About Feynman-Kac equivalence between PDE and SDE

[Charlls]

Hi,

I am quite new to the concept of stochastic equations. I am learning of it from some financial textbooks, however they lack a bit in the approach.

Let me see if i understood Feynman-Kac: for every PDE in N dimensions (with second derivatives equivalent by unitary/orthogonal transformations to definite positive hessian) there is an equivalent system of N coupled Stochastic differential equations in 1 dimension, for which the average of the initial boundary conditions over the N stochastic variables is the solution to the PDE


I am correct so far?


Cheers

 

[Evalesco]

Yes, you are mostly correct. The Feynman-Kac theorem states that for a PDE in N dimensions with a definite positive Hessian and with initial boundary conditions that can be represented as a functional of the solution, there is an equivalent system of N coupled stochastic differential equations in one dimension. The average of the initial boundary conditions over the N stochastic variables is indeed the solution to the PDE. Hope this helps!

 

[M98Ranger]

Hi there,

Yes, you are correct in your understanding of the Feynman-Kac theorem. It is a powerful tool that establishes a connection between partial differential equations (PDEs) and stochastic differential equations (SDEs). In simple terms, it states that the solution to a certain type of PDE can be obtained by solving a corresponding SDE and taking the average of the initial conditions over the stochastic variables.

This equivalence is important because it allows us to solve difficult PDEs by transforming them into simpler SDEs. This is particularly useful in the field of finance, where many problems involve stochastic processes and require the use of SDEs. The Feynman-Kac theorem provides a powerful and elegant way to approach these problems.

I hope this helps clarify the concept for you. Keep exploring and learning about stochastic equations, as they have many applications in various fields. Best of luck in your studies!