Some properties of exponential martingales and the problem of optimal stopping. (English. Ukrainian original) Zbl 0955.60055

Mishura, Yu. S.; Ol’tsik, Ya. A.

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.

Reviewer: Yu.V.Kozachenko (Kyïv)

MSC:

60G60	Random fields
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)

Keywords:

martingale; optimal stopping; Wiener process; stochastic differential equation; optimal financial strategy