Kolmogorov backward equation

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The Kolmogorov backward equation (KBE) and its adjoint the Kolmogorov forward equation (KFE) are partial differential equations that arise in the theory of continuous-time continuous-state Markov processes. Both were published by Andrey Kolmogorov in 1931. Later it was realized that the KFE was already known to physicists under the name Fokker-Planck equation; the KBE on the other hand was new.

Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state x of the system at time t (namely a probability distribution $p t (x)$ ); we want to know the probability distribution of the state at a later time $s > t$ . The adjective 'forward' refers to the fact that $p t (x)$ serves as the initial condition and the PDE is integrated forward in time. (In the common case where the initial state is known exactly $p t (x)$ is a Dirac delta function centered on the known initial state).

The Kolmogorov backward equation on the other hand is useful when we are interested at time t in whether at a future time s the system will be in a given subset of states, sometimes called the target set. The target is described by a given function $u s (x)$ which is equal to 1 if state x is in the target set and zero otherwise. We want to know for every state x at time t $(t < s)$ what is the probability of ending up in the target set at time s (sometimes called the hit probability). In this case $u s (x)$ serves as the final condition of the PDE, which is integrated backward in time, from s to t.

[edit] Formulating the Kolmogorov backward equation

Assume that the system state $x (t)$ evolves according to the stochastic differential equation

$dx(t) = \mu(x(t),t)\,dt + \sigma(x(t),t)\,dW(t)$

then the Kolmogorov backward equation is

$-\frac{\partial}{\partial t}p(x,t)=\mu(x,t)\frac{\partial}{\partial x}p(x,t) + \frac{1}{2}\sigma^2(x,t)\frac{\partial^2}{\partial x^2}p(x,t)$

for $t\le s$ , subject to the final condition $p (x, s) = u s (x)$ .

This equation can be derived from the Feynman-Kac formula by noting that the hit probability is the same as the expected value of $u s (x)$ over all paths that originate from state x at time t.