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

Francesco Russo and Lukas Wurzer
august, 2017
Publication type:
Paper in peer-reviewed journals
Journal:
Stochastics and Dynamics., vol. 17, pp. 1750030
arXiv:
images/icons/icon_arxiv.png 1407.3218
Keywords :
Backward stochastic differential equations; random terminal time; martingale problem; distributional drift; elliptic partial differential equations.
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.
BibTeX:
@article{Rus-Wur-2017,
    author={Francesco Russo and Lukas Wurzer },
    title={Elliptic PDEs with distributional drift and backward SDEs 
           driven by a càdlàg martingale with random terminal time },
    doi={10.1142/S0219493717500307 },
    journal={Stochastics and Dynamics. },
    year={2017 },
    month={8},
    volume={17 },
    pages={1750030},
}