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.

In this paper we study a class of infinite horizon fully coupled forward–backward stochastic differential equations (FBSDEs) with random coefficients that are stimulated by various continuous time future expectations models. Under standard Lipschitz and monotonicity conditions and by means of the contraction mapping principle, we establish existence and uniqueness of an adapted solution, and we obtain results regarding the dependence of this solution on the data of the problem. Making further the connection with finite horizon quasilinear backward stochastic partial differential equations via a generalization of the well known four-step-scheme, we are led to the notion of stochastic viscosity solutions. As an application of this framework, we also provide a stochastic maximum principle for the optimal control problem of such FBSDEs, which in the linear-quadratic Markovian case boils down to the solvability of an infinite horizon fully coupled system of forward-backward Ricatti differential equations.

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In this paper we study a class of infinite horizon fully coupled forward-backward stochastic differential equations (FBSDEs), that are stimulated by various continuous time future expectations models with random coefficients. Under standard Lipschitz and monotonicity conditions, and by means of the contraction mapping principle, we establish existence, uniqueness, a comparison property and dependence on a parameter of adapted solutions. Making further the connection with infinite horizon quasilinear backward stochastic partial differential equations (BSPDEs) via a generalization of the well known four-step-scheme, we are led to the notion of stationary stochastic viscosity solutions. A stochastic maximum principle for the optimal control problem of such FBSDEs is also provided as an application to this framework.MSC 2010 subject classifications: Primary 91B70, 60H15; secondary 91G80, 49L25.

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