Feynman-Kac Formula
1 Intuition
The Feynman-Kac Formula establishes a link between Parabolic 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.
Think of the PDE as describing how a quantity \(u\) is transported and smoothed backwards in time from the terminal value \(\psi\). The drift \(\mu\) moves information along, and the diffusion \(\sigma\) blurs it, exactly as in the Heat Equation. Instead of solving for \(u\) everywhere at once, we can release a particle at \(x\) at time \(t\), let it wander according to the Itô Process \(X\), and average what it collects along the way:
- \(\psi(X_T)\) is the payoff it collects at the end,
- \(f(X_s,s)\) is a reward collected continuously along the path,
- \(V(X_s,s)\) is a rate at which the collected amount is discounted (or the particle is “killed”).
The average of all of this over many paths is \(u(x,t)\).
2 Feynman-Kac Formula
Consider the PDE
\[ \begin{aligned} u_{t}(x,t) & +\mu(x,t)u_{x}(x,t)+\frac{1}{2}\sigma^2(x,t)u_{xx}(x,t) \\ & -V(x,t)u(x,t)+f(x,t)=0, \end{aligned} \]
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. The Feynman-Kac formula expresses \(u(x,t)\) as a conditional expectation under a probability measure \(\mathbb{Q}\) \[ u(x,t)=\mathbb{E}^\mathbb{Q}\left[ e^{-\int_{t}^{T}V(X_{r},r)\,dr}\,\psi(X_{T}) +\int_{t}^{T}e^{-\int_{t}^{s}V(X_{r},r)\,dr}f(X_{s},s)\,ds \,\middle|\, X_{t}=x \right], \] where \(X\) is an Itô Process satisfying \[ dX_{s}=\mu(X_{s},s)ds+\sigma(X_{s},s)dW_{s}^\mathbb{Q}, \] and \(W^\mathbb{Q}\) is Brownian Motion under \(\mathbb{Q}\).
Remark. The operator \(\mathcal{L}u=\mu u_{x}+\frac{1}{2}\sigma^2u_{xx}\) appearing in the PDE is the Infinitesimal Generator of \(X\), so the PDE can be written compactly as \(u_{t}+\mathcal{L}u-Vu+f=0\). The formula says that the solution of this backward equation is an expectation over the diffusion with that generator.
The statement is valid under standard regularity conditions (e.g. \(\mu,\sigma\) Lipschitz with linear growth, \(u\) of class \(C^{1,2}\) and \(\mathbb{E}\int_t^T\sigma^2u_x^2\,ds<\infty\) so that the stochastic integral below is a true martingale).
Proof. Fix \((x,t)\) and write \(A_{s}=\int_{t}^{s}V(X_{r},r)\,dr\), so that \(dA_{s}=V(X_{s},s)ds\). Apply the Itô Formula to the process \(e^{-A_{s}}u(X_{s},s)\) for \(s\in[t,T]\): \[ \begin{aligned} d\left(e^{-A_{s}}u(X_{s},s)\right) & = e^{-A_{s}}\left(u_{t}+\mu u_{x}+\tfrac{1}{2}\sigma^2u_{xx}-Vu\right)ds +e^{-A_{s}}\sigma u_{x}\,dW_{s}^\mathbb{Q}. \end{aligned} \] The PDE says the bracket equals \(-f(X_{s},s)\), so \[ d\left(e^{-A_{s}}u(X_{s},s)\right) =-e^{-A_{s}}f(X_{s},s)\,ds+e^{-A_{s}}\sigma u_{x}\,dW_{s}^\mathbb{Q}. \] Integrating from \(t\) to \(T\), using \(A_{t}=0\), \(X_{t}=x\) and \(u(X_{T},T)=\psi(X_{T})\), \[ e^{-A_{T}}\psi(X_{T})-u(x,t) =-\int_{t}^{T}e^{-A_{s}}f(X_{s},s)\,ds +\int_{t}^{T}e^{-A_{s}}\sigma u_{x}\,dW_{s}^\mathbb{Q}. \] Take \(\mathbb{E}^\mathbb{Q}[\,\cdot\mid X_{t}=x]\). The Itô integral is a martingale started at zero, so its expectation vanishes, and rearranging gives \[ u(x,t)=\mathbb{E}^\mathbb{Q}\left[e^{-A_{T}}\psi(X_{T}) +\int_{t}^{T}e^{-A_{s}}f(X_{s},s)\,ds\,\middle|\,X_{t}=x\right], \] which is the claim, since \(e^{-A_{s}}=e^{-\int_{t}^{s}V(X_{r},r)dr}\).
In other words, the process \[ M_{s}=e^{-A_{s}}u(X_{s},s)+\int_{t}^{s}e^{-A_{r}}f(X_{r},r)\,dr \] is a martingale, and the formula is just \(\mathbb{E}[M_{T}]=M_{t}\). The proof only shows that a smooth solution must have this form (uniqueness); existence of a smooth solution needs extra conditions on the coefficients.
3 Examples
Take \(X=W\) (so \(\mu=0\), \(\sigma=1\)), \(V=f=0\) and \(\psi(x)=x^2\). The PDE is the backward Heat Equation \[ u_{t}+\tfrac{1}{2}u_{xx}=0,\qquad u(x,T)=x^2. \] The formula gives \[ u(x,t)=\mathbb{E}\left[W_{T}^2\mid W_{t}=x\right] =x^2+(T-t), \] since \(W_{T}=x+(W_{T}-W_{t})\) with \(W_{T}-W_{t}\sim N(0,T-t)\). Check: \(u_{t}=-1\) and \(\tfrac{1}{2}u_{xx}=1\), so the PDE holds.
Take \(X=W\), \(V=0\), \(\psi=0\) and \(f=1\). Then \[ u(x,t)=\mathbb{E}\left[\int_{t}^{T}1\,ds\right]=T-t, \] which indeed solves \(u_{t}+\tfrac{1}{2}u_{xx}+1=0\) with \(u(x,T)=0\).
Take \(X=W\), \(f=0\), \(\psi=1\) and constant \(V=\lambda\). The formula gives \(u(x,t)=e^{-\lambda(T-t)}\), which solves \(u_{t}+\tfrac{1}{2}u_{xx}-\lambda u=0\). The term \(V\) is a pure discount (or killing) rate.
Let \(X=S\) follow a Geometric Brownian Motion under the risk-neutral measure, \(dS=rS\,dt+\sigma S\,dW^\mathbb{Q}\), with \(V=r\) and \(f=0\). The PDE becomes \[ u_{t}+rxu_{x}+\tfrac{1}{2}\sigma^2x^2u_{xx}-ru=0,\qquad u(x,T)=\psi(x), \] which is the Black-Scholes PDE, and the formula reads \[ u(S_{t},t)=e^{-r(T-t)}\,\mathbb{E}^\mathbb{Q}\left[\psi(S_{T})\mid S_{t}\right], \] the familiar statement that an option price is the discounted risk-neutral expectation of its payoff. For a call, \(\psi(x)=(x-K)^+\).
4 Where it is used
- Financial mathematics. The link between the Black-Scholes-Merton Model PDE and risk-neutral pricing, and its extensions to stochastic volatility, interest rate and barrier models (Barrier Options). Here \(V\) is the short rate and \(f\) a dividend or coupon stream.
- Numerical methods. In high dimensions, grid-based PDE solvers fail (curse of dimensionality) but the expectation can be estimated by Monte Carlo: simulate paths of \(X\) (SDE Simulation) and average the payoff. Conversely, low-dimensional expectations can be computed by a PDE solver.
- Quantum mechanics. With \(\mu=0\) and \(\sigma^2=\hbar/m\) (and imaginary time) the PDE is the Schrodinger Equation with potential \(V\); the formula is the Feynman path integral made rigorous by Kac, hence the name.
- Probability and PDE theory. Probabilistic representations of solutions to PDEs (e.g. the Heat Equation), killing rates and survival probabilities, and links to the backward Kolmogorov Equations and the Fokker-Planck Equation.
- Stochastic control. The same argument applied to a controlled process leads to the Hamilton-Jacobi-Bellman equation (Stochastic Control).