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. 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 f}{\partial t} %2B \mu(x,t) \frac{\partial f}{\partial x} %2B \frac{1}{2} \sigma^2(x,t) \frac{\partial^2 f}{\partial x^2} = V(x,t) f

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

\ f(x,T)=\psi(x),

where \mu,\ \sigma,\ \psi, V are known functions, \ T is a parameter and \ f is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as an expectation:

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

where \ X is an Itō process driven by the equation

dX = \mu(X,t)\,dt %2B \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. This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.

Contents

Proof

NOTE: The proof presented below is essentially that of,[1] albeit with more detail. 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) %2B e^{- \int_t^s V(X_\tau)\, d\tau}\,du(X_s,s) %2Bde^{- \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(dt) 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) %2B\mu(X_s,s)\frac{\partial u}{\partial X}%2B\frac{\partial u}{\partial s}%2B\frac{1}{2}\sigma^2(X_s,s)\frac{\partial^2 u}{\partial X^2}\right)\,ds
\;%2Be^{- \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

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).

See also

References

  1. ^ http://www.math.nyu.edu/faculty/kohn/pde_finance.html
  2. ^ 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.