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Exit problem as the generalized solution of Dirichlet problem

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  • Yuecai Han
  • Qingshuo Song
  • Gu Wang

Abstract

This paper investigates sufficient conditions for a Feynman-Kac functional up to an exit time to be the generalized viscosity solution of a Dirichlet problem. The key ingredient is to find out the continuity of exit operator under Skorokhod topology, which reveals the intrinsic connection between overfitting Dirichlet boundary and fine topology. As an application, we establish the sub and supersolutions for a class of non-stationary HJB (Hamilton-Jacobi-Bellman) equations with fractional Laplacian operator via Feynman-Kac functionals associated to $\alpha$-stable processes, which help verify the solvability of the original HJB equation.

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  • Yuecai Han & Qingshuo Song & Gu Wang, 2018. "Exit problem as the generalized solution of Dirichlet problem," Papers 1806.09302, arXiv.org, revised Jan 2019.
    • Handle: RePEc:arx:papers:1806.09302
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    1. Rama Cont & Ekaterina Voltchkova, 2005. "Integro-differential equations for option prices in exponential Lévy models," Finance and Stochastics, Springer, vol. 9(3), pages 299-325, July.
    2. Karatzas, Ioannis & Ocone, Daniel, 2002. "A leavable bounded-velocity stochastic control problem," Stochastic Processes and their Applications, Elsevier, vol. 99(1), pages 31-51, May.
    3. repec:dau:papers:123456789/332 is not listed on IDEAS
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