Controlling the occupation time of an exponential martingale

Abstract : 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 σ 1 and σ 2 , where 0 < σ 1 < σ 2. The optimal control consists in choosing the minimal volatility σ 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.
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Stefan Ankirchner, Christophette Blanchet-Scalliet, Monique Jeanblanc. Controlling the occupation time of an exponential martingale. Applied Mathematics and Optimization, Springer Verlag (Germany), 2017, 76 (2), pp.415-428. ⟨http://link.springer.com/journal/volumesAndIssues/245⟩. ⟨10.1007/s00245-016-9356-2⟩. ⟨hal-01227899v2⟩

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