[Show abstract][Hide abstract] 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.
Full-text · Article · Jun 2011 · Stochastic Processes and their Applications
[Show abstract][Hide abstract] ABSTRACT: 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.
[Show abstract][Hide abstract] ABSTRACT: This paper studies the habit-forming preference problem of maximizing total expected utility from consumption net of the standard of living, a weighted average of past consumption. We describe the effective state space of the corresponding optimal wealth and standard of living processes, identify the associated value function as a generalized utility function, and exploit the interplay between dynamic programming and Feynman-Kac results via the theory of random fields and stochastic partial differential equations (SPDEs). The resulting value random field of the optimization problem satisfies a nonlinear, backward SPDE of parabolic type, widely referred to as the stochastic Hamilton-Jacobi-Bellman equation. The dual value random field is characterized further in terms of a backward parabolic SPDE which is linear. Progressively measurable versions of stochastic feedback formulae for the optimal portfolio and consumption choices are obtained as well.
Preview · Article · Jan 2009 · SIAM Journal on Control and Optimization