Compound Poisson process
A compound Poisson process is a continuous-time 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. To be precise, a compound Poisson process, parameterised by a rate and jump size distribution G, is a process given by
where, is the counting variable of a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which are also independent of
When are non-negative integer-valued random variables, then this compound Poisson process is known as a stuttering Poisson process.
Properties of the compound Poisson process
The expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:
Making similar use of the law of total variance, the variance can be calculated as:
Lastly, using the law of total probability, the moment generating function can be given as follows:
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.
Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure
where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by
and
is a convolution of measures, and the series converges weakly.
See also
- Poisson process
- Poisson distribution
- Compound Poisson distribution
- Non-homogeneous Poisson process
- Campbell's formula for the moment generating function of a compound Poisson process
- v
- t
- e
- Additive process
- Bessel process
- Birth–death process
- Brownian motion
- Cauchy process
- Contact process
- Continuous-time random walk
- Cox process
- Diffusion process
- Dyson Brownian motion
- Empirical process
- Feller process
- Fleming–Viot process
- Gamma process
- Geometric process
- Hawkes 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
- Schramm–Loewner evolution
- Semimartingale
- Sigma-martingale
- Stable process
- Superprocess
- Telegraph process
- Variance gamma process
- Wiener process
- Wiener sausage
- Binomial options pricing model
- Black–Derman–Toy
- Black–Karasinski
- Black–Scholes
- Chan–Karolyi–Longstaff–Sanders (CKLS)
- Chen
- Constant elasticity of variance (CEV)
- Cox–Ingersoll–Ross (CIR)
- Garman–Kohlhagen
- Heath–Jarrow–Morton (HJM)
- Heston
- Ho–Lee
- Hull–White
- Korn-Kreer-Lenssen
- LIBOR market
- Rendleman–Bartter
- SABR volatility
- Vašíček
- Wilkie
- 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
- Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
- 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
- Karhunen–Loève theorem
- 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
- Stopping time
- Stratonovich integral
- Uniform integrability
- Usual hypotheses
- Wiener space
- Actuarial mathematics
- Control theory
- Econometrics
- Ergodic theory
- Extreme value theory (EVT)
- Large deviations theory
- Mathematical finance
- Mathematical statistics
- Probability theory
- Queueing theory
- Renewal theory
- Ruin theory
- Signal processing
- Statistics
- Stochastic analysis
- Time series analysis
- Machine learning
- List of topics
- Category