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| The '''Kolmogorov backward equation (KBE)''' (diffusion) and its [[Adjoint of an operator|adjoint]] sometimes known as the Kolmogorov forward equation (diffusion) are [[partial differential equation]]s (PDE) that arise in the theory of continuous-time continuous-state [[Markov process]]es. Both were published by [[Andrey Kolmogorov]] in 1931.<ref name="k31">Andrei Kolmogorov, "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" (On Analytical Methods in the Theory of Probability), 1931, [http://eudml.org/doc/159476]</ref> 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.
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| 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]] <math>p_t(x)</math>); we want to know the probability distribution of the state at a later time <math>s>t</math>. The adjective 'forward' refers to the fact that <math>p_t(x)</math> serves as the initial condition and the PDE is integrated forward in time. (In the common case where the initial state is known exactly <math>p_t(x)</math> is a [[Dirac delta function]] centered on the known initial state).
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| 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 <math>u_s(x)</math> which is equal to 1 if state ''x'' is in the target set at time ''s'', and zero otherwise. In other words, <math>u_s(x) = 1_B </math>, the indicator function for the set ''B''. We want to know for every state ''x'' at time <math>t, (t<s)</math> what is the probability of ending up in the target set at time ''s'' (sometimes called the hit probability). In this case <math>u_s(x)</math> serves as the final condition of the PDE, which is integrated backward in time, from ''s'' to ''t''.
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| ==Formulating the Kolmogorov backward equation==
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| Assume that the system state <math>x(t)</math> evolves according to the [[stochastic differential equation]]
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| :<math>dx(t) = \mu(x(t),t)\,dt + \sigma(x(t),t)\,dW(t)</math>
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| then the Kolmogorov backward equation is as follows <ref>Risken, H., "The Fokker-Planck equation: Methods of solution and applications" 1996, Springer</ref>
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| :<math>-\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)</math>
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| for <math>t\le s</math>, subject to the final condition <math>p(x,s)=u_s(x)</math>.
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| This can be derived using [[Itō's lemma]] on <math> p(x,t) </math> and setting the dt term equal to zero.
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| 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 <math>u_s(x)</math> over all paths that originate from state x at time t:
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| :<math> P(X_s \in B \mid X_t = x) = E[u_s(x) \mid X_t = x]</math>
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| Historically of course the KBE <ref name="k31">Andrei Kolmogorov, "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" (On Analytical Methods in the Theory of Probability), 1931, [http://www.springerlink.com/content/v724507673277262/fulltext.pdf]</ref> was developed before the Feynman-Kac formula (1949).
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| ==Formulating the Kolmogorov forward equation==
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| With the same notation as before, the corresponding Kolmogorov forward equation is:
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| :<math>\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)]</math>
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| for <math>s \ge t</math>, with initial condition <math>p(x,t)=p_t(x)</math>. For more on this equation see [[Fokker–Planck equation]].
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| {{more footnotes|date=June 2011}}
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| ==References==
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| *{{cite book|author=Etheridge, A.|title=A Course in Financial Calculus|publisher=Cambridge University Press|year=2002}}
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| {{reflist}}
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| [[Category:Parabolic partial differential equations]]
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| [[Category:Stochastic differential equations]]
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| [[it:Equazione retrospettiva di Kolmogorov]]
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