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# Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time

Abstract : We introduce a generalized notion of semilinear elliptic partial differential equations where the corresponding second order partial differential operator $L$ has a generalized drift. We investigate existence and uniqueness of generalized solutions of class $C^1$. The generator $L$ is associated with a Markov process $X$ which is the solution of a stochastic differential equation with distributional drift. If the semilinear PDE admits boundary conditions, its solution is naturally associated with a backward stochastic differential equation (BSDE) with random terminal time, where the forward process is $X$. Since $X$ is a weak solution of the forward SDE, the BSDE appears naturally to be driven by a martingale. In the paper we also discuss the uniqueness of a BSDE with random terminal time when the driving process is a general càdlàg martingale.
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Preprints, Working Papers, ...
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https://hal.inria.fr/hal-01023176
Contributor : Francesco Russo Connect in order to contact the contributor
Submitted on : Tuesday, June 2, 2015 - 10:58:39 AM
Last modification on : Wednesday, May 11, 2022 - 12:06:06 PM
Long-term archiving on: : Tuesday, September 15, 2015 - 9:26:14 AM

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### Identifiers

• HAL Id : hal-01023176, version 2
• ARXIV : 1407.3218

### Citation

Francesco Russo, Lukas Wurzer. Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time. 2015. ⟨hal-01023176v2⟩

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