Gamma Process

In mathematics and probability theory, a gamma process, also known as (Moran-)Gamma subordinator, is a random process with independent gamma distributed increments. Often written as $\Gamma(t;\gamma,\lambda)$, it is a pure-jump increasing Lévy process with intensity measure $\nu(x)=\gamma x^ \exp(-\lambda x),$ for positive $x$. Thus jumps whose size lies in the interval ,x+dx) occur as a Poisson process with intensity $\nu(x)\,dx.$ The parameter $\gamma$ controls the rate of jump arrivals and the scaling parameter $\lambda$ inversely controls the jump size. It is assumed that the process starts from a value 0 at ''t'' = 0.

The gamma process is sometimes also parameterised in terms of the mean ($\mu$) and variance ($v$) of the increase per unit time, which is equivalent to $\gamma = \mu^2/v$ and $\lambda = \mu/v$.

Properties

Since we use the Gamma function in these properties, we may write the process at time $t$ as $X_t\equiv\Gamma(t;\gamma, \lambda)$ to eliminate ambiguity.

Some basic properties of the gamma process are:

Marginal distribution

The marginal distribution of a gamma process at time $t$ is a gamma distribution with mean $\gamma t/\lambda$ and variance $\gamma t/\lambda^2.$

That is, its density $f$ is given by $f(x;t, \gamma, \lambda) = \frac x^e^.$

Scaling

Multiplication of a gamma process by a scalar constant $\alpha$ is again a gamma process with different mean increase rate.
:$\alpha\Gamma(t;\gamma,\lambda) \simeq \Gamma(t;\gamma,\lambda/\alpha)$

Adding independent processes

The sum of two independent gamma processes is again a gamma process.
:$\Gamma(t;\gamma_1,\lambda) + \Gamma(t;\gamma_2,\lambda) \simeq \Gamma(t;\gamma_1+\gamma_2,\lambda)$

Moments

:$\operatorname E(X_t^n) = \lambda^ \cdot \frac,\ \quad n\geq 0 ,$ where $\Gamma(z)$ is the Gamma function.

Moment generating function

:$\operatorname E\Big(\exp(\theta X_t)\Big) = \left(1- \frac\theta\lambda\right)^,\ \quad \theta<\lambda$

Correlation

:\operatorname(X_s, X_t) = \sqrt,\ s, for any gamma process $X(t) .$

The gamma process is used as the distribution for random time change in the variance gamma process.

Literature

* ''Lévy Processes and Stochastic Calculus'' by David Applebaum, CUP 2004, .

References

Lévy processes

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