## Linear-quadratic stochastic control with state constraints on finite-time horizon

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 $\mathcal D^c:=([0,T]\times\mathbb R^d)\setminus\mathcal D$, 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 $\mathcal D^c$. Fully explicit formulae are presented in some relevant examples. (Joint work with Erik Ekstr\"om, University of Uppsala, Sweden)

Area: CS29 - Advances in Stochastic Control and System Modeling (Bernardo D'Auria)

Keywords: Stochastic control, state constraints, linear-quadratic problems, Fleming transform

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