Entropy solutions to a strongly degenerate anisotropic convection-diffusion equation with application to utility theory |
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Authors: | AL Amadori |
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Institution: | Istituto per le Applicazioni del Calcolo “Mauro Picone,” Consiglio Nazionale delle Ricerche, Viale del Policlinico 137, I-00161 Roma, Italy |
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Abstract: | We study the deterministic counterpart of a backward-forward stochastic differential utility, which has recently been characterized as the solution to the Cauchy problem related to a PDE of degenerate parabolic type with a conservative first order term. We first establish a local existence result for strong solutions and a continuation principle, and we produce a counterexample showing that, in general, strong solutions fail to be globally smooth. Afterward, we deal with discontinuous entropy solutions, and obtain the global well posedness of the Cauchy problem in this class. Eventually, we select a sufficient condition of geometric type which guarantees the continuity of entropy solutions for special initial data. As a byproduct, we establish the existence of an utility process which is a solution to a backward-forward stochastic differential equation, for a given class of final utilities, which is relevant for financial applications. |
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Keywords: | Degenerate parabolic problems Conservation laws Entropy solutions Financial mathematics Utility models |
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