A class of stochastic control problems with state constraints

We obtain a probabilistic solution to linear-quadratic optimal control problems with state constraints. Given a closed set $\mathcal{D}\subseteq [0,T]\times\mathbb{R}^d$, a diffusion $X$ in $\mathbb{R}^d$ must be linearly controlled in order to keep the time-space process $(t,X_t)$ inside the set $\mathcal{C}:=([0,T]\times\mathbb{R}^d)\setminus\mathcal{D}$, while at the same time minimising an expected cost that depends on the state $(t,X_t)$ and is quadratic in the speed of the control exerted. We find a probabilistic representation for the value function and an optimal control under a set of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of the set $\mathcal{D}$. The optimally controlled dynamics is in strong form, in the sense that it is adapted to the filtration generated by the driving Brownian motion. Fully explicit formulae are presented in some relevant examples.