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<p>We consider partial differential equations (PDEs) characterized by an upper barrier that depends on the solution itself and a fixed lower barrier, while accommodating a non-local driver. First, we show a Feynman–Kac representation for the PDE when the driver is local. Specifically, we relate the non-linear Snell envelope, arising from an optimal stopping problem—where the underlying process is the first component in the solution to a stopped backward stochastic differential equation (BSDE) with jumps and a constraint on the jumps process—to a viscosity solution for the PDE. Leveraging this Feynman–Kac representation, we subsequently prove existence and uniqueness of viscosity solutions in the non-local setting by employing a contraction argument. This approach also introduces a novel form of non-linear Snell envelope and expands the probabilistic representation theory for PDEs.</p>

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Optimal stopping of BSDEs with constrained jumps and related double obstacle PDEs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年1月29日

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  • Magnus Perninge

摘要

We consider partial differential equations (PDEs) characterized by an upper barrier that depends on the solution itself and a fixed lower barrier, while accommodating a non-local driver. First, we show a Feynman–Kac representation for the PDE when the driver is local. Specifically, we relate the non-linear Snell envelope, arising from an optimal stopping problem—where the underlying process is the first component in the solution to a stopped backward stochastic differential equation (BSDE) with jumps and a constraint on the jumps process—to a viscosity solution for the PDE. Leveraging this Feynman–Kac representation, we subsequently prove existence and uniqueness of viscosity solutions in the non-local setting by employing a contraction argument. This approach also introduces a novel form of non-linear Snell envelope and expands the probabilistic representation theory for PDEs.