Additive stochastic process

A real-valued stochastic process $X = \{ {X ( t ) } : {t \in \mathbf R _ {+} } \}$ such that for each integer $n \geq 1$ and $0 \leq t _ {0} < \dots < t _ {n}$ the random variables $X ( t _ {0} ) ,X ( t _ {1} ) - X ( t _ {0} ) \dots X ( t _ {n} ) - X ( t _ {n - 1 } )$ are independent. Finite-dimensional distributions of the additive stochastic process $X$ are defined by the distributions of $X ( 0 )$ and the increments $X ( t ) - X ( s )$, $0 \leq s < t$. $X$ is called a homogeneous additive stochastic process if, in addition, the distributions of $X ( t ) - X ( s )$, $0 \leq s < t$, depend only on $t - s$. Each additive stochastic process $X$ can be decomposed as a sum (see [a1])

$$\tag{a1 } X ( t ) = f ( t ) + X _ {1} ( t ) + X _ {2} ( t ) , t \geq 0,$$

where $f$ is a non-random function, $X _ {1}$ and $X _ {2}$ are independent additive stochastic processes, $X _ {1}$ is stochastically continuous, i.e., for each $s \in \mathbf R _ {+}$ and $\epsilon > 0$, ${\mathsf P} \{ | {X _ {1} ( t ) - X _ {1} ( s ) } | > \epsilon \} \rightarrow 0$ as $t \rightarrow s$, and $X _ {2}$ is purely discontinuous, i.e., there exist a sequence $\{ {t _ {k} } : {k \geq 1 } \} \subset \mathbf R _ {+}$ and independent sequences $\{ {X _ {k} ^ {+} } : {k \geq 1 } \}$, $\{ {X _ {k} ^ {-} } : {k \geq 1 } \}$ of independent random variables such that

$$\tag{a2 } X _ {2} ( t ) = \sum _ {t _ {k} \leq t } X _ {k} ^ {-} + \sum _ {t _ {k} < t } X _ {k} ^ {+} , t \geq 0,$$

and the above sums for each $t > 0$ converge independently of the order of summands.

A stochastically continuous additive process $X$ has a modification that is right continuous with left limits, and the distributions of the increments $X ( t ) - X ( s )$, $s < t$, are infinitely divisible (cf. Infinitely-divisible distribution). They are called Lévy processes. For example, the Brownian motion with drift coefficient $b$ and diffusion coefficient $\sigma ^ {2}$ is an additive process $X$; for it $X ( t ) - X ( s )$, $s < t$, has a normal distribution (Gaussian distribution) with mean value $b ( t - s )$ and variation $\sigma ^ {2} ( t - s )$, $X ( 0 ) = 0$.

The Poisson process with parameter $\lambda$ is an additive process $X$; for it, $X ( t ) - X ( s )$, $s < t$, has the Poisson distribution with parameter $\lambda ( t - s )$ and $X ( 0 ) = 0$. A Lévy process $X$ is stable (cf. Stable distribution) if $X ( 0 ) = 0$ and if for each $s < t$ the distribution of $X ( t ) - X ( s )$ equals the distribution of $c ( t - s ) X ( 1 ) + d ( t - s )$ for some non-random functions $c$ and $d$.

If, in (a1), (a2), $f$ is a right-continuous function of bounded variation for each finite time interval and ${\mathsf P} \{ X _ {k} ^ {+} =0 \} = 1$, $k \geq 1$, then the additive process $X$ is a semi-martingale (cf. also Martingale). A semi-martingale $X$ is an additive process if and only if the triplet of predictable characteristics of $X$ 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)