On the well-posedness of a Hamilton-Jacobi-Bellman equation with transport noise

Abstract

In this paper we consider the following non-linear stochastic partial differential equation (SPDE): \begin{align*} \begin{cases} \mathrm{d}u(s,x)=\sum^n_{i=1} \mathscr{L}_i u(s,x)\circ \mathrm{d}W_i(s)+\left(V(x)+\mu\Delta u(s,x)-\frac{1}{2}\vert\nabla u(s,x)\vert^2\right)\mathrm{d}s, \quad &\text{in } (0,T)\times \mathbb{T}^n, u(0,x)=u_0(x), & \text{on } \mathbb{T}^n, \end{cases} \end{align*} where $\mathbb{T}^n$ is the $n$-dimensional torus, the functions $u_0, V: \mathbb{T}^n \to \mathbb{R}$ are given and $\{\mathscr{L}_i\}_i$ is a collection of first order linear operators. This can be seen as a Cauchy problem for a Hamilton-Jacobi-Bellman equation with transport noise in any space dimension. We introduce the concept of a strong solution from the realm of PDEs and establish the existence and uniqueness of maximal solutions (strong solutions upto a stopping time). Moreover, for a particular class of $\{\mathscr{L}_i\}_i$ we establish global well-posedness of strong solutions. The proof relies on studying an associated truncated version of the original SPDE and showing its global well-posedness in the class of strong solutions.