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Open AccessArticle

A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier

1
Department of Statistics, Madrid University Carlos III (UC3M), Avenida de la Universidad 30, 28911 Leganés (Madrid), Spain
2
UC3M-BS Institute of Financial Big Data (IFiBiD), Calle Madrid 135, 28903 Getafe (Madrid), Spain
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Department of Mathematics, The Autonomous University of Madrid (UAM), Campus de Cantoblanco, 28049 Madrid, Spain
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Institute of Mathematical Sciences (ICMAT), Campus de Cantoblanco, 28049 Madrid, Spain
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(1), 123; https://doi.org/10.3390/math8010123
Received: 13 November 2019 / Revised: 4 January 2020 / Accepted: 8 January 2020 / Published: 14 January 2020
In this paper, we study the optimal stopping-time problems related to a class of Itô diffusions, modeling for example an investment gain, for which the terminal value is a priori known. This could be the case of an insider trading or of the pinning at expiration of stock options. We give the explicit solution to these optimization problems and in particular we provide a class of processes whose optimal barrier has the same form as the one of the Brownian bridge. These processes may be a possible alternative to the Brownian bridge in practice as they could better model real applications. Moreover, we discuss the existence of a process with a prescribed curve as optimal barrier, for any given (decreasing) curve. This gives a modeling approach for the optimal liquidation time, i.e., the optimal time at which the investor should liquidate a position to maximize the gain. View Full-Text
Keywords: Hamilton–Jacobi–Bellman equation; optimal stopping time; Brownian bridge; liquidation strategy Hamilton–Jacobi–Bellman equation; optimal stopping time; Brownian bridge; liquidation strategy
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D’Auria, B.; Ferriero, A. A Class of Itô Diffusions with Known Terminal Value and Specified Optimal Barrier. Mathematics 2020, 8, 123.

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