Feynman–Kac formula explained

The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations and stochastic processes. In 1947, when Kac and Feynman were both faculty members at Cornell University, 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-valued 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$$\backslash frac(x,t)\; +\; \backslash mu(x,t)\; \backslash frac(x,t)\; +\; \backslash tfrac\; \backslash sigma^2(x,t)\; \backslash frac(x,t)\; -V(x,t)\; u(x,t)\; +\; f(x,t)\; =\; 0,$$defined for all

x \in R and t \in [ 0 , T] , subject to the terminal condition$$u(x,T)=\backslash psi(x),$$where \mu,\sigma,\psi,V,f are known functions, T is a parameter, and u: R x [ 0 ,T] \to R is the unknown. Then the Feynman–Kac formula expresses u(x,t) as a conditional expectation under the probability measure Q where X is an Itô process satisfying$$\backslash mathrmX\_t\; =\; \backslash mu(X,t)\backslash ,\backslash mathrmt\; +\; \backslash sigma(X,t)\backslash ,\backslash mathrm\; W^Q\_t,$$and W ^{Q} _{t} a Wiener process (also called Brownian motion) under Q .

Intuitive interpretation

Suppose that the position

X _{t} of a particle evolves according to the diffusion process$$dX\_t\; =\; \backslash mu(X,t)\backslash ,\backslash mathrmt\; +\; \backslash sigma(X,t)\backslash ,\backslash mathrm\; W^Q\_t.$$Let the particle incur "cost" at a rate of f(X _{s,} s) at location X _{s} at time s . Let it incur a final cost at \psi(X _{T)} .

Also, allow the particle to decay. If the particle is at location

X _{s} at time s , then it decays with rate V(X _{s,} s) . After the particle has decayed, all future cost is zero.

Then

u(x, t) is the expected cost-to-go, if the particle starts at (t, 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$$Y(s)\; =\; \backslash exp\backslash left(-\backslash int\_t^s\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)u(X\_s,s)\; +\backslash int\_t^s\; \backslash exp\backslash left(-\backslash int\_t^r\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)f(X\_r,r)\; \backslash ,\; dr$$one gets:

Since$$d\backslash left(\backslash exp\backslash left(-\; \backslash int\_t^s\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)\backslash right)\; =\; -V(X\_s,s)\; \backslash exp\backslash left(-\backslash int\_t^s\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)\; \backslash ,ds,$$ the third term is

O(dt du) and can be dropped. We also have that$$d\backslash left(\backslash int\_t^s\; \backslash exp\backslash left(-\; \backslash int\_t^r\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)f(X\_r,r)dr\backslash right)\; =\backslash exp\backslash left(-\backslash int\_t^s\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)\; f(X\_s,s)\; ds.$$

Applying Itô's lemma to

du(X _{s,s)} , it follows that

The first term contains, in parentheses, the above partial differential equation and is therefore zero. What remains is:$$dY\_s=\backslash exp\backslash left(-\backslash int\_t^s\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)\backslash sigma(X,s)\backslash frac\backslash ,dW.$$

Integrating this equation from

t to T , one concludes that:$$Y(T)\; -\; Y(t)\; =\; \backslash int\_t^T\backslash exp\backslash left(-\backslash int\_t^s\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)\backslash sigma(X,s)\backslash frac\backslash ,dW.$$

Upon taking expectations, conditioned on

X _{t} = x , and observing that the right side is an Itô integral, which has expectation zero,^[3] it follows that:$$E[Y(T)\backslash mid\; X\_t=x]\; =\; E[Y(t)\backslash mid\; X\_t=x]\; =\; u(x,t).$$

The desired result is obtained by observing that:$$E[Y(T)\backslash mid\; X\_t=x]\; =E\; \backslash left\; [\backslash exp\backslash left(-\backslash int\_t^T\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)\; u(X\_T,T)\; +\; \backslash int\_t^T\; \backslash exp\backslash left(-\backslash int\_t^r\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)f(X\_r,r)\backslash ,dr\; \backslash ,\backslash Bigg|\backslash ,\; X\_t=x\; \backslash right\; ]$$and finally$$u(x,t)\; =\; E\; \backslash left\; [\backslash exp\backslash left(-\backslash int\_t^T\; V(X\_\backslash tau,\backslash tau)\backslash ,\; d\backslash tau\backslash right)\; \backslash psi(X\_T)\; +\; \backslash int\_t^T\; \backslash exp\backslash left(-\backslash int\_t^s\; V(X\_\backslash tau,\backslash tau)\backslash ,d\backslash tau\backslash right)\; f(X\_s,s)\backslash ,ds\; \backslash ,\backslash Bigg|\backslash ,\; X\_t=x\; \backslash right\; ]$$

Remarks

• The proof above that a solution must have the given form is essentially that of ^[4] with modifications to account for

f(x,t) .

• The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding partial differential equation for

u: R ^{N x} [ 0 ,T] \to R becomes:^[5] $$\backslash frac\; +\; \backslash sum\_^N\; \backslash mu\_i(x,t)\backslash frac\; +\; \backslash frac\; \backslash sum\_^N\; \backslash sum\_^N\backslash gamma\_(x,t)\; \backslash frac\; -r(x,t)\backslash ,u\; =\; f(x,t),$$ where, $$\backslash gamma\_(x,t)\; =\; \backslash sum\_^N\; \backslash sigma\_(x,t)\backslash sigma\_(x,t),$$ i.e. \gamma = \sigma \sigma ^{T} , where \sigma ^{T} denotes the transpose of \sigma .

• More succinctly, letting

A be the infinitesimal generator of the diffusion process,$$\backslash frac\; +\; A\; u\; -r(x,t)\backslash ,u\; =\; f(x,t),$$

• 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 $$$$

\exp\left(-\int_0^t V(x(\tau))\, d\tau\right) in the case where x(τ) is some realization of a diffusion process starting at . 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

u V(x) \geq 0 , $$E\backslash left[\backslash exp\backslash left(-\; u\; \backslash int\_0^t\; V(x(\backslash tau))\backslash ,\; d\backslash tau\backslash right)\; \backslash right]\; =\; \backslash int\_^\; w(x,t)\backslash ,\; dx$$ where and $$\backslash frac\; =\; \backslash frac\; \backslash frac\; -\; u\; V(x)\; w.$$The Feynman–Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If$$I\; =\; \backslash int\; f(x(0))\; \backslash exp\backslash left(-u\backslash int\_0^t\; V(x(t))\backslash ,\; dt\backslash right)\; g(x(t))\backslash ,\; Dx$$where the integral is taken over all random walks, then $$I\; =\; \backslash int\; w(x,t)\; g(x)\backslash ,\; dx$$ where is a solution to the parabolic partial differential equation $$\backslash frac\; =\; \backslash frac\; \backslash frac\; -\; u\; V(x)\; w$$ with initial condition .

Applications

Finance

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.

For example, consider a stock price

S _{t} undergoing geometric Brownian motion$$dS\_t\; =\; (r\_t\; dt\; +\; \backslash sigma\_t\; dW\_t)\; S\_t$$where r _{t} is the risk-free interest rate and \sigma _{t} is the volatility. Equivalently, by Itô's lemma,$$d\backslash ln\; S\_t\; =\; \backslash left(r\_t\; -\; \backslash tfrac\; 1\; 2\; \backslash sigma\_t^2\backslash right)dt\; +\; \backslash sigma\_t\; \backslash ,\; dW\_t.$$Now consider a European call option on an S _{t} expiring at time T with strike K . At expiry, it is worth (X _{T} - K) ^{+.} Then, the risk-neutral price of the option, at time t and stock price x , is$$u(x,\; t)\; =\; E\backslash left[e^\{-\backslash int\_t^T\; r\_s\; ds\}\; (S\_T\; -\; K)^+\; |\; \backslash ln\; S\_t\; =\; \backslash ln\; x\; \backslash right].$$Plugging into the Feynman–Kac formula, we obtain the Black–Scholes equation:$$\backslash begin\backslash partial\_t\; u\; +\; Au\; -\; r\_t\; u\; =\; 0\; \backslash \backslash u(x,\; T)\; =\; (x-K)^+\backslash end$$where$A\; =\; (r\_t\; -\backslash sigma\_t^2/2)\backslash partial\_\; +\; \backslash frac\; 12\; \backslash sigma\_t^2\; \backslash partial\_^2\; =\; r\_t\; x\backslash partial\_x\; +\; \backslash frac\; 1\; 2\; \backslash sigma\_t^2\; x^2\; \backslash partial\_^2.$ More generally, consider an option expiring at time T with payoff g(S _{T)} . The same calculation shows that its price u(x,t) satisfies$$\backslash begin\backslash partial\_t\; u\; +\; Au\; -\; r\_t\; u\; =\; 0\; \backslash \backslash u(x,\; T)\; =\; g(x).\backslash end$$Some other options like the American option do not have a fixed expiry. Some options have value at expiry determined by the past stock prices. For example, an average option has a payoff that is not determined by the underlying price at expiry but by the average underlying price over some predetermined period of time. For these, the Feynman–Kac formula does not directly apply.

Quantum mechanics

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)
• Stochastic mechanics

Further reading

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

Notes and References

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