Formula relating stochastic processes to partial differential equations
The Feynman窶適ac 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窶適ac 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.
Theorem
Consider the partial differential equation
defined for all and , subject to the terminal condition
Suppose that the position of a particle evolves according to the diffusion process
Let the particle incur "cost" at a rate of at location at time . Let it incur a final cost at .
Also, allow the particle to decay. If the particle is at location at time , then it decays with rate . After the particle has decayed, all future cost is zero.
Then is the expected cost-to-go, if the particle starts at
Partial proof
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. The proof of that lesser result is as follows:
When originally published by Kac in 1949,[6] the Feynman窶適ac 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
in the case where x(マ) is some realization of a diffusion process starting at x(0) = 0. The Feynman窶適ac formula says that this expectation is equivalent to the integral of a solution to a diffusion equation. Specifically, under the conditions that ,
where w(x, 0) = ホエ(x) and
The Feynman窶適ac formula can also be interpreted as a method for evaluating
functional integrals of a certain form. If
where the integral is taken over all
random walks, then
For example, consider a stock price undergoing geometric Brownian motion
where is the risk-free interest rate and is the volatility. Equivalently, by Itテエ's lemma,
Now consider a European call option on an expiring at time with strike . At expiry, it is worth Then, the risk-neutral price of the option, at time and stock price , is
Plugging into the Feynman窶適ac formula, we obtain the Black窶鉄choles equation:
where
More generally, consider an option expiring at time with payoff . The same calculation shows that its price satisfies
Some other options like the
American option do not have a fixed expiry. 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. For these, the Feynman窶適ac formula does not directly apply.
Formula relating stochastic processes to partial differential equations
The Feynman窶適ac 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窶適ac 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.
Theorem
Consider the partial differential equation
defined for all and , subject to the terminal condition
Suppose that the position of a particle evolves according to the diffusion process
Let the particle incur "cost" at a rate of at location at time . Let it incur a final cost at .
Also, allow the particle to decay. If the particle is at location at time , then it decays with rate . After the particle has decayed, all future cost is zero.
Then is the expected cost-to-go, if the particle starts at
Partial proof
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. The proof of that lesser result is as follows:
When originally published by Kac in 1949,[6] the Feynman窶適ac 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
in the case where x(マ) is some realization of a diffusion process starting at x(0) = 0. The Feynman窶適ac formula says that this expectation is equivalent to the integral of a solution to a diffusion equation. Specifically, under the conditions that ,
where w(x, 0) = ホエ(x) and
The Feynman窶適ac formula can also be interpreted as a method for evaluating
functional integrals of a certain form. If
where the integral is taken over all
random walks, then
For example, consider a stock price undergoing geometric Brownian motion
where is the risk-free interest rate and is the volatility. Equivalently, by Itテエ's lemma,
Now consider a European call option on an expiring at time with strike . At expiry, it is worth Then, the risk-neutral price of the option, at time and stock price , is
Plugging into the Feynman窶適ac formula, we obtain the Black窶鉄choles equation:
where
More generally, consider an option expiring at time with payoff . The same calculation shows that its price satisfies
Some other options like the
American option do not have a fixed expiry. 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. For these, the Feynman窶適ac formula does not directly apply.