Stochastic process, differentiable

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.

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Comments[edit]

For additional references see Stochastic process.