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A real-valued [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103701.png" /> such that for each integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103702.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103703.png" /> the random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103704.png" /> are independent. Finite-dimensional distributions of the additive stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103705.png" /> are defined by the distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103706.png" /> and the increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103708.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a1103709.png" /> is called a homogeneous additive stochastic process if, in addition, the distributions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037011.png" />, depend only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037012.png" />. Each additive stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037013.png" /> can be decomposed as a sum (see [[#References|[a1]]])
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037015.png" /> is a non-random function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037017.png" /> are independent additive stochastic processes, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037018.png" /> is stochastically continuous, i.e., for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037021.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037023.png" /> is purely discontinuous, i.e., there exist a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037024.png" /> and independent sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037026.png" /> of independent random variables such that
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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]]])
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037027.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a1 }
 +
X ( t ) = f ( t ) + X _ {1} ( t ) + X _ {2} ( t ) ,  t \geq  0,
 +
$$
  
and the above sums for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037028.png" /> converge independently of the order of summands.
+
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
  
A stochastically continuous additive process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037029.png" /> has a modification that is right continuous with left limits, and the distributions of the increments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037031.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037032.png" /> and diffusion coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037033.png" /> is an additive process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037034.png" />; for it <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037036.png" />, has a [[Normal distribution|normal distribution]] (Gaussian distribution) with mean value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037037.png" /> and variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037039.png" />.
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$$ \tag{a2 }
 +
X _ {2} ( t ) = \sum _ {t _ {k} \leq  t } X _ {k}  ^ {-} + \sum _ {t _ {k} < t } X _ {k}  ^ {+} ,   t \geq  0,
 +
$$
  
The [[Poisson process|Poisson process]] with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037040.png" /> is an additive process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037041.png" />; for it, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037043.png" />, has the [[Poisson distribution|Poisson distribution]] with parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037045.png" />. A Lévy process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037046.png" /> is stable (cf. [[Stable distribution|Stable distribution]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037047.png" /> and if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037048.png" /> the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037049.png" /> equals the distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037050.png" /> for some non-random functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037052.png" />.
+
and the above sums for each $  t > 0 $
 +
converge independently of the order of summands.
  
If, in (a1), (a2), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037053.png" /> is a right-continuous function of bounded variation for each finite time interval and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037055.png" />, then the additive process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037056.png" /> is a semi-martingale (cf. also [[Martingale|Martingale]]). A semi-martingale <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037057.png" /> is an additive process if and only if the triplet of predictable characteristics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110370/a11037058.png" /> is non-random (see [[#References|[a2]]]).
+
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]]].
 
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]]].

Latest revision as of 16:09, 1 April 2020


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)
How to Cite This Entry:
Additive stochastic process. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_stochastic_process&oldid=45030
This article was adapted from an original article by B. Grigelionis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article