How to simulate a process $S\_t=\sum\_{0\leq s\leq t}\Delta\_s,$ where $\Delta\_s$ is a Poisson point process with values in $(0,\infty)$ and with characteristic measure $\Pi(dx)=\frac{\alpha}{\Gamma(1-\alpha)}x^{-1-\alpha}dx, \alpha=0.5,1,1.5.$ This means for every Borel set $B\subset (0,\infty),$ the counting process $N\_\cdot^B=Card\{s\in [0,\cdot]:\Delta\_s\in B\}$ is a Poisson process with intensity $\Pi(B).$


I am a fresh student on simulation. It would be nice if you can give me a program (Matlab or Mathematica) of a simple case. Thank you very much in advance.



How to simulate Poisson point process
I assume we know how to simulate a Poisson point process with constant intensity in an interval (e.g. by considering partial sums of i.i.d. exponential variables.)


That allows you to simulate a standard Poisson point process in a rectangle $[a,b] \times [0,d]$ by simulating an intensity $d$ Poisson process in $[a,b]$ and assigning the resulting points $\{x\_i\}\_{i=1}^T$ uniform heights $\{y\_i\}\_{i=1}^T$ in $[0,d]$ to obtain $\{(x\_i,y\_i)\}\_{i=1}^T$.


To simulate (with a small error $<\epsilon$ in total variation) a Poisson point process with slowly decreasing integrable intensity $f(x)dx$ in $[0,\infty)$, first pick an increasing sequence $\{N\_k\}\_{k \ge 0}$ with $N\_0=0$, e.g. $N\_k=2^k-1$. Then pick $m$ large enough so that $\int\_{N\_m}^\infty f(x) \, dx <\epsilon$, so we can neglect points beyond $N\_m$. Then for each $k=0,1,\ldots m-1$ independently simulate a standard Poisson process in a rectangle $[N\_k,N\_{k+1}] \times [0,f(N\_k)]$. For each $k$ this yields points $(x\_i(k),y\_i(k))$
for $1 \le i \le T\_k$. Reject those points for which $y\_i(k)>f(x\_i(k))$.
Note that with $f$ decreasing polynomially and $N\_k$ increasing geometrically, the rejection probability is bounded away from 1. The $x$ coordinates of the surviving points will yield a sample from a Poisson process with intensity $f(x)\, dx $ in $[0,N\_m]$.


