
Propagation of Uncertainty in a Continuous Time Dynamical System 219
Definition 7.1.10 A second-order stochastic process {X(t) : t ∈ T} has a
mean square derivative (or m.s. derivative)
˙
X(t) at a fixed t if
lim
s→0
E
˙
X(t) −
X(t + s) − X(t)
s
2
!
= 0.
Theorem 7.1.2 A second-order stochastic process {X(t) : t ∈ T} is m.s. dif-
ferentiable at t if and only if
∂
2
R
∂s∂˜s
exists and is fin ite at (t, t).
A second-order stochastic process is said to be m.s. differentiable in T if it
is m.s. differ entiable at any t ∈ T. By the above theorem, we see that for a
wide-sense stationary stochastic pro cess, it is m.s. differentiable if and only if
the first- and second-order