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 the real-valued case of Feynman's path integrals.
The complex case, which occurs when a particle's spin is included, is still an open question.
[2] It offers a method of solving certain partial differential equations 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 partial differential equation
μ , σ , ψ ,
as a conditional expectation under the probability measure
a Wiener process (also called Brownian motion) under
of a particle evolves according to the diffusion process
Let the particle incur "cost" at a rate of
After the particle has decayed, all future cost is zero.
is the expected cost-to-go, if the particle starts at
A proof that the above 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.
be the solution to the above partial differential equation.
The first term contains, in parentheses, the above partial differential equation and is therefore zero.
, and observing that the right side is an Itô integral, which has expectation zero,[3] it follows that:
The desired result is obtained by observing that:
The Feynman–Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form.
where w(x, t) is a solution to the parabolic partial differential equation
In quantitative finance, the Feynman–Kac formula is used to efficiently calculate solutions to the Black–Scholes equation to price options on stocks[7] and zero-coupon bond prices in affine term structure models.
undergoing geometric Brownian motion
Then, the risk-neutral price of the option, at time
Plugging into the Feynman–Kac formula, we obtain the Black–Scholes equation:
More generally, consider an option expiring at time
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.
In quantum chemistry, it is used to solve the Schrödinger equation with the pure diffusion Monte Carlo method.