Kolmogorov backward equations (diffusion)

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The Kolmogorov backward equation (KBE) (diffusion) and its adjoint sometimes known as the Kolmogorov forward equation (diffusion) 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.[1] Later it was realized that the forward equation 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 B, 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 at time s, and zero otherwise. In other words, u_s(x) = 1_B , the indicator function for the set B. 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.

Formulating the Kolmogorov backward equation[edit]

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 as follows [2]

-\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 can be derived using Itō's lemma on  p(x,t) and setting the dt 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 u_s(x) over all paths that originate from state x at time t:

 P(X_s \in B \mid X_t = x) = E[u_s(x) \mid X_t = x]

Historically of course the KBE [1] was developed before the Feynman-Kac formula (1949).

Formulating the Kolmogorov forward equation[edit]

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[edit]

  • Etheridge, A. (2002). A Course in Financial Calculus. Cambridge University Press. 
  1. ^ a b Andrei Kolmogorov, "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" (On Analytical Methods in the Theory of Probability), 1931, [1]
  2. ^ Risken, H., "The Fokker-Planck equation: Methods of solution and applications" 1996, Springer