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On the Optimal Stopping of a One-dimensional Diffusion

Damien Lamberton 1, 2 Mihail Zervos 3
2 MATHRISK - Mathematical Risk handling
Inria Paris-Rocquencourt, UPEM - Université Paris-Est Marne-la-Vallée, ENPC - École des Ponts ParisTech
Abstract : We consider a one-dimensional diffusion which solves a stochastic differential equation with Borel-measurable coefficients in an open interval. We allow for the endpoints to be inaccessible or absorbing. Given a Borel-measurable function $r$ that is uniformly bounded away from 0, we establish a new analytic representation of the $r$-potential of a continuous additive functional of the diffusion. We also characterize the value function of an optimal stopping problem with general reward function as the unique solution of a variational inequality (in the sense of distributions) with appropriate growth or boundary conditions. Furthermore, we establish several other characterisations of the solution to the optimal stopping problem, including a generalisation of the so-called "principle of smooth fit".
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Damien Lamberton, Mihail Zervos. On the Optimal Stopping of a One-dimensional Diffusion. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2013, 18 (34), pp.1-49. ⟨10.1214/EJP.v18-2182⟩. ⟨hal-00720149⟩

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