[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.
Stochastic Processes and their Applications 06/2011; 123(8). DOI:10.1016/j.spa.2013.03.001 · 1.06 Impact Factor
[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.
71st International Atlantic Economic Conference; 03/2011
[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.
SIAM Journal on Control and Optimization 01/2009; 48(2):481-520. DOI:10.1137/070686998 · 1.46 Impact Factor