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

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)+μΔu(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 $\{{\mathcal{L}}_i{\}}_i$ 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 $\{{\mathcal{L}}_i{\}}_i$ 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.