arXiv subjects

A. Popier

Publications and source records attributed to A. Popier.

3 recordsLinked to original sources

arXiv2015

BSDEs with monotone generator driven by Brownian and Poisson noises in a general filtration

We analyze multidimensional BSDEs in a filtration that supports a Brownian motion and a Poisson random measure. Under a monotonicity assumption on the driver, the paper extends several results from the literature. We establish existence and uniqueness of solutions in $L^p$ provided that the generator and the terminal condition satisfy appropriate integrability conditions. The analysis is first carried out under a deterministic time horizon, and then generalized to random time horizons given by a stopping time with respect to the underlying filtration. Moreover, we provide a comparison principle in dimension one.

math.PR↗

arXiv2015

Stochastic partial differential equations with singular terminal condition

In this paper, we first prove existence and uniqueness of the solution of a backward doubly stochastic differential equation (BDSDE) and of the related stochastic partial differential equation (SPDE) under monotonicity assumption on the generator. Then we study the case where the terminal data is singular, in the sense that it can be equal to +$\infty$ on a set of positive measure. In this setting we show that there exists a minimal solution, both for the BDSDE and for the SPDE. Note that solution of the SPDE means weak solution in the Sobolev sense.

math.PR↗

arXiv2007

Backward stochastic differential equations with random stopping time and singular final condition

In this paper we are concerned with one-dimensional backward stochastic differential equations (BSDE in short) of the following type: \[Y_t=ξ-\int_{t\wedge τ}^τY_r|Y_r|^q dr-\int_{t\wedge τ}^τZ_r dB_r,\qquad t\geq 0,\] where $τ$ is a stopping time, $q$ is a positive constant and $ξ$ is a $\mathcal{F}_τ$-measurable random variable such that $\mathbf{P}(ξ=+\infty)>0$. We study the link between these BSDE and the Dirichlet problem on a domain $D\subset \mathbb{R}^d$ and with boundary condition $g$, with $g=+\infty$ on a set of positive Lebesgue measure. We also extend our results for more general BSDE.

math.PR↗