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October 2018 Diffusion transformations, Black–Scholes equation and optimal stopping
Umut Çetin
Ann. Appl. Probab. 28(5): 3102-3151 (October 2018). DOI: 10.1214/18-AAP1385

Abstract

We develop a new class of path transformations for one-dimensional diffusions that are tailored to alter their long-run behaviour from transient to recurrent or vice versa. This immediately leads to a formula for the distribution of the first exit times of diffusions, which is recently characterised by Karatzas and Ruf [Probab. Theory Related Fields 164 (2016) 1027–1069] as the minimal solution of an appropriate Cauchy problem under more stringent conditions. A particular limit of these transformations also turn out to be instrumental in characterising the stochastic solutions of Cauchy problems defined by the generators of strict local martingales, which are well known for not having unique solutions even when one restricts solutions to have linear growth. Using an appropriate diffusion transformation, we show that the aforementioned stochastic solution can be written in terms of the unique classical solution of an alternative Cauchy problem with suitable boundary conditions. This in particular resolves the long-standing issue of non-uniqueness with the Black–Scholes equations in derivative pricing in the presence of bubbles. Finally, we use these path transformations to propose a unified framework for solving explicitly the optimal stopping problem for one-dimensional diffusions with discounting, which in particular is relevant for the pricing and the computation of optimal exercise boundaries of perpetual American options.

Citation

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Umut Çetin. "Diffusion transformations, Black–Scholes equation and optimal stopping." Ann. Appl. Probab. 28 (5) 3102 - 3151, October 2018. https://doi.org/10.1214/18-AAP1385

Information

Received: 1 February 2017; Revised: 1 January 2018; Published: October 2018
First available in Project Euclid: 28 August 2018

zbMATH: 06974775
MathSciNet: MR3847983
Digital Object Identifier: 10.1214/18-AAP1385

Subjects:
Primary: 60J60
Secondary: 60G40, 60J55, 91G80

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 5 • October 2018
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