Pierre Fatou

The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and stochastic processes. 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

${\frac {\partial u}{\partial t}}(x,t)+\mu (x,t){\frac {\partial u}{\partial x}}(x,t)+{\tfrac {1}{2}}\sigma ^{2}(x,t){\frac {\partial ^{2}u}{\partial x^{2}}}(x,t)-V(x,t)u(x,t)+f(x,t)=0$,

defined for all x in R and t in [0, T], subject to the terminal condition

$u(x,T)=\psi (x),$

where μ, σ, ψ, V are known functions, T is a parameter and $u:\mathbb {R} \times [0,T]\to \mathbb {R}$ is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a conditional expectation

$u(x,t)=E^{Q}\left[\int _{t}^{T}e^{-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau }f(X_{r},r)dr+e^{-\int _{t}^{T}V(X_{\tau },\tau )\,d\tau }\psi (X_{T}){\Bigg |}X_{t}=x\right]$

under the probability measure Q such that X is an Itō process driven by the equation

$dX=\mu (X,t)\,dt+\sigma (X,t)\,dW^{Q},$

with W^Q(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

$Y(s)=e^{-\int _{t}^{s}V(X_{\tau })\,d\tau }u(X_{s},s)+\int _{t}^{s}e^{-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau }f(X_{r},r)dr$

one gets

$dY=de^{-\int _{t}^{s}V(X_{\tau })\,d\tau }u(X_{s},s)+e^{-\int _{t}^{s}V(X_{\tau })\,d\tau }\,du(X_{s},s)+de^{-\int _{t}^{s}V(X_{\tau })\,d\tau }du(X_{s},s)+d\int _{t}^{s}e^{-\int _{t}^{r}V(X_{\tau })\,d\tau }f(X_{r},r)dr$

Since

$de^{-\int _{t}^{s}V(X_{\tau })\,d\tau }=-V(X_{s})e^{-\int _{t}^{s}V(X_{\tau })\,d\tau }\,ds,$

the third term is $o(dtdu)$ and can be dropped. We also have that

$d\int _{t}^{s}e^{-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau }f(X_{r},r)dr=e^{-\int _{t}^{s}V(X_{\tau })\,d\tau }f(X_{s},s)ds.$

Applying Itō's lemma once again to $du(X_{s},s)$, it follows that

$dY=e^{-\int _{t}^{s}V(X_{\tau })\,d\tau }\,\left(-V(X_{s})u(X_{s},s)+f(X_{s},s)+\mu (X_{s},s){\frac {\partial u}{\partial X}}+{\frac {\partial u}{\partial s}}+{\tfrac {1}{2}}\sigma ^{2}(X_{s},s){\frac {\partial ^{2}u}{\partial X^{2}}}\right)\,ds+e^{-\int _{t}^{s}V(X_{\tau })\,d\tau }\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.$

The first term contains, in parentheses, the above PDE and is therefore zero. What remains is

$dY=e^{-\int _{t}^{s}V(X_{\tau })\,d\tau }\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.$

Integrating this equation from t to T, one concludes that

$Y(T)-Y(t)=\int _{t}^{T}e^{-\int _{t}^{s}V(X_{\tau })\,d\tau }\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.$

Upon taking expectations, conditioned on X_t = x, and observing that the right side is an Itō integral, which has expectation zero, it follows that

$E[Y(T)|X_{t}=x]=E[Y(t)|X_{t}=x]=u(x,t).$

The desired result is obtained by observing that

$E[Y(T)|X_{t}=x]=E\left[e^{-\int _{t}^{T}V(X_{\tau })\,d\tau }u(X_{T},T)+\int _{t}^{T}e^{-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau }f(X_{r},r)dr{\Bigg |}X_{t}=x\right]$

and finally

$u(x,t)=E\left[e^{-\int _{t}^{T}V(X_{\tau })\,d\tau }\psi (X_{T}))+\int _{t}^{T}e^{-\int _{t}^{s}V(\tau )d\tau }f(X_{s},s)ds{\Bigg |}X_{t}=x\right]$

Remarks

• The proof above is essentially that of with modifications to account for $f(x,t)$.

• The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding PDE for $u:\mathbb {R} ^{N}\times [0,T]\to \mathbb {R}$ becomes (see H. Pham book below):

${\frac {\partial u}{\partial t}}+\sum _{i=1}^{N}\mu _{i}(x,t){\frac {\partial u}{\partial x_{i}}}+{\tfrac {1}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}\gamma _{ij}(x,t){\frac {\partial ^{2}u}{\partial x_{i}x_{j}}}-r(x,t)u=f(x,t),$

where,

$\gamma _{ij}(x,t)=\sum _{k=1}^{N}\sigma _{ik}(x,t)\sigma _{jk}(x,t),$

i.e. γ = σσ′, where σ′ denotes the transpose matrix of σ).

• This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.

• 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

$e^{-\int _{0}^{t}V(x(\tau ))\,d\tau }$

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 $uV(x)\geq 0$,

$E\left[e^{-u\int _{0}^{t}V(x(\tau ))\,d\tau }\right]=\int _{-\infty }^{\infty }w(x,t)\,dx$

where w(x, 0) = δ(x) and

${\frac {\partial w}{\partial t}}={\tfrac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-uV(x)w.$

The Feynman–Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If

$I=\int f(x(0))e^{-u\int _{0}^{t}V(x(t))\,dt}g(x(t))\,Dx$

where the integral is taken over all random walks, then

$I=\int w(x,t)g(x)\,dx$

where w(x, t) is a solution to the parabolic partial differential equation

${\frac {\partial w}{\partial t}}={\tfrac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-uV(x)w$

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

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.