Stochastic Burgers PDEs with random coefficients and a generalization of the Cole-Hopf transformation

TL;DR

This paper extends the Cole-Hopf transformation to stochastic Burgers equations with random coefficients, linking them to stochastic heat equations and FBSDEs, with applications in finance and control.

Contribution

It generalizes the Cole-Hopf transformation for backward stochastic Burgers equations and establishes new links with FBSDEs and stochastic heat equations.

Findings

Explicit solutions for stochastic Burgers equations are constructed.

The generalized Cole-Hopf transformation's applicability range is characterized.

Stochastic Feynman-Kac representations are derived for solutions.

Abstract

This paper studies forward and backward versions of random Burgers equation (RBE) with stochastic coefficients. Firstly, the celebrated Cole-Hopf transformation reduces the forward RBE to a forward random heat equation (RHE) that can be treated pathwise. Next we provide a connection between the backward Burgers equation and a system of FBSDEs. Exploiting this connection, we derive a generalization of the Cole-Hopf transformation which links the backward RBE with the backward RHE and investigate the range of its applicability. Stochastic Feynman-Kac representations for the solutions are provided. Explicit solutions are constructed and applications in stochastic control and mathematical finance are discussed.