Kolmogorov Backward equation and the

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Kolmogorov Backward equation and the
Fokker-Planck equation
Remarkable feature of diffusion processes their
transition probability density, though
non-Gaussian, is already completely determined
by its first two differential moments.
PROOF derive an evolution equation for p(y, t
Ix,s), solely on the basis of conditions (a -
c).
Transform Chapman-Kolmogorov into a linear PDE.
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Auxiliary Result Let v be continuous , bounded
s.t.
4.35
has continuous derivatives
and
Then
4.36
With boundary conditions
4.37
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PROOF
Consider s1 and s2 with
According to the definition of u(x,s) (4.35)
In the first term used C-K (4.8) and in second ,
multiplied by 1.
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Decompose state space in 2 parts J1 and J2 and
bound them separately
Since v is bounded, u(x,s) will be too
And by a), Eq (4.16)
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For the first ter, use Taylor
The error term is bounded by
where
One uses now b) and c) (4.19-20) of the
diffusion process definition
One obtains (4.36) by
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Recalling
One can write it as
Since v is arbitrary, this means
This is the Kolmogorov Backwards Equation (KBE)
Let us now deduce the forward equation
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PROVE FIRST
(Using an s-independent function in 4.38)
PROOF OF FORWARD EQUATION
If 4.16,19,20 if the derivatives exist and are
continuous, then
4.45
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Let v(x) be an arbitrary twice continuously
differentiable function which vanishes
identically outside a certain bounded interval.
Then we have
This can be also obtained from C-K for the first
term
And renaming y ?gt z
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The boundary terms vanish since v(y) is zero
outside a bounded interval. Taking into account
that v(y) is arbitrary, the forward equation
(4.45) follows. This evolution equation for p(y,
t Ix,s) is linear in the transition
probability density, in contrast to the
Chapman-Kolmogorov equation from which it
follows. It is known in the mathematical
literature as the Kolmogorov forward equation
since the varialion-is_with respect to the
"future" state y and time argument t. In the
physical literature this equation is called the
Fokker-Planck equation (FPE). The KBE and FPE
show that a Markov diffusion process is indeed
already completely defined by its first two
differential moments the KBE and FPE are partial
differential equations for the transition
probability densities, whose coefficients are
given by the driftf and the diffusion g2. Hence
the transition probability density, as a solution
of the KBE or the FPE, is completely and uniquely
defined by its first two differential Moments..
It is this surprising property that makes the
concept of diffusion processes so powerful.
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Multiplying the FPE (4.45)
on both sides by the one-dimensional probability
density p(x,s) and then integrating over x, we
obtain that the one-dimensional probability
density p(y, t) of a diffusion process also obeys
an equation of type (4.45)
4.46
The inverse is of course not true. If the
one-dimensional probability density p(y, t) of
some random process obeys an equation of the form
(4.46), then the process is not necessarily
Markovian 4.4. If the diffusion process is
time homogeneous, i. e., p(y, t Ix,s) p(y, t -s
I x,O) p(y, r Ix), then as remarked above, the
drift and diffusion are time independent.
Furthermore,
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For time-homogeneous Markov processes, KBE and
FPE read (writing t for f)
Let us now write down the FPE and KBE for the two
examples of diffusion processes we have
encountered so far. Brownian motion Vb WI is a
diffusion process and one might say trivially,
since diffusion processes were defined as
processes that look locally in time like a Wiener
process. For this simple example FPE and KBE
coincide and they are given by the classical
diffusion equation
The solution to this equation is of course given
by (2.99). For the O-V process, the two equations
read, using (4.31,32)
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PAWULA THEOREM
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