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Using Langevin's equation one has
dx
D ˛.x;t/
dt
C
w
.t/
(7.14)
Assume that function y D f.x/where f is twice differentiable. It holds that
1
2
f
00
.x/
dx
2
dy
D
f.xC
dx
/ f.x/D f
0
.x/
dx
C
C D
1
f
0
.x/Œ˛.x;t/
dt
C b.x;t/
dw
.t/ C
2
f
00
.x/b
2
.x;t/.
dw
.t//
2
C D
Œf
0
.x/˛.x;t/ C
1
2
f
00
.x/b
2
.x;t/
dt
C f
0
.x/b.x;t/
dw
.t/ C
(7.15)
The relation
f
0
.x/˛.x;t/ C
2
f
00
.x/b
2
.x;t/
dt
Cf
0
.x/b.x;t/
dw
.t/
1
df Œx.t/ D
(7.16)
is Itô's formula and can be generalized to the multi-dimensional case.
7.3
Fokker-Planck's Partial Differential Equation
7.3.1
The Fokker-Planck Equation
Fokker-Planck's equation stands for a basic tool for the solution of stochastic
differential equation. Instead of attempting an analytical solution of the
stochastic differential equation one can attempt to find the equivalent solution
of Fokker-Planck equation which stands for a partial differential equation.
It is assumed that .x;t/ denotes the probability density function the stochastic
variable X to take the value x at time instant t. One can obtain the Fokker-Planck
equation after a series of transformations applied to Langevin's equation, thus con-
firming that the two relations are equivalent. Thus one starts from a Langevin-type
equation
(7.17)
dx
D f.x;t/
dt
C g.x;t/
dt
Next the transformation y D h.x/ is introduced, where h is an arbitrary function
that is twice differentiable. It holds that
dh
.x/ D h
0
.x/f.x;t/
dt
C h
00
g
2
.x;t/=2
dt
C h
0
.x/g
dw
(7.18)
By computing mean values one gets
d
dt
Efh.x/gDEfh
0
.x/f.x;t/gCEfh
00
.x/g
2
.x;t/=2g
(7.19)
