Feynman–Kac formula

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, f 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(t) = x.

Proof

Let u(x, t) be the solution to above PDE. Applying Itō's lemma to the process

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

one gets

${\begin{aligned}dY={}&d\left(e^{-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau }\right)u(X_{s},s)+e^{-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau }\,du(X_{s},s)\\[6pt]&{}+d\left(e^{-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau }\right)du(X_{s},s)+d\left(\int _{t}^{s}e^{-\int _{t}^{r}V(X_{\tau },\tau )\,d\tau }f(X_{r},r)\,dr\right)\end{aligned}}$

Since

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

the third term is $O(dt\,du)$ and can be dropped. We also have that

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

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

${\begin{aligned}dY={}&e^{-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau }\,\left(-V(X_{s},s)u(X_{s},s)+f(X_{s},s)+\mu (X_{s},s){\frac {\partial u}{\partial X}}+{\frac {\partial u}{\partial s}}+{\tfrac {1}{2}}\sigma ^{2}(X_{s},s){\frac {\partial ^{2}u}{\partial X^{2}}}\right)\,ds\\[6pt]&{}+e^{-\int _{t}^{s}V(X_{\tau },\tau )\,d\tau }\sigma (X,s){\frac {\partial u}{\partial X}}\,dW.\end{aligned}}$

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

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

Integrating this equation from t to T, one concludes that

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

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

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

The desired result is obtained by observing that

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

and finally

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

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):

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

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

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

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

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

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

• Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.
• Hall, B. C. (2013). Quantum Theory for Mathematicians. Springer.
• Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag.