We will derive first the Kolmogorov backward equation then the Kolmogorov forward equatin, which are fundamental to the PDE analysis of stochastic processes.

First we derive backward Kolmogorov equation: Start with Chapman-Kolmogorov equation using times

We'll use the Chapman-Kolmogorov equation in terms of probability transition density which we assume to exist.

This can be proven rigorously under certain conditions.

Kolmogorov equations

Tuesday, April 10, 20074:04 PM

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Substitute this into the integral in the Chapman-kolmogorov equation.

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Boundary conditions in space for the parabolic equation? Intuitively need 2 boundary conditions because it's second order in x. If the state space is bounded, then have to apply absorbing or reflecting or Robin.boundary conditions on the state space. What about Cauchy problem where the state space is unbounded, then have to apply growth/decay conditions at infinity.

Forward Kolmogorov equation derivation: PDE w.r.t. target variables

It turns out that if one just tries to repeat the argument using the Chapman-kolmogorov equation using times

doesnt work!

The point is that when you substitute the Taylor expansion into the C-K equation, the integrals you have to do are not clearly related to the drift and diffusion coefficient like they were in the derivation for the backward Kolmogorov equation above. The usual way to derive the forward Kolmogorov equation is to use the backward Kolmogorov equation and do an adjoint operation.

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To prepare for the limit, where the second factor of the probability transition density will become a delta function, we integrate by parts to avoid worrying about differentiaing a delta function (but this is not strictly necessary if one is careful about handling derivatives of delta functions.)

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Why do we have two PDE's for the same function? It turns out that each equation is useful in practice for answering certain questions. But for now:

Backward Kolmogorov equation: Valid under more general conditions than the forward Kolmogorov equation. The reason is that the probability transition density is really just a nice function of the source variables but it is sort of measure-valued with respect to the target variable.

Forward Kolmogorov equation: More intuitive.

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