Gamma process

The gamma process is often abbreviated as $X(t)\backslash equiv\backslash Gamma(t;\backslash gamma,\; \backslash lambda)$ where $t$ represents the time from 0. The shape parameter $\backslash gamma$ (inversely) controls the jump size, and the rate parameter $\backslash lambda$ controls the rate of jump arrivals, analogously with the gamma distribution.{{sfn|Sánchez-Silva|Klutke|2016|p=133}} Both $\backslash gamma$ and $\backslash lambda$ must be greater than 0. We use the gamma function and gamma distribution in this article, so the reader should distinguish between $\backslash Gamma(\backslash cdot)$ (the gamma function), $\backslash Gamma(\backslash gamma,\; \backslash lambda)$ (the gamma distribution), and $\backslash Gamma(t;\backslash gamma,\; \backslash lambda)$ (the gamma process).

The process is a pure-jump increasing Lévy process with intensity measure $\backslash nu(x)=\backslash gamma\; x^\{-1\}\; \backslash exp(-\backslash lambda\; x),$ for all positive $x$. It is assumed that the process starts from a value 0 at $t=0$ meaning $X(0)=0$. Thus jumps whose size lies in the interval $[x,x+dx)$ occur as a Poisson process with intensity $\backslash nu(x)\backslash ,dx.$

The process can also be defined as a stochastic process $X(t),\; t\; \backslash leq\; 0$ with $X(0)\; =\; 0$ and independent increments, whose marginal distribution $f$ of the random variable $X(t)\; -\; X(s)$ for an increment $l\; =\; t-s\; \backslash geq\; 0$ is given by{{sfn|Sánchez-Silva|Klutke|2016|p=133}}

$$f(x;l,\; \backslash gamma,\; \backslash lambda)\; =\; \backslash Gamma(\backslash gamma\; l,\; \backslash lambda)\; =\; \backslash frac\; \{\backslash lambda^\{\backslash gamma\; l\}\}\{\backslash Gamma\; (\backslash gamma\; l)\}\; x^\{\backslash gamma\; l-1\}e^\{-\backslash lambda\; x\}.$$

It is also possible to allow the shape parameter $\backslash gamma$ to vary as a function of time, $\backslash gamma(t)$.{{sfn|Sánchez-Silva|Klutke|2016|p=133}}

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Because the value at each time $t$ has mean $\backslash gamma\; t/\backslash lambda$ and variance $\backslash gamma\; t/\backslash lambda^2,${{sfn|Sánchez-Silva|Klutke|2016|p=94}} the gamma process is sometimes also parameterised in terms of the mean ($\backslash mu$) and variance ($v$) of the increase per unit time. These satisfy $\backslash gamma\; =\; \backslash mu^2/v$ and $\backslash lambda\; =\; \backslash mu/v$.

Multiplication of a gamma process by a scalar constant $\backslash alpha$ is again a gamma process with different mean increase rate.

$$\backslash alpha\backslash Gamma(t;\backslash gamma,\backslash lambda)\; \backslash simeq\; \backslash Gamma(t;\backslash gamma,\backslash lambda/\backslash alpha)$$

The sum of two independent gamma processes is again a gamma process.

$$\backslash Gamma(t;\backslash gamma\_1,\backslash lambda)\; +\; \backslash Gamma(t;\backslash gamma\_2,\backslash lambda)\; \backslash simeq\; \backslash Gamma(t;\backslash gamma\_1+\backslash gamma\_2,\backslash lambda)$$

The moment function helps mathematicians find expected values, variances, skewness, and kurtosis.

$$\backslash operatorname\; E(X\_t^n)\; =\; \backslash lambda^\{-n\}\; \backslash cdot\; \backslash frac\{\backslash Gamma(\backslash gamma\; t+n)\}\{\backslash Gamma(\backslash gamma\; t)\},\backslash \; \backslash quad\; n\backslash geq\; 0\; ,$$ where $\backslash Gamma(z)$ is the Gamma function.

The moment generating function is the expected value of $\backslash exp(tX)$ where X is the random variable.

Correlation displays the statistical relationship between any two gamma processes.

The gamma process is used as the distribution for random time change in the variance gamma process. Specifically, combining Brownian motion with a gamma process produces a variance gamma process,{{sfn|Applebaum|2004|pp=58-59}} and a variance gamma process can be written as the difference of two gamma processes.{{sfn|Fu|Madan|2007|p=38}}

• Moran process

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• {{cite book |last1=Applebaum |first1=David |title=Lévy processes and stochastic calculus |date=2004 |publisher=Cambridge University Press |location=Cambridge, UK ; New York |isbn=0-521-83263-2}}
• {{cite book |last1=Fu |first1=Michael |last2=Madan |first2=Dilip B. |title=Advances in mathematical finance |date=2007 |publisher=Birkhauser |location=Boston |isbn=978-0-8176-4545-8}}
• {{cite book |last1=Klenke |first1=Achim |title=Probability theory: a comprehensive course |date=2008 |publisher=Springer |location=London |isbn=978-1-84800-048-3|doi=10.1007/978-1-84800-048-3_24}}
• {{cite book |last1=Sánchez-Silva |first1=Mauricio |last2=Klutke |first2=Georgia-Ann |title=Reliability and life-cycle analysis of deteriorating systems |date=2016 |publisher=Springer |location=Cham |isbn=978-3-319-20946-3}}

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{{Stochastic processes}}

Category:Lévy processes