Nash Equilibrium and the Legendre Transform in Optimal Stopping Games with One Dimensional Diffusions
نویسندگان
چکیده
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 in [19]. 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 in [19] 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 in [18]. 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.
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