Weak solution for a class of fully nonlinear stochastic Hamilton–Jacobi–Bellman equations

Abstract

This paper is concerned with the stochastic Hamilton-Jacobi-Bellman equation with controlled leading coefficients, which is a type of fully nonlinear backward stochastic partial differential equation (BSPDE for short). In order to formulate the weak solution for such kind of BSPDEs, a class of regular random parabolic potentials are introduced in the backward stochastic framework. The existence and uniqueness of weak solution is proved, and for the partially non-Markovian case, we obtain the associated gradient estimate. As a byproduct, the existence and uniqueness of solution for a class of degenerate reflected BSPDEs is discussed as well. (2010): 60H15, 49L20, 93E20, 35D30 Mathematics Subject Classification

Pseudo-Markovian viscosity solutions of fully nonlinear degenerate PPDEs

“…When σ does not depend on α, namely there is no diffusion control, [15] established its wellposedness in Sobolev sense. The recent work [23] extended the result to the general case with diffusion control, also in terms of Sobolev solutions.…”

“…We shall characterize the value function as the unique viscosity solution to this degenerate PPDE. We note that in the recent work Qiu [23] viewed the stochastic HJB equation as a backward SPDE and proved its wellposedness in the sense of Sobolev solutions. The second example is the path dependent Isaccs equations, induced from the path dependent zero sum game as in Pham & Zhang [22].…”

“…Our goal is to characterize u 0 as the unique pseudo Markovian viscosity solution of (6.4). We remark that, the Sobolev theory in [15,23] requires the special structure of HJB equation, and may not be easy to extend to more general cases like Isaacs equations induced from games. Our viscosity solution characterization, however, can be easily extended as we will see in next section.…”

Pricing Options Under Rough Volatility with Backward SPDEs

“…for which the solution is a triple (u, ψ, µ) with µ being a regular random radon measure. A solution theory may be developed by generalizing the regular stochastic potential and capacity theory in [Qiu17,QW14]; nevertheless, we would not seek such a generality in this paper, in order to put more efforts in the numerical approximations.…”

The Relationship between Maximum Principle and Dynamic Programming Principle for Stochastic Recursive Control Problem with Random Coefficients

“…As in the deterministic case, the existence of the solution for stochastic HJB equation is a very hard problem. The solvability has only been proved for a few cases, see [22,24,18,19,28] for instance. One contribution of our paper is to show that the value function of the recursive optimal control problem will be the classical solution of stochastic HJB equation, if the needed regularity is satisfied.…”