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Modeling and Inverse Problems in the Presence of Uncertainty
book

Modeling and Inverse Problems in the Presence of Uncertainty

by H. T. Banks, Shuhua Hu, W. Clayton Thompson
April 2014
Intermediate to advanced content levelIntermediate to advanced
405 pages
13h
English
Chapman and Hall/CRC
Content preview from Modeling and Inverse Problems in the Presence of Uncertainty
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)
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Publisher Resources

ISBN: 9781482206432