## Minimizing or maximizing the expected time to reach zero.(English)Zbl 0613.93067

The authors consider stochastic control systems described by the Ito differential equation $$dx(t)=a(t)\cdot dt+b(t)\cdot dw(t)$$ with nonanticipative controls a(t) and b(t) to be chosen in an admissible set. Deriving an improved verification lemma of its own interest, they solve the problems of finding optimal controls which minimize or maximize the expected time to reach the zero state. They also discuss an application to a portfolio problem.
Reviewer: A.Kistner

### MSC:

 93E20 Optimal stochastic control 60G40 Stopping times; optimal stopping problems; gambling theory 60J60 Diffusion processes 49K45 Optimality conditions for problems involving randomness 60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) 91G80 Financial applications of other theories
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