) t>=0, where {Nt,t>=0} is a poisson process and Y1,Y2...are iid independent of {Nt,t>=0}.
Algorithm:
step 1, set your parameters, e.g., i) lambda, ii) parameters of gamma distributions: shape-nu, scale-alpha.
step 2, set the sampling interval [0,T].
step 3, set the initial time t=0 and the initial value of x x=0.
step 4, generate Delta~Exponential(lambda).
step 5, obtain the jump time of Poisson process t=t+Delta
step 6, generate y~gamma(nu,alpha)
step 7, obtain the value of x at time t, x<-x+y
step 8, repeat steps 4-7, till t>=T.
R code for a detailed example.
lambda<-0.1
nu<-2
alpha<-3
T<-100
time<-0
x<-0
while(time[length(time)]<T){
Delta<-rexp(1,rate=lambda)
time<-c(time,time[length(time)]+Delta)
x<-c(x,x[length(x)]+rgamma(1,nu,rate=alpha))
}
plot(time, x, type="s", main="", xlab="t", ylab="")
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