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

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

2011-06-10

arxiv.org/abs/1106.1976v1

Mathematics

Probability

Submitted for publication

Scientific paper

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 paper introduces and studies in detail general versions of stochastic Burgers equation with random coefficients, in both forward and backward sense. Concerning the former, the Cole-Hopf transformation still applies and we reduce a forward stochastic Burgers equation to a forward stochastic heat equation that can be treated in a "pathwise" manner. 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. In both settings, forward and backward, stochastic Feynman-Kac formulae are derived for the solutions of the respective stochastic Burgers equations, as well. Finally, applications that illustrate the obtained results are presented to a controllability problem and to a pricing/hedging problem arising from mathematical finance.

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