Feynman–Kac formula

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 a real-valued analogy to Feynman's path integrals. The complex case, needed in quantum mechanics, is still an open question.^[2]

The formula offers a method of solving certain partial differential equations by simulating random paths of a stochastic process. Conversely, it can be used to compute an important class of expectations of random processes by deterministic means.

Theorem

Consider the partial differential equation ${\frac {\partial }{\partial t}}u(x,t)+\mu (x,t){\frac {\partial }{\partial x}}u(x,t)+{\tfrac {1}{2}}\sigma ^{2}(x,t){\frac {\partial ^{2}}{\partial x^{2}}}u(x,t)-V(x,t)u(x,t)+f(x,t)=0,$ defined for all $x\in \mathbb {R}$ and $t\in [0,T]$ , subject to the terminal condition $u(x,T)=\psi (x),$ where $\mu ,\sigma ,\psi ,V,f$ are known functions, $T$ is a parameter, and $u:\mathbb {R} \times [0,T]\to \mathbb {R}$ is the unknown function. Then the Feynman–Kac formula expresses $u(x,t)$ as a conditional expectation of a certain random variable:

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

where $X_{t}$ is an Itô process satisfying the stochastic differential equation $dX_{t}=\mu (X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t},$ and $W_{t}$ is the Wiener process (also called Brownian motion).

Intuitive interpretation

Suppose that $X_{t}$ describes the position at time $t$ of a particle that evolves according to the diffusion process $dX_{t}=\mu (X_{t},t)\,dt+\sigma (X_{t},t)\,dW_{t}.$ 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 time $T$ of $\psi (X_{T})$ .

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 away, 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 expected-value 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:

Derivation

Assume that $u(x,t)$ satisfies the PDE:

{\frac {\partial u}{\partial t}}+\mu {\frac {\partial u}{\partial x}}+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}-Vu+f=0

with terminal condition $u(x,T)=\psi (x).$ Further let $X_{t}$ be a stochastic process as defined above.

Let $g(t,s)=e^{-\int _{t}^{s}V(X_{r},r)dr}$ . Its differential satisfies:

dg(t,s)=-V(X_{s},s)g(t,s)\,ds.

Define the stochastic process:

Y_{s}=g(t,s)u(X_{s},s)+\int _{t}^{s}g(t,\tau )f(X_{\tau },\tau )\,d\tau

for $s\in [t,T]$ . At boundary times:

Y_{t}=u(X_{t},t),\quad Y_{T}=g(t,T)\psi (X_{T})+\int _{t}^{T}g(t,\tau )f(X_{\tau },\tau )\,d\tau .

If $Y_{s}$ is a martingale, then we have

u(x,t)=\mathbb {E} [Y_{T}\mid Y_{t}=u(x,t)]=\mathbb {E} [Y_{T}\mid X_{t}=x]

which is what we want to show. So we just need to prove $Y_{s}$ is a martingale.

We assume $X_{s}$ follows the SDE

dX_{s}=\mu (X_{s},s)\,ds+\sigma (X_{s},s)\,dW_{s}.

By Itô's lemma:

du(X_{s},s)=\left({\frac {\partial u}{\partial s}}+\mu {\frac {\partial u}{\partial x}}+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}\right)ds+\sigma {\frac {\partial u}{\partial x}}dW_{s}.

Differentiate $Y_{s}$ :

dY_{s}=d\left[g(t,s)u(X_{s},s)\right]+g(t,s)f(X_{s},s)\,ds.

Expand $d[gu]$ :

d[gu]=g\,du+u\,dg+\underbrace {d[g,u]} _{=0}.

Substitute $dg=-Vg\,ds$ and $du$ :

d[gu]=g\left({\frac {\partial u}{\partial s}}+\mu {\frac {\partial u}{\partial x}}+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}\right)ds

\qquad \quad +\;g\sigma {\frac {\partial u}{\partial x}}\,dW_{s}\;-\;Vgu\,ds.

Add the integral term:

dY_{s}=\left[g\left({\frac {\partial u}{\partial s}}+\mu {\frac {\partial u}{\partial x}}+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}\right)-Vgu+gf\right]ds+g\sigma {\frac {\partial u}{\partial x}}dW_{s}.

For $Y_{s}$ to be a martingale, the drift term must vanish:

{\frac {\partial u}{\partial s}}+\mu {\frac {\partial u}{\partial x}}+{\frac {1}{2}}\sigma ^{2}{\frac {\partial ^{2}u}{\partial x^{2}}}-Vu+f=0.

Remarks

The proof above that a solution must have the given form is essentially that of ^[3] 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 $u:\mathbb {R} ^{N}\times [0,T]\to \mathbb {R}$ becomes:^[4] ${\frac {\partial u}{\partial t}}+\sum _{i=1}^{N}\mu _{i}(x,t){\frac {\partial u}{\partial x_{i}}}+{\frac {1}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}\gamma _{ij}(x,t){\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}-r(x,t)\,u=f(x,t),$ where, $\gamma _{ij}(x,t)=\sum _{k=1}^{N}\sigma _{ik}(x,t)\sigma _{jk}(x,t),$ i.e. $\gamma =\sigma \sigma ^{\mathrm {T} }$ , where $\sigma ^{\mathrm {T} }$ denotes the transpose of $\sigma$ .
More succinctly, letting $A$ be the infinitesimal generator of the diffusion process, ${\frac {\partial u}{\partial t}}+Au-r(x,t)\,u=f(x,t).$
This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.

Original formulation about Wiener functionals

When originally published by Kac in 1949,^[5] the formula was presented as a means 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(t) 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 corresponding diffusion equation. Specifically, under the conditions that $V(x)\geq 0$ , $\mathbb {E} \left[\exp \left(-\int _{0}^{t}V(x(\tau ))\,d\tau \right)\right]=\int _{-\infty }^{\infty }w(x,t)\,dx$ where $w (x, t)$ is the solution to the parabolic partial differential equation ${\frac {\partial w}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-V(x)w$ with initial condition $w (x, 0) = δ (x)$ .

The Feynman–Kac formula can also be used to evaluate functional integrals of a certain form. If, for example, $I=\int f(x(0))\exp \left(-\int _{0}^{t}V(x(\tau ))\,d\tau \right)g(x(t))\,Dx$ where the notation $Dx$ refers to integration taken over all random walks $x:[0,t]\to \mathbb {R}$ , then $I=\int _{-\infty }^{\infty }w(x,t)g(x)\,dx$ where $w (x, t)$ is the solution to ${\frac {\partial w}{\partial t}}={\frac {1}{2}}{\frac {\partial ^{2}w}{\partial x^{2}}}-V(x)w$ with initial condition $w (x, 0) = f (x)$ .

Example

In practical applications, the Feynman–Kac formula can be used with numerical methods like Euler-Maruyama to numerically approximate solutions to partial differential equations. For instance, it can be applied to the convection–diffusion partial differential equation (PDE):

{\frac {\partial }{\partial t}}u(x,t)+b{\frac {\partial }{\partial x}}u(x,t)=-\sigma ^{2}{\frac {\partial ^{2}}{\partial x^{2}}}u(x,t)

Consider the convection–diffusion PDE with parameters $b=1$ , $\sigma =1$ and terminal condition $u(x,T)=e^{-x^{2}}$ with $T=1$ . Then the PDE can be solved analytically:

u(x,t)={\frac {1}{\sqrt {5-4t}}}\exp \left(-{\frac {(1-t+x)^{2}}{5-4t}}\right)

Applying the Feynman-Kac formula, the solution can also be written as the conditional expectation:

u(x,t)=\mathbb {E} \left[e^{-X_{T}^{2}}\,{\bigl |}\,X_{t}=x\right]

where $X$ is an Itô process governed by the SDE $dX_{t}=dt+{\sqrt {2}}dW_{t}$ and $W_{t}$ is a Wiener process. Then using the Euler-Maruyama method, the SDE can be numerically integrated forwards in time from the initial condition $(x_{0},t_{0})$ till the terminal time $T,$ yielding simulated values of $X_{T}$ . To approximate the expectation in the Feynman-Kac method, the simulation is repeated $N$ times. These are often called realizations. The solution is then estimated by the Monte Carlo average

u(x_{0},t_{0})\approx {\frac {1}{N}}\sum _{i=1}^{N}\exp \left(-\left(X_{T}^{(i)}\right)^{2}\right)

The figure below compares the analytical solution with the numerical approximation obtained using the Euler–Maruyama method with $N=1000$ . The left-hand plots show vertical slices of the gradient plot on the right, with each vertical line on the surface corresponding to a colored curve on the left. While the numerical solution exhibits some noise, it closely follows the shape of the exact solution. Increasing the number of simulations $N$ or decreasing the Euler–Maruyama time step improves the accuracy and reduces the variance of the approximation.

This example illustrates how stochastic simulation, enabled by the Feynman–Kac formula and numerical methods like Euler–Maruyama, can approximate PDE solutions. In practice, such stochastic approaches are especially valuable when dealing with high-dimensional systems or complex geometries where traditional PDE solvers become computationally prohibitive. One key advantage of the SDE-based method is its natural parallelism—each simulation, or realization, can be computed independently—making it well-suited for high-performance computing environments. While stochastic simulations introduce variance, this can be mitigated by increasing the number of realizations or refining the time discretization. Thus, stochastic differential equations provide a flexible and scalable alternative to deterministic PDE solvers, particularly in contexts where uncertainty is intrinsic or dimensionality poses a computational barrier. In contrast to traditional PDE solvers, which typically require solving for the entire solution over a grid, this method enables direct computation at specific points in space and time. This targeted approach allows computational resources to be focused on regions of interest, potentially resulting in substantial efficiency gains.

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^[6] and zero-coupon bond prices in affine term structure models.

For example, consider a stock price $S_{t}$ undergoing geometric Brownian motion $dS_{t}=\left(r_{t}dt+\sigma _{t}dW_{t}\right)S_{t}$ where $r_{t}$ is the risk-free interest rate and $\sigma _{t}$ is the volatility. Equivalently, by Itô's lemma, $d\ln S_{t}=\left(r_{t}-{\tfrac {1}{2}}\sigma _{t}^{2}\right)dt+\sigma _{t}\,dW_{t}.$ 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 $u(x,t)=\operatorname {E} \left[e^{-\int _{t}^{T}r_{s}ds}(S_{T}-K)^{+}|\ln S_{t}=\ln x\right].$ Plugging into the Feynman–Kac formula, we obtain the Black–Scholes equation: ${\begin{cases}\partial _{t}u+Au-r_{t}u=0\\u(x,T)=(x-K)^{+}\end{cases}}$ where ${\textstyle A=(r_{t}-\sigma _{t}^{2}/2)\partial _{\ln x}+{\frac {1}{2}}\sigma _{t}^{2}\partial _{\ln x}^{2}=r_{t}x\partial _{x}+{\frac {1}{2}}\sigma _{t}^{2}x^{2}\partial _{x}^{2}.}$ 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 ${\begin{cases}\partial _{t}u+Au-r_{t}u=0\\u(x,T)=g(x).\end{cases}}$ 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.^[7]

References

↑ Kac, Mark (1987). Enigmas of Chance: An Autobiography. University of California Press. pp. 115–16. ISBN 0-520-05986-7.
↑ 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.
↑ "PDE for Finance".
↑ See Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag. ISBN 978-3-642-10044-4.
↑ 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.
↑ 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.
↑ 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.

v t e Richard Feynman
Career	Feynman diagram Feynman–Kac formula Wheeler–Feynman absorber theory Bethe–Feynman formula Hellmann–Feynman theorem Feynman slash notation Feynman parametrization Path integral formulation Parton model Sticky bead argument One-electron universe Quantum cellular automaton Rogers Commission Report Feynman checkerboard Feynman sprinkler
Works	"There's Plenty of Room at the Bottom" (1959) The Feynman Lectures on Physics (1964) The Character of Physical Law (1965) QED: The Strange Theory of Light and Matter (1985) Surely You're Joking, Mr. Feynman! (1985) What Do You Care What Other People Think? (1988) Feynman's Lost Lecture: The Motion of Planets Around the Sun (1997) The Meaning of It All (1999) The Pleasure of Finding Things Out (1999) Perfectly Reasonable Deviations from the Beaten Track (2005)
Family	Joan Feynman (sister) Charles Hirshberg (nephew)
Related	Namesakes One-loop Feynman diagram Quantum Man: Richard Feynman's Life in Science Tuva or Bust! Feynman Prize in Nanotechnology Infinity (1996 film) QED (2001 play)}} The Challenger Disaster (2013 film)

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Airy process Bessel process Birth–death process pure birth Brownian motion Bridge Dyson Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Quasimartingale Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Determinantal Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin theorem Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Kosambi–Karhunen–Loève theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time/Hitting time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category

Theorem

Intuitive interpretation

Partial proof

Derivation

Remarks

Original formulation about Wiener functionals

Example

Applications

Finance

Quantum mechanics

See also

References

Further reading

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