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: $$Y_t=\xi -\int_{t\wedge \tau}^{\tau}Y_r|Y_r|^q dr-\int_{t\wedge \tau}^{\tau}Z_r dB_r,\qquad t\geq 0,$$ where $\tau$ is a stopping time, $q$ is a positive constant and $\xi$ is a $\mathcal{F}_{\tau}$-measurable random variable such that $\mathbf{P}(\xi =+\infty)>0$. 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.Comment: Published at http://dx.doi.org/10.1214/009117906000000746 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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

Backward stochastic differential equations with singular terminal condition

In this paper, we are concerned with backward stochastic differential equations (BSDE for short) of the following type: where q is a positive constant and [xi] is a random variable such that . We study the link between these BSDE and the associated Cauchy problem with terminal data g, where g=+[infinity] on a set of positive Lebesgue measure.Backward stochastic differential equation Non-integrable data Viscosity solutions of partial differential equations

Limit behaviour of BSDE with jumps and with singular terminal condition

We study the behaviour at the terminal time T of the minimal solution of a backward stochastic differential equation when the terminal data can take the value + ∞ with positive probability. In a previous paper [T. Kruse and A. Popier, Stoch. Process. Appl. 126 (2016) 2554–2592], we have proved existence of this minimal solution (in a weak sense) in a quite general setting. But two questions arise in this context and were still open: is the solution right continuous with left limits on [0,T]? In other words does the solution have a left limit at time T? The second question is: is this limit equal to the terminal condition? In this paper, under additional conditions on the generator and the terminal condition, we give a positive answer to these two questions

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

Un gisement moustérien dans le Massif Central à Saint-Maurice-sur-Loire (Loire) Prise de date et observations préliminaires

Combier Jean, Larue M., Popier A. Un gisement moustérien dans le Massif Central à Saint-Maurice-sur-Loire (Loire) Prise de date et observations préliminaires. In: Bulletin de la Société préhistorique de France, tome 54, n°11-12, 1957. pp. 763-769

Backward stochastic differential equations with non-Markovian singular terminal values

We solve a class of BSDE with a power function f (y) = y(q), q > 1, driving its drift and with the terminal boundary condition xi = infinity . 1( B(m,r)c )(for which q > 2 is assumed) or xi = infinity . 1B(m,r), where B(m, r) is the ball in the path space C([0,T]) of the underlying Brownian motion centered at the constant function m and radius r. The solution involves the derivation and solution of a related heat equation in which f serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain, the BSDE has continuous sample paths with the prescribed terminal value