# Additive stochastic process

A real-valued stochastic process such that for each integer and the random variables are independent. Finite-dimensional distributions of the additive stochastic process are defined by the distributions of and the increments , . is called a homogeneous additive stochastic process if, in addition, the distributions of , , depend only on . Each additive stochastic process can be decomposed as a sum (see [a1])

(a1) |

where is a non-random function, and are independent additive stochastic processes, is stochastically continuous, i.e., for each and , as , and is purely discontinuous, i.e., there exist a sequence and independent sequences , of independent random variables such that

(a2) |

and the above sums for each converge independently of the order of summands.

A stochastically continuous additive process has a modification that is right continuous with left limits, and the distributions of the increments , , are infinitely divisible (cf. Infinitely-divisible distribution). They are called Lévy processes. For example, the Brownian motion with drift coefficient and diffusion coefficient is an additive process ; for it , , has a normal distribution (Gaussian distribution) with mean value and variation , .

The Poisson process with parameter is an additive process ; for it, , , has the Poisson distribution with parameter and . A Lévy process is stable (cf. Stable distribution) if and if for each the distribution of equals the distribution of for some non-random functions and .

If, in (a1), (a2), is a right-continuous function of bounded variation for each finite time interval and , , then the additive process is a semi-martingale (cf. also Martingale). A semi-martingale is an additive process if and only if the triplet of predictable characteristics of is non-random (see [a2]).

The method of characteristic functions (cf. Characteristic function) and the factorization identities are main tools for the investigation of properties of additive stochastic processes (see [a3]). The theory of additive stochastic processes can be extended to stochastic processes with values in a topological group. A general reference for this area is [a1].

#### References

[a1] | A.V. Skorokhod, "Random processes with independent increments" , Kluwer Acad. Publ. (1991) (In Russian) |

[a2] | B. Grigelionis, "Martingale characterization of stochastic processes with independent increments" Lietuvos Mat. Rinkinys , 17 (1977) pp. 75–86 (In Russian) |

[a3] | N.S. Bratijchuk, D.V. Gusak, "Boundary problems for processes with independent increments" , Naukova Dumka (1990) (In Russian) |

**How to Cite This Entry:**

Additive stochastic process. B. Grigelionis (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Additive_stochastic_process&oldid=15176