- Art Gallery -

# .

The Feynman–Kac formula, named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (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. Consider the PDE

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

defined for all real x and t in the interval [0, T] , subject to the terminal condition

$$u(x,T)=\psi(x), where \( \mu,\ \sigma,\ \psi, V$$ are known functions, $$\ T$$ is a parameter and $$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 an conditional expectation: \(u(x,t) = E\left[ \int_t^T e^{- \int_t^s V(X_\tau)\, d\tau}f(X_s,s)ds + e^{- \int_t^T V(X_\tau)\, d\tau}\psi(X_T) | X_t=x \right]$$

where X is an Itō process driven by the equation

$$dX = \mu(X,t)\,dt + \sigma(X,t)\,dW,$$

with W(t) is a Wiener process (also called Brownian motion) and the initial condition for X(t) is X(0) = x.

Proof

NOTE: The proof presented below is essentially that of [1] where it is assumed that f(x, t)=0 .

Let u(x, t) be the solution to above PDE. Applying Itō's lemma to the process $$Y(s) = e^{- \int_t^s V(X_\tau)\, d\tau} u(X_s,s)$$ one gets

$$dY = de^{- \int_t^s V(X_\tau)\, d\tau} u(X_s,s) + e^{- \int_t^s V(X_\tau)\, d\tau}\,du(X_s,s) +de^{- \int_t^s V(X_\tau)\, d\tau}du(X_s,s)$$

Since $$de^{- \int_t^s V(X_\tau)\, d\tau} =-V(X_s) e^{- \int_t^s V(X_\tau)\, d\tau} \,ds$$ , the third term is o(dtdu) and can be dropped. Applying Itō's lemma once again to du(X_s,s), it follows that

$$dY=e^{- \int_t^s V(X_\tau)\, d\tau}\,\left(-V(X_s) u(X_s,s) +\mu(X_s,s)\frac{\partial u}{\partial X}+\frac{\partial u}{\partial s}+\frac{1}{2}\sigma^2(X_s,s)\frac{\partial^2 u}{\partial X^2}\right)\,ds \;+e^{- \int_t^s V(X_\tau)\, d\tau}\sigma(X,s)\frac{\partial u}{\partial X}\,dW.$$

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

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

Integrating this equation from t to T , one concludes that

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

Upon taking expectations, conditioned on $$X_t = x , and observing that the right side is an Itō integral, which has expectation zero, it follows that \( E[Y(T)| X_t=x] = E[Y(t)| X_t=x] = u(x,t).$$ The desired result is obtained by observing that

$$E[Y(T)| X_t=x] = E[e^{- \int_t^T V(X_\tau)\, d\tau} u(X_T,T)| X_t=x] = E[e^{- \int_t^T V(X_\tau)\, d\tau} \psi(X_T))| X_t=x]$$

Remarks

The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding PDE for $$u:\mathbb{R}^N\times[0,T]\to\mathbb{R}$$ becomes (see H. Pham book below):

$$\frac{\partial u}{\partial t} + \sum_{i=1}^N \mu_i(x,t)\frac{\partial u}{\partial x_i} + \frac{1}{2} \sum_{i=1}^N\sum_{j=1}^N\gamma_{ij}(x,t) \frac{\partial^2 u}{\partial x_i x_j} -r(x,t) u = f(x,t),$$

where,

$$\gamma_{ij}(x,t) = \sum_{k=1}^N\sigma_{ik}(x,t)\sigma_{jk}(x,t),$$

i.e. $$\gamma =\sigma\,\sigma^\prime$$ , where $$\sigma^\prime$$ denotes the transpose matrix of \sigma).

This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.

When originally published by Kac in 1949,[2] 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

$$e^{-\int_0^t V(x(\tau))\, d\tau}$$

in the case where $$\ x(\tau) is some realization of a diffusion process starting at \( \ x(0) = 0$$ . The Feynman–Kac formula says that this expectation is equivalent to the integral of a solution to a diffusion equation. Specifically, under the conditions that $$\ u V(x) \geq 0,$$

$$E\left( e^{- u \int_0^t V(x(\tau))\, d\tau} \right) = \int_{-\infty}^{\infty} w(x,t)\, dx$$

where $$\ w(x,0) = \delta(x)$$ and

$$\frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2 w}{\partial x^2} - u V(x) w.$$

The Feynman–Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. If

$$I = \int f(x(0)) e^{-u\int_0^t V(x(t))\, dt} g(x(t))\, Dx$$

where the integral is taken over all random walks, then

$$I = \int w(x,t) g(x)\, dx$$

where $$\ w(x,t)$$ is a solution to the parabolic partial differential equation

$$\frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2 w}{\partial x^2} - u V(x) w$$

with initial condition $$\ w(x,0) = f(x).$$

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

References

Simon, Barry (1979). Functional Integration and Quantum Physics. Academic Press.
Pham, Huyên (2009). Continuous-time stochastic control and optimisation with financial applications. Springer-Verlag.

^ http://www.math.nyu.edu/faculty/kohn/pde_finance.html
^ 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.

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