Backward Stochastic Burgers PDEs with Random Coefficients

Burgers equation is a quasilinear partial differential equation, proposed in 1930's to model the evolution of turbulent fluid motion, which can be linearized to the heat equation via the celebrated Cole-Hopf transformation. This work introduces and studies in detail general versions of backward stochastic Burgers equation with random coefficients. In case of deterministic coefficients, we obtain a probabilistic representation of the Cole-Hopf transformation by associating the backward Burgers equation with a system of forward-backward stochastic differential equations. Returning to random coefficients, we exploit this representation in order to establish a stochastic version of the Cole-Hopf transformation. This generalized transformation allows us to find solutions to a backward stochastic Burgers equation through a backward stochastic heat equation, subject to additional constraints that reflect the presence of randomness in the coefficients. Finally, an application that illustrates the obtained results is presented to a pricing/hedging problem in a tax regulated financial market with a money market and a stock.