BSDE with quadratic growth and unbounded terminal value

In this paper, we study the existence of solution to BSDE with quadratic growth and unbounded terminal value. We apply a localization procedure together with a priori bounds. As a byproduct, we apply the same method to extend a result on BSDEs with integrable terminal condition

Simulation of BSDEs by Wiener Chaos Expansion

International audienceWe present an algorithm to solve BSDEs based on Wiener Chaos Expansion and Picard's iterations. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. We use the Malliavin derivative to compute $Z$. Concerning the error, we derive explicit bounds with respect to the number of chaos and the discretization time step. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy

Stability of BSDEs with Random Terminal Time and Homogenization of Semilinear Elliptic PDEs

AbstractIn this paper, we extend the probabilistic method for homogenization of semi-linear parabolic PDEs, developed by Buckdahn, Hu, and Peng to the case of elliptic PDEs. First, we give a stability result for BSDEs with random terminal time which are related to elliptic PDEs as shown in Peng (Stochastics Stochastics Rep.37(1991), 61–74). In the one dimensional case, we also partially relax the monotonicity assumption on the coefficient. Then, we use these stability results for BSDEs with random terminal time to study homogenization of systems of semilinear elliptic PDEs

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

BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

This paper is devoted to the study of the differentiability of solutions to real-valued backward stochastic differential equations (BSDEs for short) with quadratic generators driven by a cylindrical Wiener process. The main novelty of this problem consists in the fact that the gradient equation of a quadratic BSDE has generators which satisfy stochastic Lipschitz conditions involving BMO martingales. We show some applications to the nonlinear Kolmogorov equations

An asymptotical method to estimate the parameters of a deteriorating system under condition-based maintenance

In this paper, we develop a new method to estimate the parameters of a deteriorating system under perfect condition-based maintenance. This method is based on the asymptotical behavior of the system, which is studied by using the renewal process theory. We obtain a Central Limit Theorem (CLT in the following) for the parameters. We compare the accuracy and the speed of the method with the maximum likelihood one (ML method in the following) on different examples

A simple constructive approach to quadratic BSDEs with or without delay

Abstract This paper provides a simple approach for the consideration of quadratic BSDEs with bounded terminal conditions. Using solely probabilistic arguments, we retrieve the existence and uniqueness result derived via PDE-based methods by Kobylanski [11]. This approach is related to the study of quadratic BSDEs presented by Tevzadz