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Some properties of exponential martingales and the problem of optimal stopping. (English. Ukrainian original) Zbl 0955.60055
Theory Probab. Math. Stat. 60, 159-164 (2000); translation from Teor. Jmovirn. Mat. Stat. 60, 143-148 (1999).
The authors study the optimal stopping of stochastic processes $$Y_{t}=Z_{t} A_{t}$$, $$t\in [0,T],$$ where $$A_{t}$$ is a nonnegative random process and $$Z_{t}=C\cdot\exp \bigl\{ M_{t}-{1\over 2}\langle M\rangle_{t}\bigl\}$$, $$C>0.$$ The process $$M_{t}$$ is a bounded mean oscillation martingale. That is, $$M_{t}$$ is a local martingale and $\sup\limits_{t\in [0,T]} \biggl\|E(\langle M\rangle_{T}- \langle M\rangle_{t}/{\mathcal F}_{t})^{1/2}\biggr\|_{\infty}<\infty.$ The main result of this paper is the following theorem: Let $$A_{t}$$ be a nonnegative random process and let exist the stopping $$\sigma \in [0,T]$$ such that $$A_{\sigma}\geq A_{\tau}$$ for any stopping $$\tau \in [0,T].$$ Then $$\sigma$$ is an optimal stopping of the process $$Y_{t}.$$ This result is used for investigating the optimal financial strategy with wealth process governed by a stochastic differential equation.
##### MSC:
 60G60 Random fields 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)