Compound Poisson process

A compound Poisson process is a continuous-time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. A compound Poisson process, parameterised by a rate $\lambda >0$ and jump size distribution G, is a process $\{\,Y(t):t\geq 0\,\}$ given by

Y(t) = \sum_{i=1}^{N(t)} D_i

where, $\{\,N(t):t\geq 0\,\}$ is a Poisson process with rate $\lambda$ , and $\{\,D_{i}:i\geq 1\,\}$ are independent and identically distributed random variables, with distribution function G, which are also independent of $\{\,N(t):t\geq 0\,\}.\,$

When $D_i$ are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process which has the feature that two or more events occur in a very short time .

Properties of the compound Poisson process

Using conditional expectation, the expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:

\,E(Y(t)) = E(E(Y(t)|N(t))) = E(N(t)E(D)) = E(N(t))E(D) = \lambda t E(D).

Making similar use of the law of total variance, the variance can be calculated as:

\begin{align} \operatorname{var}(Y(t)) &= E(\operatorname{var}(Y(t)|N(t))) + \operatorname{var}(E(Y(t)|N(t))) \\ &= E(N(t)\operatorname{var}(D)) + \operatorname{var}(N(t)E(D)) \\ &= \operatorname{var}(D)E(N(t)) + E(D)^2 \operatorname{var}(N(t)) \\ &= \operatorname{var}(D)\lambda t + E(D)^2\lambda t \\ &= \lambda t(\operatorname{var}(D) + E(D)^2) \\ &= \lambda t E(D^2). \end{align}

Lastly, using the law of total probability, the moment generating function can be given as follows:

\,\Pr(Y(t)=i) = \sum_{n} \Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n)

\begin{align} E(e^{sY}) & = \sum_i e^{si} \Pr(Y(t)=i) \\ & = \sum_i e^{si} \sum_{n} \Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n) \\ & = \sum_n \Pr(N(t)=n) \sum_i e^{si} \Pr(Y(t)=i|N(t)=n) \\ & = \sum_n \Pr(N(t)=n) \sum_i e^{si}\Pr(D_1 + D_2 + \cdots + D_n=i) \\ & = \sum_n \Pr(N(t)=n) M_D(s)^n \\ & = \sum_n \Pr(N(t)=n) e^{n\ln(M_D(s))} \\ & = M_{N(t)}(\ln(M_D(s))) \\ & = e^{\lambda t \left ( M_D(s) - 1\right ) }. \end{align}

Exponentiation of measures

Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.

\mu(A) = \Pr(D \in A).\,

Let δ₀ be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure

\exp(\lambda t(\mu - \delta_0))\,

where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by

\exp(\nu) = \sum_{n=0}^\infty {\nu^{*n} \over n!}

and

\nu^{*n} = \underbrace{\nu * \cdots *\nu}_{n \text{ factors}}

is a convolution of measures, and the series converges weakly.

Stochastic processes

Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding

Continuous time	Bessel process Birth–death process Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Point process Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage

Both	Branching process Galves–Löcherbach model Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise

Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph

Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model

Financial models	Black–Derman–Toy Black–Karasinski Black–Scholes Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie

Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson

Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c

Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible

Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem

Inequalities	Burkholder–Davis–Gundy Doob's martingale Kunita–Watanabe

Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract

Disciplines	Actuarial mathematics Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Statistics Stochastic analysis Time series analysis Machine learning

List of topics Category

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Compound Poisson process

Properties of the compound Poisson process

Exponentiation of measures

See also