
Propagation of Uncertainty in a Continuous Time Dynamical System 289
Theorem 7.4.10 Consider the stochastic differen tial equation
dX(t) = g(t, X(t))dt + σ(t, X(t))dW (t), (7.153)
where g and σ are non-random fun ctions of t and x. If the equality
∂
∂x
σ(t, x)
1
σ
2
(t, x)
∂σ
∂t
(t, x) −
∂
∂x
g
σ
(t, x) +
1
2
∂
2
σ
∂x
2
(t, x)
= 0
holds, then (7.153) can be reduced to the linear stochastic differential equation
dY (t) = ¯g(t)dt + ¯σ(t)dW (t).
Here ¯σ is some non-random fun ction determined from
d
dt
¯σ(t) = ¯σ(t)σ(t, x)
1
σ
2
(t, x)
∂σ
∂t
(t, x) −
∂
∂x
g
σ
(t, x) +
1
2
∂
2
σ
∂x
2
(t, x)
,
and ¯g is some deterministic function given by
¯g(t) =
∂h
∂t
(t, x) +
∂h
∂x
(t, x)g(t, x) +
1
2
∂
2
h
∂x
2
(t, x)