Gamma Process Increments and Their Distribution Parameters

Summary

The document explains how to determine the distribution of an increment in a gamma process. It specifies a process whose value at time t has a gamma distribution with shape parameter proportional to time and a shared rate parameter. The answer relies on the defining properties of a Lévy process: increments are independent and stationary, so the increment from s to t has the same distribution as the process value over an interval of length t minus s.

Because gamma variables with a common rate parameter add by summing their shape parameters, the increment is gamma distributed with shape a times the interval length and rate b. This gives the parameters needed to simulate increments over chosen time steps. The result depends on the stated parameterization and on the process having the gamma Lévy process properties described; it is a distributional explanation, not a broader treatment of simulation choices or calibration.

Key ideas

• A Lévy process has stationary and independent increments.
• The increment from time s to time t has the same distribution as the process value over an interval of length t minus s.
• For the stated gamma process, an increment has shape parameter a times the interval length and rate parameter b.
• Gamma variables with a common rate add by summing their shape parameters.

Tags

# Simulation of Gamma process (distribution of increments)


# Simulation of Gamma process (distribution of increments)












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?

## Answer by Bob Jansen (score 1, accepted)

https://quant.stackexchange.com/a/63042

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.