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 an open question.^[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 ${\displaystyle \mu ,\sigma ,\psi ,V,f}$  are known functions, ${\displaystyle T}$  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 ${\displaystyle Q}$  such that ${\displaystyle X}$  is an Itô process driven by the equation

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

with ${\displaystyle W^{Q}(t)}$  is a Wiener process (also called Brownian motion) under ${\displaystyle Q}$ , and the initial condition for ${\displaystyle X(t)}$  is ${\displaystyle X(t)=x}$ .

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 ${\displaystyle u(x,t)}$  be the solution to the above partial differential equation. Applying the product rule for Itô processes to the process

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

one gets

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

Since

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

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

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

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

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

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

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

Integrating this equation from ${\displaystyle t}$  to ${\displaystyle T}$ , one concludes that

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

Upon taking expectations, conditioned on ${\displaystyle X_{t}=x}$ , 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

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

and finally

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

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

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

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

   
  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] and zero-coupon bond prices in affine term structure models.

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

See also

• Itô's lemma
• Kunita–Watanabe inequality
• Girsanov theorem
• Kolmogorov forward equation (also known as Fokker–Planck equation)

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. "PDE for Finance".
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.

Further reading

• Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.
• Hall, B. C. (2013). Quantum Theory for Mathematicians. Springer.