
284 Modeling and Inverse Problems in the Presence of Uncertainty
being some positive constants. In addition, α, γ a nd ̺ = (̺
0
, ̺
1
, . . . , ̺
m−1
)
T
are non-random functions of t, and Z ·̺(t) denotes the dot product of Z and
̺(t) (that is, Z ·̺(t) = Z
T
̺(t) =
m−1
X
j=0
Z
j
̺
j
(t)). The corresp onding stochastic
differential equation takes the for m
dX(t) = [α(t)X(t) + ξ(t)]dt + η(t)dW (t), X(0) = X
0
, (7.140)
where ξ and η are some non-random functions of t, and {W (t) : t ≥ 0} is a
Wiener process.
The conditions for the pointwise equivalence between the RDE (7.139) and
the SDE (7.140) are stated in the following theorem.
Theorem 7.4.6 If functions ξ, η and γ,