Feynman-Kac formula
Encyclopedia
The Feynman–Kac formula, named after Richard Feynman
and Mark Kac
, establishes a link between parabolic partial differential equation
s (PDEs) and stochastic process
es. 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,
defined for all real
and
in the interval 
,
subject to the terminal condition
where
are known functions, 
is a parameter and 
is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as an expectation
:
where
is an Itō process driven by the equation
with
is a Wiener process
(also called Brownian motion
) and the initial condition for
is 
. This expectation can then be approximated using Monte Carlo
or quasi-Monte Carlo method
s.
Let
be the solution to above PDE. Applying Itō's lemma
to the process
one gets
Since
, the third term is 
and can be dropped. Applying Itō's lemma
once again to
, it follows that
The first term contains, in parentheses, the above PDE and is therefore zero. What remains is
Integrating this equation from
to 
, one concludes that
Upon taking expectations, conditioned on
, and observing that the right side is an Itō integral, which has expectation zero, it follows that 
. The desired result is obtained by observing that
in the case where
is some realization of a diffusion process starting at 
. 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
,
where
and
The Feynman–Kac 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 walk
s, then
where
is a solution to the parabolic partial differential equation
with initial condition
.
Richard Feynman
Richard Phillips Feynman was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics...
and Mark Kac
Mark Kac
Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D...
, establishes a link between parabolic partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s (PDEs) and stochastic process
Stochastic process
In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...
es. 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,
defined for all real
and in the interval ,subject to the terminal condition
where
are known functions, is a parameter and is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as an expectationExpected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
:
where
is an Itō process driven by the equationwith
is a Wiener processWiener process
In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...
(also called Brownian motion
Brownian motion
Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...
) and the initial condition for
is . This expectation can then be approximated using Monte CarloMonte Carlo method
Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute their results. Monte Carlo methods are often used in computer simulations of physical and mathematical systems...
or quasi-Monte Carlo method
Quasi-Monte Carlo method
In numerical analysis, a quasi-Monte Carlo method is a method for the computation of an integral that is based on low-discrepancy sequences...
s.
Proof
NOTE: The proof presented below is essentially that of, albeit with more detail.Let
be the solution to above PDE. Applying Itō's lemmaIto's lemma
In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...
to the process
one getsSince
, the third term is and can be dropped. Applying Itō's lemmaIto's lemma
In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...
once again to
, it follows thatThe first term contains, in parentheses, the above PDE and is therefore zero. What remains is
Integrating this equation from
to , one concludes thatUpon taking expectations, conditioned on
, and observing that the right side is an Itō integral, which has expectation zero, it follows that . The desired result is obtained by observing thatRemarks
When originally published by Kac in 1949, 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 functionin the case where
is some realization of a diffusion process starting at . The Feynman–Kac formula says that this expectation is equivalent to the integral of a solution to adiffusion equation. Specifically, under the conditions that
,where
andThe Feynman–Kac 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 walk
Random walk
A random walk, sometimes denoted RW, is a mathematical formalisation of a trajectory that consists of taking successive random steps. For example, the path traced by a molecule as it travels in a liquid or a gas, the search path of a foraging animal, the price of a fluctuating stock and the...
s, then
where
is a solution to the parabolic partial differential equationParabolic partial differential equation
A parabolic partial differential equation is a type of second-order partial differential equation , describing a wide family of problems in science including heat diffusion, ocean acoustic propagation, in physical or mathematical systems with a time variable, and which behave essentially like heat...
with initial condition
.See also
- Itō's lemmaIto's lemmaIn mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...
- Kunita–Watanabe theoremKunita–Watanabe theoremIn stochastic calculus, the Kunita-Watanabe theorem or Kunita-Watanabe inequality is a generalization of the Cauchy Schwarz inequality to integrals of stochastic processes.- Statement of the Theorem :...
- Girsanov theoremGirsanov theoremIn probability theory, the Girsanov theorem describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure...
- Kolmogorov forward equation (also known as Fokker–Planck equation)