##
**Discontinuous solutions of deterministic optimal stopping time problems.**
*(English)*
Zbl 0629.49017

The deterministic optimal stopping time problem with continuous control as well consists of minimizing the cost
\[
J(x,v,\vartheta,\psi)\equiv \int^{\vartheta}_{0}f(y_ x(s),v(s))e^{-\lambda s}ds+\psi (y_ x(\vartheta))e^{-\lambda \vartheta}
\]
over stopping times \(\vartheta\geq 0\) and controls \(V(\cdot)\in L^{\infty}({\mathbb{R}}^+;V)\) subject to the state y evolving according to \(dy_ x(s)+b(y_ x(s),v(s))ds=0\), \(y_ x(0)=x\in {\mathbb{R}}^ N\). The obstacle is \(\psi\) (\(\cdot)\) and the value function for a given obstacle is denoted \(u[\psi](x)=\inf_{\vartheta,v}J(x,v,\vartheta,\psi)\). When \(\psi\) is bounded and continuous, it is well known that u is characterized as the unique viscosity solution of the Bellman variational inequality \(\max \{H(x,u,Du),u-\psi \}=0\) in \({\mathbb{R}}^ N\) where \(H(x,t,p)=\sup \{b(x,v)\cdot p+\lambda t-f(x,v);v\in V\}\) The objective of this paper is to obtain a similar characterization for u if the obstacle is any arbitrary, possibly discontinuous, but bounded function. An application in which the obstacle is discontinuous is the problem of minimum exit time from a domain.

The study proceeds in this paper by using the upper and lower semicontinuous envelopes of the value function.

The study proceeds in this paper by using the upper and lower semicontinuous envelopes of the value function.

Reviewer: E.Barron

### MSC:

49L20 | Dynamic programming in optimal control and differential games |

35F20 | Nonlinear first-order PDEs |

49J40 | Variational inequalities |

60G40 | Stopping times; optimal stopping problems; gambling theory |

35D05 | Existence of generalized solutions of PDE (MSC2000) |

49J45 | Methods involving semicontinuity and convergence; relaxation |

### Keywords:

discontinuous obstacle; deterministic optimal stopping time problem; viscosity solution; Bellman variational inequality; minimum exit time from a domain
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\textit{G. Barles} and \textit{B. Perthame}, RAIRO, Modélisation Math. Anal. Numér. 21, 557--579 (1987; Zbl 0629.49017)

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