Additive stochastic process

A real-valued stochastic process $X=\{X(t):t\in {\mathbf{R}}_{+}\}$ such that for each integer $n\ge 1$ and $0\le t_0<\cdots <t_n$ 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 $X(t)-X(s)$, $0\le s<t$. $X$ is called a homogeneous additive stochastic process if, in addition, the distributions of $X(t)-X(s)$, $0\le s<t$, depend only on $t-s$. Each additive stochastic process $X$ can be decomposed as a sum (see [a1])

$$\begin{array}{}\text{(a1)}& X(t)=f(t)+X_1(t)+X_2(t),t\ge 0,\end{array}$$

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 $ϵ>0$, $\mathsf{P}\{|X_1(t)-X_1(s)|>ϵ\}\to 0$ as $t\to s$, and $X_2$ is purely discontinuous, i.e., there exist a sequence $\{t_k:k\ge 1\}\subset {\mathbf{R}}_{+}$ and independent sequences $\{X_k^{+}:k\ge 1\}$, $\{X_k^{-}:k\ge 1\}$ of independent random variables such that

$$\begin{array}{}\text{(a2)}& X_2(t)=\sum _{t_k\le t}{X}_k^{-}+\sum _{t_k<t}{X}_k^{+},t\ge 0,\end{array}$$

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 $\mathsf{P}\{X_k^{+}=0\}=1$, $k\ge 1$, 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].

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