
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