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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
314 Modeling and Inverse Problems in the Presence of Uncertainty
then by (8.11 ) and (8.12) we have
Prob{X(t) = x}
= Prob{X(0) = x} +
Z
t
0
l
X
j=1
λ
j
(x v
j
)Prob{X(s) = x v
j
}ds
Z
t
0
l
X
j=1
λ
j
(x)Prob{X(s) = x}ds.
Let Φ(t, x) = Prob{X(t) = x}. Then we have
Φ(t, x) = Φ(0, x) +
Z
t
0
l
X
j=1
λ
j
(x v
j
)Φ(s, x v
j
)ds
Z
t
0
l
X
j=1
λ
j
(x)Φ(s, x)ds.
Differentiating the above equation with respect to t yields that
d
dt
Φ(t, x) =
l
X
j=1
λ
j
(x v
j
)Φ(t, x v
j
)
l
X
j=1
λ
j
(x)Φ(t, x), (8.13)
which is often called Kolmogorov’s forward equation o r the chemical master
equation.
Ideally one would like to dir ectly solve Kolmogorov’s forward equation
(8.13) with given initial conditions to obtain
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Publisher Resources

ISBN: 9781482206432