Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time - Archive ouverte HAL Access content directly
Journal Articles Stochastics and Dynamics Year : 2017

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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Dates and versions

hal-01023176 , version 1 (11-07-2014)
hal-01023176 , version 2 (02-06-2015)

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Francesco Russo, Lukas Wurzer. Elliptic PDEs with distributional drift and backward SDEs driven by a càdlàg martingale with random terminal time. Stochastics and Dynamics, 2017, 17, pp.1750030. ⟨10.1142/S0219493717500307⟩. ⟨hal-01023176v2⟩
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