# 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. In 1947, when Kac and Feynman were both Cornell faculty, Kac attended a presentation of Feynman's and remarked that the two of them were working on the same thing from different directions.[1] The Feynman–Kac formula resulted, which proves rigorously the real case of Feynman's path integrals. The complex case, which occurs when a particle's spin is included, is still[when?] unproven.[2]

It offers a method of solving certain partial differential equations by simulating random paths of a stochastic process. Conversely, an important class of expectations of random processes can be computed by deterministic methods.

## Theorem

Consider the partial differential equation

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

defined for all ${\displaystyle x\in \mathbb {R} }$ and ${\displaystyle t\in [0,T]}$, subject to the terminal condition

${\displaystyle u(x,T)=\psi (x),}$

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

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

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

${\displaystyle dX_{t}=\mu (X,t)\,dt+\sigma (X,t)\,dW_{t}^{Q},}$

with WQ(t) is a Wiener process (also called Brownian motion) under Q, and the initial condition for X(t) is X(t) = x.

## Partial proof

A proof that the above formula is a solution of the differential equation is long, difficult and not presented here. It is however reasonably straightforward to show that, if a solution exists, it must have the above form. The proof of that lesser result is as follows:

Let u(x, t) be the solution to the above partial differential equation. Applying the product rule for Itô processes to the process

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

one gets

{\displaystyle {\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

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

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

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

Applying Itô's lemma to ${\displaystyle du(X_{s},s)}$, it follows that

{\displaystyle {\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 partial differential equation and is therefore zero. What remains is

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

Integrating this equation from t to T, one concludes that

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

Upon taking expectations, conditioned on Xt = x, and observing that the right side is an Itô integral, which has expectation zero,[3] it follows that

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

The desired result is obtained by observing that

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

and finally

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

## Remarks

• The proof above that a solution must have the given form is essentially that of [4] with modifications to account for ${\displaystyle f(x,t)}$.
• The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding partial differential equation for ${\displaystyle u:\mathbb {R} ^{N}\times [0,T]\to \mathbb {R} }$ becomes:[5]
${\displaystyle {\frac {\partial u}{\partial t}}+\sum _{i=1}^{N}\mu _{i}(x,t){\frac {\partial u}{\partial x_{i}}}+{\frac {1}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}\gamma _{ij}(x,t){\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}-r(x,t)\,u=f(x,t),}$
where,
${\displaystyle \gamma _{ij}(x,t)=\sum _{k=1}^{N}\sigma _{ik}(x,t)\sigma _{jk}(x,t),}$
i.e. ${\displaystyle \gamma =\sigma \sigma ^{\mathrm {T} }}$, where ${\displaystyle \sigma ^{\mathrm {T} }}$ denotes the transpose of ${\displaystyle \sigma }$.
• This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.
• When originally published by Kac in 1949,[6] 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
${\displaystyle e^{-\int _{0}^{t}V(x(\tau ))\,d\tau }}$
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 ${\displaystyle uV(x)\geq 0}$,
${\displaystyle E\left[e^{-u\int _{0}^{t}V(x(\tau ))\,d\tau }\right]=\int _{-\infty }^{\infty }w(x,t)\,dx}$
where w(x, 0) = δ(x) and
${\displaystyle {\frac {\partial w}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-uV(x)w.}$
The Feynman–Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If
${\displaystyle I=\int f(x(0))e^{-u\int _{0}^{t}V(x(t))\,dt}g(x(t))\,Dx}$
where the integral is taken over all random walks, then
${\displaystyle I=\int w(x,t)g(x)\,dx}$
where w(x, t) is a solution to the parabolic partial differential equation
${\displaystyle {\frac {\partial w}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-uV(x)w}$
with initial condition w(x, 0) = f(x).

## Applications

In quantitative finance, the Feynman–Kac formula is used to efficiently calculate solutions to the Black–Scholes equation to price options on stocks.[7]

In quantum chemistry, it is used to solve the Schrödinger equation with the Pure Diffusion Monte Carlo method.[8]

## References

1. ^ Kac, Mark (1987). Enigmas of Chance: An Autobiography. University of California Press. pp. 115–16. ISBN 0-520-05986-7.
2. ^ Glimm, James; Jaffe, Arthur (1987). Quantum Physics: A Functional Integral Point of View (2 ed.). New York, NY: Springer. pp. 43–44. doi:10.1007/978-1-4612-4728-9. ISBN 978-0-387-96476-8. Retrieved 13 April 2021.
3. ^ Øksendal, Bernt (2003). "Theorem 3.2.1.(iii)". Stochastic Differential Equations. An Introduction with Applications (6th ed.). Springer-Verlag. p. 30. ISBN 3540047581.
4. ^
5. ^ See Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag. ISBN 978-3-642-10044-4.
6. ^ Kac, Mark (1949). "On Distributions of Certain Wiener Functionals". Transactions of the American Mathematical Society. 65 (1): 1–13. doi:10.2307/1990512. JSTOR 1990512. This paper is reprinted in Baclawski, K.; Donsker, M. D., eds. (1979). Mark Kac: Probability, Number Theory, and Statistical Physics, Selected Papers. Cambridge, Massachusetts: The MIT Press. pp. 268–280. ISBN 0-262-11067-9.
7. ^ Paolo Brandimarte (6 June 2013). "Chapter 1. Motivation". Numerical Methods in Finance and Economics: A MATLAB-Based Introduction. John Wiley & Sons. ISBN 978-1-118-62557-6.
8. ^ Caffarel, Michel; Claverie, Pierre (15 January 1988). "Development of a pure diffusion quantum Monte Carlo method using a full generalized Feynman–Kac formula. I. Formalism". The Journal of Chemical Physics. 88 (2): 1088–1099. Bibcode:1988JChPh..88.1088C. doi:10.1063/1.454227.