Controlling the Occupation Time of an Exponential Martingale.
- Resource Type
- Article
- Authors
- Ankirchner, Stefan; Blanchet-Scalliet, Christophette; Jeanblanc, Monique
- Source
- Applied Mathematics & Optimization. Oct2017, Vol. 76 Issue 2, p415-428. 14p.
- Subject
- *MARTINGALES (Mathematics)
*EXPONENTIAL functions
*MATHEMATICAL proofs
*INTEGRALS
*MAXIMUM likelihood statistics
- Language
- ISSN
- 0095-4616
We consider the problem of maximizing the expected amount of time an exponential martingale spends above a constant threshold up to a finite time horizon. We assume that at any time the volatility of the martingale can be chosen to take any value between $$\sigma _1$$ and $$\sigma _2$$ , where $$0 < \sigma _1 < \sigma _2$$ . The optimal control consists in choosing the minimal volatility $$\sigma _1$$ when the process is above the threshold, and the maximal volatility if it is below. We give a rigorous proof using classical verification and provide integral formulas for the maximal expected occupation time above the threshold. [ABSTRACT FROM AUTHOR]