Feynman–Kac formula

The Feynman–Kac formula named after Richard Feynman and Mark Kac, establishes a link between parabolic partial differential equations (PDEs) and . 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 x in R and t in [0, T], subject to the terminal condition

where μ, σ, ψ, V, f are known functions, T is a parameter and ${\displaystyle u:\mathbb {R} \times [0,T]\to \mathbb {R} }$ is the unknown. Then the Feynman–Kac formula tells us that the solution can be written as a conditional expectation

under the probability measure Q such that X is an Itō process driven by the equation

with W^Q(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

one gets

Since

the third term is ${\displaystyle O(dt\,du)}$ and can be dropped. We also have that

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