Controlled Ordinary Differential Equations with Random Path-Dependent Coefficients and Stochastic Path-Dependent Hamilton-Jacobi Equations

Abstract

This paper is devoted to the stochastic optimal control problem of ordinary differential equations allowing for both path-dependence and measurable randomness. As opposed to the deterministic path-dependent cases, the value function turns out to be a random field on the path space and it is characterized by a stochastic path-dependent Hamilton-Jacobi (SPHJ) equation. A notion of viscosity solution is proposed and the value function is proved to be the unique viscosity solution to the associated SPHJ equation.

Stochastic Path-Dependent Hamilton-Jacobi-Bellman Equations and Controlled Stochastic Differential Equations with Random Path-Dependent Coefficients

“…Finally, we compare the present work with the accompanying one [24]. In fact, when σ(ω, t, x t , θ(t))(≡ σ(ω, t)) is path-invariant and uncontrolled in (1.2), we may take X(t) = X(t) − ξ(t) with ξ(t) = t 0 σ(s) dW (s) for t ∈ [0, T ], and then the optimization (1.1)-(1.2) is equivalent to the following one:…”

“…The paper [24] is devoted to the control problem (1.8)-(1.9) and the existence and uniqueness of viscosity solution is addressed for the associated stochastic path-dependent Hamilton-Jacobi equation which, we note, is first-order. In contrast, our SPHJB (1.5) is second-order, and this leads to the different methods and contents for the viscosity solution theory.…”

“…In contrast, our SPHJB (1.5) is second-order, and this leads to the different methods and contents for the viscosity solution theory. For instance, to deal with the lacking of local compactness of the path space, subspaces of Lipschitz functions are used for treating viscosity solutions in [24], while we use subspaces of Hölder functions herein because of the controlled stochastic integrals in the state process (1.2). Two Lipschitz functions over two successive time intervals with a joint point and an identical Lipschitz constant may be pieced together as a new Lipschitz function with the same Lipschitz constant, which, however, does not hold for Hölder functions.…”