\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

Gamma processes provide the classic subordinator models, i.e. non-decreasing Lévy processes. By “gamma process” in fact I mean specifically a Lévy process with gamma increments. Other processes that happen to have gamma marginals, e.g. (Wolpert 2006) are not germane.

Ground zero for these processes appears to be (Ferguson 1974; Ferguson and Klass 1972).

Tutorial introductions to gamma processes can be found in in (Applebaum 2009; Asmussen and Glynn 2007; Rubinstein and Kroese 2016; Kyprianou 2014). Existence proofs etc are deferred to those sources. You could also see Wikipedia.

The marginal distribution of the an increment of duration \(t\) is given by the gamma distribution, which we had better cover first.

## Gamma distribution

Let us take a brief divergence into the gamma distribution which is the increment distribution the gamma process (Indeed, all divisible distributions induce Lévy processes.)

The density \(g(x;t,\alpha, \lambda )\) of the univariate gamma is

\[
g(x; \alpha, \lambda)=
\frac{ \lambda^{\alpha} }{ \Gamma (\alpha) } x^{\alpha\,-\,1}e^{-\lambda x},
x\geq 0.
\]
This is the *shape-rate* parameterisation, with rate \(\lambda\)
and shape \(\alpha,\).
We can think of the gamma distribution as the distribution
at time 1 of a gamma process.

If \(\rv{g}\sim \operatorname{Gamma}(\alpha, \lambda)\) then \(\bb E(\rv{g})=\alpha/\lambda\) and \(\var(\rv{g})=\alpha/\lambda^2.\)

We use various facts about the gamma distribution which quantify its divisibility properties.

- If \(\rv{g}_1\sim \operatorname{Gamma}(\alpha_1, \lambda),\,\rv{g}_2\sim \operatorname{Gamma}(\alpha_2, \lambda),\) and \(\rv{g}_1\perp \rv{g}_2,\) then \(\rv{g}_1+\rv{g}_2\sim \operatorname{Gamma}(\alpha_1+\alpha_2, \lambda)\) (additive rule)
- If \(\rv{g}\sim \operatorname{Gamma}(\alpha, \lambda)\) then \(c \rv{g}\sim \operatorname{Gamma}(\alpha, \lambda/c)\) (multiplicative rule)
- If \(\rv{g}_1\sim \operatorname{Gamma}(\alpha_1, \lambda)\perp \rv{g}_1\sim \operatorname{Gamma}(\alpha_2, \lambda)\) then \(\frac{\rv{g}_1}{\rv{g}_1+\rv{g}_2}\sim \operatorname{Beta}(\alpha_1, \alpha_2)\) independent of \(\rv{g}_1+\rv{g}_2\) (stick-breaking rule)

### Moments

Also note that the moment generating function of the gamma distribution is

\[ \bb{E}[\exp(\rv{g} s)]=\left(1-{\frac {s}{\lambda }}\right)^{-\alpha }{\text{ for }}s<\lambda\] which gives us expressions for all moments fairly easily;

\[\begin{aligned} \bb{E}[\rv{g}]&=\left.\frac{\dd}{\dd s}\bb{E}[\exp(\rv{g} t)]\right|_{s=0}\\ &=\left.\frac{\dd}{\dd s}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha }\right|_{s=0}\\ &=\left.\frac{\alpha}{\lambda}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -1}\right|_{s=0}\\ &=\alpha/\lambda\\ \bb{E}[\rv{g}^2] &=\left.\frac{\dd}{\dd s}\frac{\alpha}{\lambda}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -1}\right|_{s=0}\\ &=\left.\frac{\alpha^2+\alpha}{\lambda^2}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -2}\right|_{s=0}\\ &=\frac{\alpha(\alpha+1)}{\lambda^2}\\ \bb{E}[\rv{g}^3] &=\left.\frac{\dd}{\dd s}\frac{\alpha^2+\alpha}{\lambda^2}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -2}\right|_{s=0}\\ &=\frac{\alpha(\alpha+1)(\alpha+2)}{\lambda^3}\left(1-{\frac {s}{\lambda }}\right)^{-\alpha -2}\\ &=\frac{\alpha(\alpha+1)(\alpha+2)}{\lambda^3}\\ \bb{E}[\rv{g}^4] &=\frac{\alpha(\alpha+1)(\alpha+2)(\alpha+3)}{\lambda^4}\\ &\dots\\ \bb{E}[\rv{g}^n] &=\frac{\langle \alpha \rangle_{n}}{\lambda^n}\\ \end{aligned}\]

Here \(\langle \alpha \rangle_{n}:=\frac{\Gamma(\alpha+n)}{\Gamma(\alpha)}\) is the rising factorial.

## The Gamma process

The univariate gamma *process* \(\{\rv{g}(t;\alpha,\lambda)\}_t\)
is an independent-increment process,
with time index \(t\) and parameters by \(\alpha, \lambda.\)
We assume it is started at \(\rv{g}(0)=0\).

The marginal density \(g(x;t,\alpha, \lambda )\) of the process at time \(t\) is a gamma RV, specifically, \[ g(x;t, \alpha, \lambda) =\frac{ \lambda^{\alpha t} } { \Gamma (\alpha t) } x^{\alpha t\,-\,1}e^{-\lambda x}, x\geq 0. \] That is, \(\rv{g}(t) \sim \operatorname{Gamma}(\alpha(t_{i+1}-t_{i}), \lambda)\).

which corresponds to increments per unit time in terms of \(\bb E(\rv{g}(1))=\alpha/\lambda\) and \(\var(\rv{g}(1))=\alpha/\lambda^2.\)

Note that if \(\alpha t=1,\) then \(\rv{g}(t;\alpha ,\lambda )\sim \operatorname{Exp}(\lambda).\)

This leads to a method for simulating a path of a gamma process at a sequence of increasing times, \(\{t_1, t_2, t_3, t_L\}.\) Given \(\rv{g}(t_1;\alpha, \lambda),\) we know that the increments are distributed as independent variates \(\rv{g}_i:=\rv{g}(t_{i+1})-\rv{g}(t_{i})\sim \operatorname{Gamma}(\alpha(t_{i+1}-t_{i}), \lambda)\). Presuming we may simulate from the Gamma distribution, it follows that

\[\rv{g}(t_i)=\sum_{j \lt i}\left( \rv{g}(t_{i+1})-\rv{g}(t_{i})\right)=\sum_{j \lt i} \rv{g}_j.\]

A standard \(d\)-dimensional gamma process is the concatenation of \(d\) independent univariate gamma processes.

## Gamma bridge

Consider a univariate gamma process, \(\rv{g}(t)\) with \(\rv{g}(0)=0.\) The gamma bridge, analogous to the Brownian bridge, is the gamma process conditional upon attaining a fixed the value \(S=\rv{g}(1)\) at terminal time \(1.\) We write \(\rv{g}_{S}:=\{\rv{g}(t)\mid \rv{g}(1)=S\}_{0\lt t \lt 1}\) for the paths of this process.

We can simulate from the gamma bridge easily. Given the increments of the process are independent, if we have a gamma process \(\rv{g}\) on the index set \([0,1]\) such that \(\rv{g}(1)=S\), then we can simulate from the bridge paths which connect these points at intermediate time \(t,\, 0<t<1\) by recalling that we have known distributions for the increments; in particular \(\rv{g}(t)\sim\operatorname{Gamma}(\alpha, \lambda)\) and \(\rv{g}(1)-\rv{g}(t)\sim\operatorname{Gamma}(\alpha (1-t), \lambda)\) and these increments, as increments over disjoints sets, are themselves independent. Then, by the stick breaking rule,

\[\frac{\rv{g}(t)}{\rv{g}(1)}\sim\operatorname{Beta}(\alpha t, \alpha(1-t))\] independent of \(\rv{g}(1).\) We can therefore sample from a path of the bridge \(\rv{g}_{S}(t)\) for some \(t\lt 1\) by simulating \(\rv{g}_{S}(t)=B S,\) where \(B\sim \operatorname{Beta}(\alpha (t),\alpha (1-t)).\)

## Multivariate gamma processes with dependence

I am not sure what general correlations are possible here, but one obvious possibility is to choose a transform matrix \(M\) with non-negative entries. Then the process \(\{M\rv{g}(t)\}\) is still marginally a gamma process, but the components of the vector are no longer independent. Is this the most general possible gamma process? What is the covariance structure of that process? 🏗

## Time-warped gamma process

Çinlar (1980) walks us through the mechanics of (deterministically) time-warping Gamma processes, which ends up being not too unpleasant. See (Singpurwalla 1997) for an application. Why bother? Linear superpositions of gamma processes can be hard work. 🏗

## Matrix gamma processes

I am tempted to look at matrix-valued extensions, much as the Wishart distribution is a matrix valued extension of the gamma distribution.
“Wishart processes” are indeed a thing (Pfaffel 2012) but the common definition, unlike the common definition of the gamma process, is not necessarily (element-wise, or in any other sense) monotonic.
That is, it generalises the *square Bessel process*, which
is indeed marginally gamma distributed (specifically \(\chi^2\) distributed)
but also non-monotonic, so it is not a natural extension of what we have here.

## Superpositions of gamma processes

This does not come out as simply as with Gaussian processes; sums of Gamma processes are messy (Mathai 1982; Moschopoulos 1985). 🏗

## Centred gamma process

Define \(H_t:=\rv{g}_t-\alpha t /\lambda.\) Then we have a mean-zero process, in fact, a martingale, since we have subtracted the compensator from it.

We call the marginal distribution at time \(t=1\) a *centred gamma distribution*, and will recycle the letter \(H=G-\alpha /\lambda\).
As a linear transform of a random variable, the MGF of the centred gamma distribution is (ignoring questions of region of convergence)

\[\begin{aligned} \bb{E}[\exp(H s)]&=\exp (\alpha s/\lambda)\bb{E}[\exp(G s)]\\ &=\exp (\alpha s/\lambda)\left(1-{\frac {s}{\lambda }}\right)^{-\alpha }\\ &=\exp (\alpha s/\lambda - \alpha \log(1-s/\lambda))\end{aligned}\]

## As a Lévy process

For arguments \(x, t>0\) and parameters \(\alpha, \lambda>0,\) we have the increment density as simply a gamma density:

\[ p_{X}(t, x)=\frac{\lambda^{\alpha t} x^{\alpha t-1} \mathrm{e}^{-x \lambda}}{ \Gamma(\alpha t)}, \]

This gives us a spectrally positive Lévy measure

\[\pi_{X}(x)=\frac{\alpha}{x} \mathrm{e}^{-\lambda x} \]

and Laplace exponent

\[\Phi_{X}(z)=\alpha \ln (1+ z/\lambda), z \geq 0. \]

That is, the Poisson rate, with respect to time \(t\) of jumps whose size is in the range \([x, x+dx)\), is \(\pi(x)dx.\) We think of this as an infinite superposition of Poisson processes driving different sized jumps, where the jumps are mostly tiny. This is how I think about Lévy process theory, at least.

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