Kolmogorov backward equation

The Kolmogorov backward equation (KBE) and its adjoint the Kolmogorov forward equation (KFE) are partial differential equations (PDE) 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

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, using Ito's lemma on $p(x,t)$:

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

for $tle\; s$, 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::$P(X\_s\; in\; B\; |\; X\_t\; =\; x)\; =\; E\; [u\_s(x)\; |\; X\_t\; =\; x]$

Formulating the Kolmogorov forward equation

With the same notation as before, the corresponding Kolmogorov forward equation is:

:$frac\{partial\}\{partial\; s\}p(x,s)=-frac\{partial\}\{partial\; x\}\; [mu(x,s)p(x,s)]\; +\; frac\{1\}\{2\}frac\{partial^2\}\{partial\; x^2\}\; [sigma^2(x,s)p(x,s)]$

for $s\; ge\; t$, with initial condition $p(x,t)=p\_t(x)$. For more on this equation see Fokker–Planck equation.

References

*cite book|author=Etheridge, E.|title=A Course in Financial Calculus|publisher=Cambridge University Press|year=2002