Feynman–Kac formula: Difference between revisions
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The first term in parentheses is the above PDE and is zero by hypothesis. Integrating both sides one gets |
The first term in parentheses is the above PDE and is zero by hypothesis. Integrating both sides one gets |
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:<math>\int_t^T df=f( |
:<math>\int_t^T df=f(X_T,T)-f(x,t)=\int_t^T\sigma(x,t)\frac{\partial f}{\partial x}\,dW.</math> |
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Reorganising and taking the [[expectation]] of both sides: |
Reorganising and taking the [[expectation]] of both sides: |
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:<math>f(x,t)=\textrm{E}\left[f( |
:<math>f(x,t)=\textrm{E}\left[f(X_T,T)\right]-\textrm{E}\left[\int_t^T\sigma(x,t)\frac{\partial f}{\partial x}\,dW\right]</math> |
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Since the expectation of an [[Itō integral]] with respect to a [[Wiener process]] <math>\ W</math> is zero, one gets the desired result: |
Since the expectation of an [[Itō integral]] with respect to a [[Wiener process]] <math>\ W</math> is zero, one gets the desired result: |
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:<math>f(x,t)=\textrm{E}\left[f( |
:<math>f(x,t)=\textrm{E}\left[f(X_T,T)\right]=\textrm{E}\left[\psi(x)\right]=\textrm{E}\left[\psi(X_T)|X_t=x\right]</math> |
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== Remarks == |
== Remarks == |
Revision as of 15:32, 6 June 2007
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 process.
Suppose we are given the PDE
subject to the terminal condition
where are known functions, is a parameter and is the unknown. This is known as the (one-dimensional) Fokker-Planck equation or Kolmogorov forward equation. Then the Feynman-Kac formula tells us that the solution can be written as an expectation:
where is an Itō process driven by the equation
where is a Wiener process (also called Brownian motion) and the initial condition for is . This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.
Proof
Applying Itō's lemma to the unknown function one gets
The first term in parentheses is the above PDE and is zero by hypothesis. Integrating both sides one gets
Reorganising and taking the expectation of both sides:
Since the expectation of an Itō integral with respect to a Wiener process is zero, one gets the desired result:
Remarks
When originally published by Kac in 1949, the Feynman-Kac formula was presented as a formula for determining the distribution of certain Wiener functionals. Suppose we wish to find the expected value of the function
in the case where is some realization of a diffusion process starting at . The Feynman-Kac formula says that this expectation is equivalent to the integral of a solution to a diffusion equation. Specifically, under the conditions that ,
where and
The Feynman-Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If
where the integral is taken over all random walks, then
where is a solution to the parabolic partial differential equation
with initial condition .
References
- Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.