On the monotone stability approach to BSDEs with jumps: Extensions, concrete criteria and examples

We show a concise extension of the monotone stability approach to backward stochastic differential equations (BSDEs) that are jointly driven by a Brownian motion and a random measure for jumps, which could be of infinite activity with a non-deterministic and time inhomogeneous compensator. The BSDE generator function can be non convex and needs not to satisfy global Lipschitz conditions in the jump integrand. We contribute concrete criteria, that are easy to verify, for results on existence and uniqueness of bounded solutions to BSDEs with jumps, and on comparison and a-priori $L^{\infty}$-bounds. Several examples and counter examples are discussed to shed light on the scope and applicability of different assumptions, and we provide an overview of major applications in finance and optimal control.Comment: 28 pages. Added DOI https://link.springer.com/chapter/10.1007%2F978-3-030-22285-7_1 for final publication, corrected typo (missing gamma) in example 4.1

Quadratic BSDEs with convex generators and unbounded terminal conditions

In a previous work, we proved an existence result for BSDEs with quadratic generators with respect to the variable z and with unbounded terminal conditions. However, no uniqueness result was stated in that work. The main goal of this paper is to fill this gap. In order to obtain a comparison theorem for this kind of BSDEs, we assume that the generator is convex with respect to the variable z. Under this assumption of convexity, we are also able to prove a stability result in the spirit of the a priori estimates stated in the article of N. El Karoui, S. Peng and M.-C. Quenez. With these tools in hands, we can derive the nonlinear Feynman--Kac formula in this context

Second Order Backward Stochastic Differential Equations with Quadratic Growth

We extend the wellposedness results for second order backward stochastic differential equations introduced by Soner, Touzi and Zhang \cite{stz} to the case of a bounded terminal condition and a generator with quadratic growth in the $z$ variable. More precisely, we obtain uniqueness through a representation of the solution inspired by stochastic control theory, and we obtain two existence results using two different methods. In particular, we obtain the existence of the simplest purely quadratic 2BSDEs through the classical exponential change, which allows us to introduce a quasi-sure version of the entropic risk measure. As an application, we also study robust risk-sensitive control problems. Finally, we prove a Feynman-Kac formula and a probabilistic representation for fully nonlinear PDEs in this setting.Comment: 31 page

FBSDEs with time delayed generators:L-P-solutions, differentiability, representation formulas and path regularity

AbstractWe extend the work of Delong and Imkeller (2010) [6,7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general Lp-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in Lp. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L2-path regularity to delay FBSDEs