Stopping of functionals with discontinuity at the boundary of an open set

• Published in: Stochastic processes and their applications Vol. 121; no. 10; pp. 2361 - 2392
• Main Authors: Palczewski, Jan, Stettner, Łukasz
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01.10.2011; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Discontinuous functional; Exact sciences and technology; Feller–Markov process; Markov processes; Mathematics; Optimal stopping; Optimal stopping Feller-Markov process Discontinuous functional Penalty method; Penalty method; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Sequential methods; Statistics; Stochastic analysis; Stochastic processes; Optimal stopping; Discontinuous functional; Feller–Markov process; Penalty method; Markov process; Discontinuity; Stopping time; Feller-Markov process; Continuous function; Numerical method; Discontinuous functional: Penalty method; Stochastic process; Algorithm; Numerical approximation; Functional; Value function; Payoff function
• ISSN: 0304-4149; 1879-209X; 1879-209X
• DOI: 10.1016/j.spa.2011.05.013

We explore properties of the value function and existence of optimal stopping times for functionals with discontinuities related to the boundary of an open (possibly unbounded) set O . The stopping horizon is either random, equal to the first exit from the set O , or fixed (finite or infinite). The payoff function is continuous with a possible jump at the boundary of O . Using a generalization of the penalty method, we derive a numerical algorithm for approximation of the value function for general Feller–Markov processes and show existence of optimal or ε -optimal stopping times. ► Optimal stopping of Feller–Markov processes with discontinuous functionals. ► Properties of the value function and existence of optimal stopping times. ► A generalization of the penalty method. ► Numerical algorithm for approximation of the value function and optimal stopping times.