Kolmogorov backward equation


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 (t 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.

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


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