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Discounted optimal stopping problems for the maximum process

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Jesper Lund Pedersen
Jesper Lund Pedersen*
Affiliation:
University of Aarhus
*
∗ Postal address: Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, 8000 Aarhus C, Denmark. Email address: [email protected]
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Abstract

The maximality principle [6] is shown to be valid in some examples of discounted optimal stopping problems for the maximum process. In each of these examples explicit formulas for the value functions are derived and the optimal stopping times are displayed. In particular, in the framework of the Black-Scholes model, the fair prices of two lookback options with infinite horizon are calculated. The main aim of the paper is to show that in each considered example the optimal stopping boundary satisfies the maximality principle and that the value function can be determined explicitly.

Keywords

Optimal stoppingdiscountingvalue functionmaximum processdiffusionmaximality principlefree boundary problemprinciple of smooth fitinfinitesimal operator(reflected) Brownian motionBessel processBlack-Scholes modelgeometric Brownian motionlookback option

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory 60H30: Applications of stochastic analysis (to PDE, etc.)
Secondary: 60G44: Martingales with continuous parameter
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 4 , December 2000 , pp. 972 - 983
Copyright
Copyright © by the Applied Probability Trust 2000 

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