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 $mathcal C:=([0,T]\times\mathbb R^d)\setminus\mathcal D$, while at the same time minimising an expected cost that depends on the state $(t,X_t)$ and it is quadratic in the speed of the control exerted. We find an explicit probabilistic representation for the value function and the optimal control under a set of mild sufficient conditions concerning the coefficients of the underlying dynamics and the regularity of the set $\mathcal C$. Fully explicit formulae are presented in some relevant examples.

(Joint work with Erik Ekstr\"om, University of Uppsala, Sweden)