View source for Additive stochastic process

<!--
a1103701.png
$#A+1 = 58 n = 0
$#C+1 = 58 : ~/encyclopedia/old_files/data/A110/A.1100370 Additive stochastic process
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

A real-valued [[Stochastic process|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 [[#References|[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|Infinitely-divisible distribution]]). They are called Lévy processes. For example, the [[Brownian motion|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|normal distribution]] (Gaussian distribution) with mean value  $  b ( t - s ) $
and variation  $  \sigma  ^ {2} ( t - s ) $, 
$  X ( 0 ) = 0 $.

The [[Poisson process|Poisson process]] with parameter  $  \lambda $
is an additive process  $  X $; 
for it,  $  X ( t ) - X ( s ) $, 
$  s < t $, 
has the [[Poisson distribution|Poisson distribution]] with parameter  $  \lambda ( t - s ) $
and  $  X ( 0 ) = 0 $. 
A Lévy process  $  X $
is stable (cf. [[Stable distribution|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|Martingale]]). A semi-martingale  $  X $
is an additive process if and only if the triplet of predictable characteristics of  $  X $
is non-random (see [[#References|[a2]]]).

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  A.V. Skorokhod,   "Random processes with independent increments" , Kluwer Acad. Publ.  (1991)  (In Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  B. Grigelionis,   "Martingale characterization of stochastic processes with independent increments"  ''Lietuvos Mat. Rinkinys'' , '''17'''  (1977)  pp. 75–86  (In Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  N.S. Bratijchuk,   D.V. Gusak,   "Boundary problems for processes with independent increments" , Naukova Dumka  (1990)  (In Russian)</TD></TR></table>