Existence and Uniqueness of Stochastic PDEs associated with the Forward Equations: An Approach using Alternate Norms

TL;DR

This paper proves existence and uniqueness of solutions for certain stochastic PDEs and related PDEs using a new monotonicity inequality, also providing a stochastic representation of the solutions.

Contribution

It introduces a novel monotonicity inequality for operators, enabling the proof of strong solution existence and uniqueness for a class of stochastic PDEs and PDEs.

Findings

Established a monotonicity inequality for operator pairs

Proved existence and uniqueness of strong solutions

Provided a stochastic representation for solutions

Abstract

We consider stochastic PDEs \[dY_t = L(Y_t)\, dt + A(Y_t).\, dB_t, t > 0\] and associated PDEs \[du_t = L u_t\, dt, t > 0\] with regular initial conditions. Here, $L$ and $A$ are certain partial differential operators involving multiplication by smooth functions and are of the order two and one respectively, and in special cases are associated with finite dimensional diffusion processes. This PDE also includes Kolmogorov's Forward Equation (Fokker-Planck Equation) as a special case. We first prove a Monotonicity inequality for the pair $(L, A)$ and using this inequality, we obtain the existence and uniqueness of strong solutions to the Stochastic PDE and the PDE. In addition, a stochastic representation for the solution to the PDE is also established.