Formulating The Kolmogorov Backward Equation
Assume that the system state evolves according to the stochastic differential equation
then the Kolmogorov backward equation is as follows
for, subject to the final condition . This can be derived using Ito's lemma on and setting the dW(t) term equal to zero.
This equation can also be derived from the Feynman-Kac formula by noting that the hit probability is the same as the expected value of over all paths that originate from state x at time t:
Historically of course the KBE was developed before the Feynman-Kac formula (1949).
Read more about this topic: Kolmogorov Backward Equations (diffusion)
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