Uniqueness of Viscosity Solutions of Stochastic Hamilton-Jacobi Equations

Abstract

This paper is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.

Mean field games with common noise and degenerate idiosyncratic noise

“…Equations like (1.8) should typically be satisfied by the value function of optimal control problems with random coefficients adapted to a given filtration, as studied by Peng [27], who proved existence and uniqueness of solutions for smooth data and uniformly elliptic a. They can also be studied with the theory of pathdependent viscosity solutions, a notion which involves taking derivatives on the path space; see for instance Qiu [28] and Qiu and Wei [29]. This theory allows for the treatment of more general equations than (1.8), for example, with non-constant volatility β, and, other than standard growth and regularity assumptions, few conditions are required for the Hamiltonian.…”

Mean Field Game System with a Common Noise

“…To explain this problem, we first consider the HJ equation separately, that is, we look at (1.2), which is a backward SPDE (BSPDE for short) associated with an optimal control problem with random coefficients. It follows from the work of Peng [32] that (1.2) has a unique solution provided that the noise satisfies a nondegeneracy assumption, which, roughly speaking, means that the β in front of ∆u t is greater than the β in front of the terms involving v. More recently, (1.2) was studied by Qiu [33] and Qiu and Wei [34], who introduced a notion of viscosity solution involving derivatives on the path space and proved its existence and uniqueness. The equations studied in the last references are more general than (1.2), in particular, the volatility is not constant, and require few conditions on the Hamiltonian other than the standard growth and regularity.…”

A Minimum Principle for Stochastic Optimal Control Problem with Interval Cost Function

“…Moreover, A Stochastic maximum principle was introduced by Ji [9]. For study of other applications and developments in this area, we refer the readers to [12,13,[15][16][17][18]. As mentioned earlier, some researchers have tried to model some uncertainties in cost function or unpredictability in state space, while considering both of them together, is a less studied subject in the optimal control research yet.…”