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Stochastic process, differentiable

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A stochastic process such that the limit

exists; it is called the derivative of the stochastic process . One distinguishes between differentiation with probability 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

Namely, exists if and only if the limit

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

A sufficient condition for the existence of a process equivalent to a given one with continuously differentiable trajectories is that its mean square-derivative is continuous and has 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=48857
This article was adapted from an original article by Yu.A. Rozanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article