In this paper we consider the following non-linear stochastic partial differential equation (SPDE): \(\begin{aligned} {\left\{ \begin{array}{ll} \textrm{d}u(s,x)=\sum ^n_{i=1} \mathscr {L}_i u(s,x)\circ \textrm{d}W_i(s)\\ \qquad \qquad +\left( V(x)+\mu \Delta u(s,x)-\frac{1}{2}\vert \nabla u(s,x)\vert ^2\right) \textrm{d}s, \quad & \text {in } (0,T)\times {\mathbb {T}}^n,\\ u(0,x)=u_0(x),\quad \text {on } {\mathbb {T}}^n, \end{array}\right. } \end{aligned}\) where \({\mathbb {T}}^n\) is the n-dimensional torus, the functions \(u_0, V: {\mathbb {T}}^n \rightarrow \mathbb {R}\) 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.
