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The '''Feynman–Kac formula''', named after [[Richard Feynman]] and [[Mark Kac]], establishes a link between parabolic [[partial differential equation]]s (PDEs) and [[stochastic process]]es. It offers a method of solving certain PDEs by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods. Consider the PDE
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:<math>\frac{\partial u}{\partial t}(x,t) + \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2}(x,t) -V(x,t) u(x,t) + f(x,t) = 0 </math>,
 
defined for all ''x'' in '''R''' and ''t'' in [0, ''T''], subject to the terminal condition
 
:<math>u(x,T)=\psi(x), </math>
 
where μ, σ, ψ, ''V'' are known functions, ''T'' is a parameter and <math> u:\mathbb{R}\times[0,T]\to\mathbb{R}</math> is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a [[conditional expectation]]
 
:<math> u(x,t) = E^Q\left[ \int_t^T e^{-  \int_t^r V(X_\tau,\tau)\, d\tau}f(X_r,r)dr + e^{-\int_t^T V(X_\tau,\tau)\, d\tau}\psi(X_T) \Bigg| X_t=x \right] </math>
 
under the [[probability measure]] Q such that ''X'' is an [[Itō process]] driven by the equation
 
:<math>dX = \mu(X,t)\,dt + \sigma(X,t)\,dW^Q,</math>
 
with ''W<sup>Q</sup>''(''t'') is a [[Wiener process]] (also called [[Brownian motion]]) under ''Q'', and the initial condition for ''X''(''t'') is ''X''(0) = ''x''.
 
== Proof ==
Let ''u''(''x'', ''t'') be the solution to above PDE.  Applying [[Itō's lemma]] to the process
 
:<math> Y(s) = e^{-  \int_t^s V(X_\tau)\, d\tau} u(X_s,s)+ \int_t^s e^{-  \int_t^r V(X_\tau,\tau)\, d\tau}f(X_r,r)dr</math>
 
one gets
 
:<math>dY = de^{-  \int_t^s V(X_\tau)\, d\tau} u(X_s,s) + e^{-  \int_t^s V(X_\tau)\, d\tau}\,du(X_s,s) +de^{-  \int_t^s V(X_\tau)\, d\tau}du(X_s,s) + d\int_t^s e^{-  \int_t^r V(X_\tau)\, d\tau}  f(X_r,r)dr</math>
 
Since
 
:<math>de^{-  \int_t^s V(X_\tau)\, d\tau} =-V(X_s) e^{-  \int_t^s V(X_\tau)\, d\tau} \,ds,</math>
the third term is <math> o(dtdu) </math> and can be dropped. We also have that
:<math> d\int_t^s e^{-  \int_t^r V(X_\tau,\tau)\, d\tau}f(X_r,r)dr = e^{-  \int_t^s V(X_\tau)\, d\tau} f(X_s,s) ds.</math>
 
Applying [[Itō's lemma]] once again to <math>du(X_s,s)</math>, it follows that  
 
:<math> dY=e^{-  \int_t^s V(X_\tau)\, d\tau}\,\left(-V(X_s) u(X_s,s) +f(X_s,s)+\mu(X_s,s)\frac{\partial u}{\partial X}+\frac{\partial u}{\partial s}+\tfrac{1}{2}\sigma^2(X_s,s)\frac{\partial^2 u}{\partial X^2}\right)\,ds + e^{- \int_t^s V(X_\tau)\, d\tau}\sigma(X,s)\frac{\partial u}{\partial X}\,dW.</math>
 
The first term contains, in parentheses, the above PDE and is therefore zero. What remains is
 
:<math>dY=e^{-  \int_t^s V(X_\tau)\, d\tau}\sigma(X,s)\frac{\partial u}{\partial X}\,dW.</math>
 
Integrating this equation from ''t'' to ''T'', one concludes that
 
:<math> Y(T) - Y(t) = \int_t^T e^{-  \int_t^s V(X_\tau)\, d\tau}\sigma(X,s)\frac{\partial u}{\partial X}\,dW.</math>
 
Upon taking expectations, conditioned on ''X<sub>t</sub>'' = ''x'', and observing that the right side is an [[Itō integral]], which has expectation zero, it follows that
 
:<math>E[Y(T)|X_t=x] =  E[Y(t)|X_t=x] = u(x,t).</math> 
 
The desired result is obtained by observing that
 
:<math>E[Y(T)| X_t=x] = E \left [e^{-  \int_t^T V(X_\tau)\, d\tau} u(X_T,T) + \int_t^T e^{-  \int_t^r V(X_\tau,\tau)\, d\tau}f(X_r,r)dr \Bigg| X_t=x \right ]</math>
and finally
:<math> u(x,t) = E \left [e^{-  \int_t^T V(X_\tau)\, d\tau} \psi(X_T)) + \int_t^T e^{-\int_t^s V(\tau)d\tau} f(X_s,s)ds \Bigg| X_t=x \right ]</math>
 
== Remarks ==
* The proof above is essentially that of <ref>http://www.math.nyu.edu/faculty/kohn/pde_finance.html</ref> with modifications to account for <math>f(x,t)</math>.
 
* The expectation formula above is also valid for ''N''-dimensional Itô diffusions. The corresponding PDE for <math> u:\mathbb{R}^N\times[0,T]\to\mathbb{R}</math> becomes (see H. Pham book below):
 
::<math>\frac{\partial u}{\partial t} + \sum_{i=1}^N \mu_i(x,t)\frac{\partial u}{\partial x_i} + \tfrac{1}{2} \sum_{i=1}^N\sum_{j=1}^N\gamma_{ij}(x,t) \frac{\partial^2 u}{\partial x_i x_j} -r(x,t) u = f(x,t), </math>
 
:where,
 
::<math> \gamma_{ij}(x,t) = \sum_{k=1}^N\sigma_{ik}(x,t)\sigma_{jk}(x,t),</math>
 
:i.e. γ = σσ′, where σ′ denotes the transpose matrix of σ).
 
* This expectation can then be approximated using [[Monte Carlo method|Monte Carlo]] or [[quasi-Monte Carlo method]]s.
 
* When originally published by Kac in 1949,<ref>{{cite journal|last=Kac|first=Mark|title=On Distributions of Certain Wiener Functionals|journal=Transactions of the American Mathematical Society|authorlink=Mark Kac|volume=65|issue=1|pages=1–13|jstor=1990512|year=1949|doi=10.2307/1990512}}</ref> 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
 
::<math> e^{-\int_0^t V(x(\tau))\, d\tau} </math>
 
:in the case where ''x''(τ) is some realization of a diffusion process starting at ''x''(0) = 0. 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 <math>u V(x) \geq 0</math>,
 
::<math> E\left[ e^{- u \int_0^t V(x(\tau))\, d\tau} \right] = \int_{-\infty}^{\infty} w(x,t)\, dx </math>
 
:where ''w''(''x'', 0) = δ(''x'') and
 
::<math>\frac{\partial w}{\partial t} = \tfrac{1}{2} \frac{\partial^2 w}{\partial x^2} - u V(x) w.</math>
 
:The Feynman–Kac formula can also be interpreted as a method for evaluating [[functional integral]]s of a certain form. If
 
::<math> I = \int f(x(0)) e^{-u\int_0^t V(x(t))\, dt} g(x(t))\, Dx </math>
 
:where the integral is taken over all [[random walk]]s, then
 
::<math> I = \int w(x,t) g(x)\, dx </math>
 
:where ''w''(''x'', ''t'') is a solution to the [[parabolic partial differential equation]]
 
::<math> \frac{\partial w}{\partial t} = \tfrac{1}{2} \frac{\partial^2 w}{\partial x^2} - u V(x) w </math>
 
:with initial condition ''w''(''x'', 0) = ''f''(''x'').
 
== See also ==
* [[Itō's lemma]]
* [[Kunita–Watanabe theorem]]
* [[Girsanov theorem]]
* [[Kolmogorov forward equation]] (also known as Fokker–Planck equation)
 
== References ==
* {{cite book|last=Simon|first=Barry|authorlink=Barry Simon|title=Functional Integration and Quantum Physics|year=1979|publisher=Academic Press}}
* {{cite book |last = Hall |first = B. C. |title = Quantum Theory for Mathematicians | year = 2013 |publisher = Springer}}
* {{cite book|last=Pham|first=Huyên|title=Continuous-time stochastic control and optimisation with financial applications|year=2009|publisher=Springer-Verlag}}
{{reflist}}
 
{{DEFAULTSORT:Feynman-Kac Formula}}
[[Category:Stochastic processes]]
[[Category:Parabolic partial differential equations]]
[[Category:Articles containing proofs]]

Latest revision as of 05:10, 31 October 2014

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