## 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.
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
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### References:

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