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.


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

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Équation de Chapman-Kolmogorov — En théorie des probabilités, et plus spécifiquement dans la théorie des processus stochastiques markoviens, l équation de Chapman Kolmogorov est une égalité qui met en relation les lois jointes de différents points de la trajectoire d un… …   Wikipédia en Français

  • Chapman–Kolmogorov equation — In mathematics, specifically in probability theory and in particular the theory of Markovian stochastic processes, the Chapman–Kolmogorov equation is an identity relating the joint probability distributions of different sets of coordinates on a… …   Wikipedia

  • Chapman-Kolmogorov equation — In mathematics, specifically in probability theory, and yet more specifically in the theory of Markovian stochastic processes, the Chapman Kolmogorov equation can be viewed as an identity relating the joint probability distributions of different… …   Wikipedia

  • Andrey Kolmogorov — Infobox Scientist name = Andrey Kolmogorov birth date = birth date|1903|4|25 birth place = Tambov, Imperial Russia nationality = Russian death date = death date and age|1987|10|20|1903|4|25 death place = Moscow, USSR field = Mathematician work… …   Wikipedia

  • Fokker–Planck equation — [ thumb|A solution to the one dimensional Fokker–Planck equation, with both the drift and the diffusion term. The initial condition is a Dirac delta function in x = 1, and the distribution drifts towards x = 0.] The Fokker–Planck equation… …   Wikipedia

  • Itō diffusion — In mathematics mdash; specifically, in stochastic analysis mdash; an Itō diffusion is a solution to a specific type of stochastic differential equation. Itō diffusions are named after the Japanese mathematician Kiyoshi Itō.OverviewA (time… …   Wikipedia

  • List of mathematics articles (K) — NOTOC K K approximation of k hitting set K ary tree K core K edge connected graph K equivalence K factor error K finite K function K homology K means algorithm K medoids K minimum spanning tree K Poincaré algebra K Poincaré group K set (geometry) …   Wikipedia

  • Feynman-Kac formula — The Feynman Kac formula, named after Richard Feynman and Mark Kac, establishes a link between partial differential equations (PDEs) and stochastic processes. It offers a method of solving certain PDEs by simulating random paths of a stochastic… …   Wikipedia

  • Infinitesimal generator (stochastic processes) — In mathematics mdash; specifically, in stochastic analysis mdash; the infinitesimal generator of a stochastic process is a partial differential operator that encodes a great deal of information about the process. The generator is used in… …   Wikipedia

  • Motoo Kimura — Born November 13, 1924 Okazaki, Japan Died Nove …   Wikipedia

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.