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{{MSC|60G51}}
 
{{MSC|60G51}}
  
 
[[Category:Stochastic processes]]
 
[[Category:Stochastic processes]]
  
A [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s0902401.png" /> such that for any natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s0902402.png" /> and all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s0902403.png" />, the increments
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A [[Stochastic process|stochastic process]] $  X ( t) $
 +
such that for any natural number $  n $
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and all real $  0 \leq  \alpha _ {1} < \beta _ {1} \leq  \alpha _ {2} < \beta _ {2} \leq  \dots \leq  \alpha _ {n} < \beta _ {n} $,  
 +
the increments
  
<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/s/s090/s090240/s0902404.png" /></td> </tr></table>
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$$
 +
X ( \beta _ {1} ) - X ( \alpha _ {1} )
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\dots X ( \beta _ {n} ) - X ( \alpha _ {n} )
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$$
  
are mutually-independent random variables. A stochastic process with independent increments is called homogeneous if the probability distribution of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s0902405.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s0902406.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s0902407.png" />, depends only on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s0902408.png" /> and not on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s0902409.png" />. Since the result of adding any non-random function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024011.png" /> is again a stochastic process with independent increments, the realizations of such processes can be arbitrarily irregular. However, by suitably "centering" the process (say by subtracting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024012.png" /> the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024013.png" /> defined by the relation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024014.png" />), one can make more definite judgements about the structure of the "centred" process. There are at most countably-many (non-random) points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024015.png" /> at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024016.png" /> has random jumps
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are mutually-independent random variables. A stochastic process with independent increments is called homogeneous if the probability distribution of $  X ( \alpha + h ) - X ( \alpha ) $,
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0 \leq  \alpha $,  
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0 < h $,  
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depends only on $  h $
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and not on $  \alpha $.  
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Since the result of adding any non-random function $  A ( t) $
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to $  X ( t) $
 +
is again a stochastic process with independent increments, the realizations of such processes can be arbitrarily irregular. However, by suitably "centering" the process (say by subtracting from $  X ( t) $
 +
the function $  f ( t) $
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defined by the relation $  {\mathsf E} \textrm{ arctan  } ( X ( t) - f ( t) ) = 0 $),  
 +
one can make more definite judgements about the structure of the "centred" process. There are at most countably-many (non-random) points $  t _ {j} $
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at which $  X ( t) $
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has random jumps
  
<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/s/s090/s090240/s09024017.png" /></td> </tr></table>
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$$
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X _ {j}  = X ( t _ {j} + 0 ) - X ( t _ {j} - 0 ) ,
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$$
  
 
and the difference
 
and the difference
  
<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/s/s090/s090240/s09024018.png" /></td> </tr></table>
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$$
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Y ( t)  = X ( t) - \sum _ {t _ {j} < t } X _ {j}  $$
  
is a stochastically-continuous stochastic process with independent increments: for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024020.png" />,
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is a stochastically-continuous stochastic process with independent increments: for any $  \epsilon > 0 $
 +
and $  t  ^  \prime  \rightarrow t $,
  
<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/s/s090/s090240/s09024021.png" /></td> </tr></table>
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$$
 +
{\mathsf P} \{ | Y ( t  ^  \prime  ) - Y ( t) | > \epsilon \}  \rightarrow  0 .
 +
$$
  
A [[Wiener process|Wiener process]] and a [[Poisson process|Poisson process]] are examples of stochastically-continuous stochastic processes with independent increments (and realizations of the first are continuous with probability one, while realizations of the second are step functions with jumps equal to one). An important example of a stochastic process with independent increments is that of a stable process (cf. [[Stable distribution|Stable distribution]]). Realizations of a stochastically-continuous stochastic process with independent increments can only have discontinuities of the first kind, with probability one. The distribution of the values of such a process for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090240/s09024022.png" /> is infinitely divisible (see [[Infinitely-divisible distribution|Infinitely-divisible distribution]]). In studying stochastic processes with independent increments one can apply the method of characteristic functions (cf. [[Characteristic function|Characteristic function]]). Problems on the probability of a process crossing a boundary and on the probability distribution of the first crossing time are solved using the so-called [[Factorization identities|factorization identities]].
+
A [[Wiener process|Wiener process]] and a [[Poisson process|Poisson process]] are examples of stochastically-continuous stochastic processes with independent increments (and realizations of the first are continuous with probability one, while realizations of the second are step functions with jumps equal to one). An important example of a stochastic process with independent increments is that of a stable process (cf. [[Stable distribution|Stable distribution]]). Realizations of a stochastically-continuous stochastic process with independent increments can only have discontinuities of the first kind, with probability one. The distribution of the values of such a process for any $  t $
 +
is infinitely divisible (see [[Infinitely-divisible distribution|Infinitely-divisible distribution]]). In studying stochastic processes with independent increments one can apply the method of characteristic functions (cf. [[Characteristic function|Characteristic function]]). Problems on the probability of a process crossing a boundary and on the probability distribution of the first crossing time are solved using the so-called [[Factorization identities|factorization identities]].
  
 
====References====
 
====References====

Latest revision as of 08:23, 6 June 2020


2020 Mathematics Subject Classification: Primary: 60G51 [MSN][ZBL]

A stochastic process $ X ( t) $ such that for any natural number $ n $ and all real $ 0 \leq \alpha _ {1} < \beta _ {1} \leq \alpha _ {2} < \beta _ {2} \leq \dots \leq \alpha _ {n} < \beta _ {n} $, the increments

$$ X ( \beta _ {1} ) - X ( \alpha _ {1} ) \dots X ( \beta _ {n} ) - X ( \alpha _ {n} ) $$

are mutually-independent random variables. A stochastic process with independent increments is called homogeneous if the probability distribution of $ X ( \alpha + h ) - X ( \alpha ) $, $ 0 \leq \alpha $, $ 0 < h $, depends only on $ h $ and not on $ \alpha $. Since the result of adding any non-random function $ A ( t) $ to $ X ( t) $ is again a stochastic process with independent increments, the realizations of such processes can be arbitrarily irregular. However, by suitably "centering" the process (say by subtracting from $ X ( t) $ the function $ f ( t) $ defined by the relation $ {\mathsf E} \textrm{ arctan } ( X ( t) - f ( t) ) = 0 $), one can make more definite judgements about the structure of the "centred" process. There are at most countably-many (non-random) points $ t _ {j} $ at which $ X ( t) $ has random jumps

$$ X _ {j} = X ( t _ {j} + 0 ) - X ( t _ {j} - 0 ) , $$

and the difference

$$ Y ( t) = X ( t) - \sum _ {t _ {j} < t } X _ {j} $$

is a stochastically-continuous stochastic process with independent increments: for any $ \epsilon > 0 $ and $ t ^ \prime \rightarrow t $,

$$ {\mathsf P} \{ | Y ( t ^ \prime ) - Y ( t) | > \epsilon \} \rightarrow 0 . $$

A Wiener process and a Poisson process are examples of stochastically-continuous stochastic processes with independent increments (and realizations of the first are continuous with probability one, while realizations of the second are step functions with jumps equal to one). An important example of a stochastic process with independent increments is that of a stable process (cf. Stable distribution). Realizations of a stochastically-continuous stochastic process with independent increments can only have discontinuities of the first kind, with probability one. The distribution of the values of such a process for any $ t $ is infinitely divisible (see Infinitely-divisible distribution). In studying stochastic processes with independent increments one can apply the method of characteristic functions (cf. Characteristic function). Problems on the probability of a process crossing a boundary and on the probability distribution of the first crossing time are solved using the so-called factorization identities.

References

[GS] I.I. Gihman, A.V. Skorohod, "The theory of stochastic processes" , 2 , Springer (1975) (Translated from Russian) MR0375463 Zbl 0305.60027
[S] A.V. Skorohod, "Random processes with independent increments" , Kluwer (1991) (Translated from Russian) MR1155400

Comments

For additional references see Stochastic process.

How to Cite This Entry:
Stochastic process with independent increments. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_process_with_independent_increments&oldid=26958
This article was adapted from an original article by Yu.V. Prokhorov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article