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
∂ u ∂ t ( x , t ) + μ ( x , t ) ∂ u ∂ x ( x , t ) + 1 2 σ 2 ( x , t ) ∂ 2 u ∂ x 2 ( x , t ) − V ( x , t ) u ( x , t ) + f ( x , t ) = 0 , defined for all x in R and t in [0, T], subject to the terminal condition
u ( x , T ) = ψ ( x ) , where μ, σ, ψ, V, f are known functions, T is a parameter and u : R × [ 0 , T ] → R is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a conditional expectation
u ( x , t ) = E Q [ ∫ t T e − ∫ t r V ( X τ , τ ) d τ f ( X r , r ) d r + e − ∫ t T V ( X τ , τ ) d τ ψ ( X T ) | X t = x ] under the probability measure Q such that X is an Itō process driven by the equation
d X = μ ( X , t ) d t + σ ( X , t ) d W Q , with WQ(t) is a Wiener process (also called Brownian motion) under Q, and the initial condition for X(t) is X(t) = x.
Let u(x, t) be the solution to above PDE. Applying Itō's lemma to the process
Y ( s ) = e − ∫ t s V ( X τ , τ ) d τ u ( X s , s ) + ∫ t s e − ∫ t r V ( X τ , τ ) d τ f ( X r , r ) d r one gets
d Y = d ( e − ∫ t s V ( X τ , τ ) d τ ) u ( X s , s ) + e − ∫ t s V ( X τ , τ ) d τ d u ( X s , s ) + d ( e − ∫ t s V ( X τ , τ ) d τ ) d u ( X s , s ) + d ( ∫ t s e − ∫ t r V ( X τ , τ ) d τ f ( X r , r ) d r ) Since
d ( e − ∫ t s V ( X τ , τ ) d τ ) = − V ( X s , s ) e − ∫ t s V ( X τ , τ ) d τ d s , the third term is O ( d t d u ) and can be dropped. We also have that
d ( ∫ t s e − ∫ t r V ( X τ , τ ) d τ f ( X r , r ) d r ) = e − ∫ t s V ( X τ , τ ) d τ f ( X s , s ) d s . Applying Itō's lemma once again to d u ( X s , s ) , it follows that
d Y = e − ∫ t s V ( X τ , τ ) d τ ( − V ( X s , s ) u ( X s , s ) + f ( X s , s ) + μ ( X s , s ) ∂ u ∂ X + ∂ u ∂ s + 1 2 σ 2 ( X s , s ) ∂ 2 u ∂ X 2 ) d s + e − ∫ t s V ( X τ , τ ) d τ σ ( X , s ) ∂ u ∂ X d W . The first term contains, in parentheses, the above PDE and is therefore zero. What remains is
d Y = e − ∫ t s V ( X τ , τ ) d τ σ ( X , s ) ∂ u ∂ X d W . Integrating this equation from t to T, one concludes that
Y ( T ) − Y ( t ) = ∫ t T e − ∫ t s V ( X τ , τ ) d τ σ ( X , s ) ∂ u ∂ X d W . Upon taking expectations, conditioned on Xt = 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 − ∫ t T V ( X τ , τ ) d τ u ( X T , T ) + ∫ t T e − ∫ t r V ( X τ , τ ) d τ f ( X r , r ) d r | X t = x ] and finally
u ( x , t ) = E [ e − ∫ t T V ( X τ , τ ) d τ ψ ( X T ) + ∫ t T e − ∫ t s V ( X τ , τ ) d τ f ( X s , s ) d s | X t = x ] The proof above is essentially that of with modifications to account for f ( x , t ) .The expectation formula above is also valid for N-dimensional Itô diffusions. The corresponding PDE for u : R N × [ 0 , T ] → R becomes (see H. Pham book below):where,i.e. γ = σσ′, where σ′ denotes the
transpose matrix of σ).
This expectation can then be approximated using Monte Carlo or quasi-Monte Carlo methods.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
x(τ) 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 ) ≥ 0 ,where
w(
x, 0) = δ(
x) andThe Feynman–Kac formula can also be interpreted as a method for evaluating functional integrals of a certain form. Ifwhere the integral is taken over all
random walks, thenwhere
w(
x,
t) is a solution to the parabolic partial differential equationwith initial condition
w(
x, 0) =
f(
x).
