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Journal of Convex Analysis 22 (2015), No. 1, 177--218
Copyright Heldermann Verlag 2015

Nash Equilibrium and the Legendre Transform in Optimal Stopping Games with One Dimensional Diffusions

Jenny Sexton
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England

We show that the value function of an optimal stopping game driven by a one dimensional diffusion can be characterised using the extension of the Legendre transform introduced by G. Peskir ["A duality principle for the Legendre transform", J. Convex Analysis 19 (2012) 609-630]. It is shown that under certain integrability conditions, a Nash equilibrium of the optimal stopping game can be derived from this extension of the Legendre transform. This result is an analytical complement to the results of Peskir cited above, where the "duality" between a concave biconjugate which is modified to remain below an upper barrier and a convex biconjugate which is modified to remain above a lower barrier is proven by appealing to the probabilistic result of G. Peskir in "Optimal stopping games and Nash equilibrium" [Theory Probab. Appl. 53 (2009) 558-571]. The main contribution of this paper is to show that, for optimal stopping games driven by a one dimensional diffusion, the semiharmonic characterisation of the value function may be proven using only results from convex analysis.

Keywords: Optimal stopping, convex analysis.

MSC: 26A51, 60G40, 91A15, 91G80

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