# Kolmogorov backward equation

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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&ndash;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 $p_t\left(x\right)$); we want to know the probability distribution of the state at a later time $s>t$. The adjective 'forward' refers to the fact that $p_t\left(x\right)$ serves as the initial condition and the PDE is integrated forward in time. (In the common case where the initial state is known exactly $p_t\left(x\right)$ 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 $u_s\left(x\right)$ 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 $x\left(t\right)$ evolves according to the stochastic differential equation

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

then the Kolmogorov backward equation is, using Ito's lemma on $p\left(x,t\right)$:

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

for $tle s$, subject to the final condition $p\left(x,s\right)=u_s\left(x\right)$.

This equation can be derived from the Feynman-Kac formula by noting that the hit probability is the same as the expected value of $u_s\left(x\right)$ over all paths that originate from state x at time t::$P\left(X_s in B | X_t = x\right) = E \left[u_s\left(x\right) | X_t = x\right]$

Formulating the Kolmogorov forward equation

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

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

for $s ge t$, with initial condition $p\left(x,t\right)=p_t\left(x\right)$. For more on this equation see Fokker&ndash;Planck equation.

References

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

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