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A [[Stochastic process|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====
<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====
For additional references see [[Stochastic process|Stochastic process]].