Stochastic process, differentiable

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

$lim_{Δ t \to 0} \frac{X (t + Δ t) - X (t)}{Δ t} = X^{'} (t)$

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

$B (t_{1}, t_{2}) = E X (t_{1}) X (t_{2}) .$

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

$B^{''} (t_{1}, t_{2}) =$

$= lim_{\begin{matrix} Δ t_{1} \to 0 \\ Δ t_{2} \to 0 \end{matrix}} \frac{B (t_{1} + Δ t_{1}, t_{2} + Δ t_{2}) - B (t_{1} + Δ t_{1}, t_{2}) - B (t_{1}, t_{2} + Δ t_{2}) + B (t_{1}, t_{2})}{Δ t_{1} Δ t_{2}}$

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

$X (t) = X (t_{0}) + \int_{t_{0}}^{t} X^{'} (s) d s, t \geq t_{0} .$

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[edit]

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

Comments[edit]

For additional references see Stochastic process.