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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.
University of Piraeus
- Department of Financial Management and Banking
Le Pirée, Attica, Greece