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The gamma process is a Levy process $X$, where $X_t$ has gamma distribution with parameters $at,b>0$ and density $$f\left(x\right)=\frac{b^{at}}{\Gamma\left(at\right)}x^{at-1}e^{-bx}$$

I want to simulate gamma process by increments but what is the distribution of $X_t - X_s$? Of course gamma but with what parameters?

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A Lévy process is defined as (Lévy process and Stochastic Calculus, David Applebaum):

Suppose that we are given a probability space $(\Omega, \mathcal{F}, P)$. A Lévy process $X = (X (t), t \geq 0)$ taking values in $\mathbb{R}^d$ is essentially a stochastic process having stationary and independent increments; we always assume that $X (0) = 0$ with probability 1. So:

  • each $X (t) : \Omega \to \mathbb{R}^d$;
  • given any selection of distinct time-points $0 \leq t_1 < t_2 < \ldots < t_n$, the random vectors $X(t_1), X(t_2) − X(t_1), X(t_3) − X(t_2), \ldots, X (t_n) − X(t_{n−1})$ are all independent;
  • given any two distinct times $0 \leq s < t < \infty$, the probability distribution of $X(t) − X(s)$ coincides with that of $X(t − s)$.

The Gamma distribution is scale invariant under summation, i.e. $$\sum_{i=1}^N X_i = \mathrm{Gamma}\left(\sum_{i=1}^N k_i, \theta\right)$$ so thanks to the third property $X_t - X_s$ has parameters $a (t - s), b$.

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