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].

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