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A [[Stochastic process|stochastic process]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090210/s0902101.png" /> such that the limit
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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/s/s090/s090210/s0902102.png" /></td> </tr></table>
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exists; it is called the derivative of the stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090210/s0902103.png" />. One distinguishes between differentiation with probability <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090210/s0902105.png" /> and mean-square differentiation, according to how this limit is interpreted. The condition of mean-square differentiability can be naturally expressed in terms of the correlation function
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A [[Stochastic process|stochastic process]]  $  X ( t) $
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such that the limit
  
<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/s090210/s0902106.png" /></td> </tr></table>
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$$
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\lim\limits _ {\Delta t \rightarrow 0 } \
  
Namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090210/s0902107.png" /> exists if and only if the limit
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\frac{X ( t + \Delta t ) - X ( t) }{\Delta t }
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  = \
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X  ^  \prime  ( t)
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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/s/s090/s090210/s0902108.png" /></td> </tr></table>
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exists; it is called the derivative of the stochastic process  $  X ( t) $.
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One distinguishes between differentiation with probability  $  1 $
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and mean-square differentiation, according to how this limit is interpreted. The condition of mean-square differentiability can be naturally expressed in terms of the correlation function
  
<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/s090210/s0902109.png" /></td> </tr></table>
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$$
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B ( t _ {1} , t _ {2} )  = \
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{\mathsf E} X ( t _ {1} ) X ( t _ {2} ) .
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$$
  
exists. A stochastic process having a mean-square derivative is absolutely continuous. More precisely, for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090210/s09021010.png" /> and with probability 1,
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Namely,  $  X  ^  \prime  ( t) $
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exists if and only if the limit
  
<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/s090210/s09021011.png" /></td> </tr></table>
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$$
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B  ^ {\prime\prime} ( t _ {1} , t _ {2} ) =
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$$
  
A sufficient condition for the existence of a process equivalent to a given one with continuously differentiable trajectories is that its mean square-derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090210/s09021012.png" /> is continuous and has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090210/s09021013.png" /> as its correlation function. For Gaussian processes this condition is also necessary.
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$$
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= \
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\lim\limits _ {\begin{array}{c}
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\Delta t _ {1} \rightarrow 0 \\
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\Delta t _ {2} \rightarrow 0
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\end{array}
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}
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\frac{B ( t _ {1} + \Delta t _ {1} , t _ {2} + \Delta t _ {2} ) - B ( t _ {1} + \Delta t _ {1} , t _ {2} ) - B (
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t _ {1} , t _ {2} + \Delta t _ {2} ) + B ( t _ {1} , t _ {2} ) }{\Delta t _ {1} \Delta t _ {2} }
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$$
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exists. A stochastic process having a mean-square derivative is absolutely continuous. More precisely, for every  $  t $
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and with probability 1,
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$$
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X ( t)  =  X ( t _ {0} ) +
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\int\limits _ {t _ {0} } ^ { {t }  }
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X  ^  \prime  ( s)  d s ,\  t \geq  t _ {0} .
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$$
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A sufficient condition for the existence of a process equivalent to a given one with continuously differentiable trajectories is that its mean square-derivative $  X  ^  \prime  ( t) $
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is continuous and has $  B  ^ {\prime\prime} ( t _ {1} , t _ {2} ) $
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as its correlation function. For Gaussian processes this condition is also necessary.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Gikhman,  A.V. Skorokhod,  "Introduction to the theory of stochastic processes" , Saunders  (1967)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Gikhman,  A.V. Skorokhod,  "Introduction to the theory of stochastic processes" , Saunders  (1967)  (Translated from Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For additional references see [[Stochastic process|Stochastic process]].
 
For additional references see [[Stochastic process|Stochastic process]].

Latest revision as of 08:23, 6 June 2020


A stochastic process $ X ( t) $ such that the limit

$$ \lim\limits _ {\Delta t \rightarrow 0 } \ \frac{X ( t + \Delta t ) - X ( t) }{\Delta t } = \ X ^ \prime ( t) $$

exists; it is called the derivative of the stochastic process $ X ( t) $. One distinguishes between differentiation with probability $ 1 $ and mean-square differentiation, according to how this limit is interpreted. The condition of mean-square differentiability can be naturally expressed in terms of the correlation function

$$ B ( t _ {1} , t _ {2} ) = \ {\mathsf E} X ( t _ {1} ) X ( t _ {2} ) . $$

Namely, $ X ^ \prime ( t) $ exists if and only if the limit

$$ B ^ {\prime\prime} ( t _ {1} , t _ {2} ) = $$

$$ = \ \lim\limits _ {\begin{array}{c} \Delta t _ {1} \rightarrow 0 \\ \Delta t _ {2} \rightarrow 0 \end{array} } \frac{B ( t _ {1} + \Delta t _ {1} , t _ {2} + \Delta t _ {2} ) - B ( t _ {1} + \Delta t _ {1} , t _ {2} ) - B ( t _ {1} , t _ {2} + \Delta t _ {2} ) + B ( t _ {1} , t _ {2} ) }{\Delta t _ {1} \Delta t _ {2} } $$

exists. A stochastic process having a mean-square derivative is absolutely continuous. More precisely, for every $ t $ and with probability 1,

$$ X ( t) = X ( t _ {0} ) + \int\limits _ {t _ {0} } ^ { {t } } X ^ \prime ( s) d s ,\ t \geq t _ {0} . $$

A sufficient condition for the existence of a process equivalent to a given one with continuously differentiable trajectories is that its mean square-derivative $ X ^ \prime ( t) $ is continuous and has $ B ^ {\prime\prime} ( t _ {1} , t _ {2} ) $ as its correlation function. For Gaussian processes this condition is also necessary.

References

[1] I.I. Gikhman, A.V. Skorokhod, "Introduction to the theory of stochastic processes" , Saunders (1967) (Translated from Russian)

Comments

For additional references see Stochastic process.

How to Cite This Entry:
Stochastic process, differentiable. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Stochastic_process,_differentiable&oldid=18818
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article