Feynman–Kac formula

Feynman–Kac Formula

The Feynman–Kac formula is a fundamental theorem in quantum mechanics and mathematical finance, providing a mathematical link between stochastic processes and partial differential equations (PDEs). Named after Richard Feynman and Mark Kac, this formula has become a cornerstone in the field of mathematical physics, particularly in the study of quantum field theory and the valuation of financial derivatives.

Overview

The Feynman–Kac formula offers a method to solve certain types of PDEs, specifically the parabolic PDEs, by converting them into an expectation value of a functional of a stochastic process. This approach is particularly useful in scenarios where direct solutions of the PDE are difficult to obtain. The formula essentially bridges the gap between deterministic processes, described by PDEs, and stochastic processes, described by Brownian motion and martingales.

Mathematical Formulation

Consider a parabolic PDE of the form:

\[ \frac{\partial u}{\partial t} + \frac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\partial x^2} + \mu(x,t) \frac{\partial u}{\partial x} - r(x,t)u = 0 \]

where \(u(x,t)\) is the unknown function to be solved for, \(\sigma(x,t)\) and \(\mu(x,t)\) are known functions representing the volatility and drift of the process, and \(r(x,t)\) is a known function representing the discount rate.

The Feynman–Kac formula states that the solution \(u(x,t)\) can be represented as the expected value of a certain functional of the stochastic process \(X_t\) that satisfies the stochastic differential equation (SDE):

\[ dX_t = \mu(X_t, t)dt + \sigma(X_t, t)dW_t \]

where \(W_t\) is a standard Wiener process. Specifically,

\[ u(x,t) = E\left[ e^{-\int_t^T r(X_s, s) ds} f(X_T) \mid X_t = x \right] \]

for some terminal condition \(f(X_T)\) at time \(T\).

Applications

1. 1. 1. Quantum Mechanics

In quantum mechanics, the Feynman–Kac formula is used to solve the Schrödinger equation for a particle in a potential field. It provides a path integral formulation of quantum mechanics, offering a different perspective from the traditional wave function approach.

1. 1. 1. Financial Mathematics

In financial mathematics, the formula is applied in the pricing of options and other financial derivatives. It allows for the valuation of options by modeling the underlying asset's price as a stochastic process and then computing the expected value of the option's payoff.

See Also

• Stochastic process
• Partial differential equation
• Brownian motion
• Martingale
• Quantum field theory
• Schrödinger equation
• Option