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

Hamilton Jacobi Bellman equations in infinite dimensions with quadratic and superquadratic Hamiltonian

We consider Hamilton Jacobi Bellman equations in an inifinite dimensional Hilbert space, with quadratic (respectively superquadratic) hamiltonian and with continuous (respectively lipschitz continuous) final conditions. This allows to study stochastic optimal control problems for suitable controlled Ornstein Uhlenbeck process with unbounded control processes

Quadratic BSDEs driven by a continuous martingale and application to utility maximization problem

In this paper, we study a class of quadratic Backward Stochastic Differential Equations (BSDEs) which arises naturally when studying the problem of utility maximization with portfolio constraints. We first establish existence and uniqueness results for such BSDEs and then, we give an application to the utility maximization problem. Three cases of utility functions will be discussed: the exponential, power and logarithmic ones

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

Adaptive importance sampling with forward-backward stochastic differential equations

We describe an adaptive importance sampling algorithm for rare events that is based on a dual stochastic control formulation of a path sampling problem. Specifically, we focus on path functionals that have the form of cumulate generating functions, which appear relevant in the context of, e.g.~molecular dynamics, and we discuss the construction of an optimal (i.e. minimum variance) change of measure by solving a stochastic control problem. We show that the associated semi-linear dynamic programming equations admit an equivalent formulation as a system of uncoupled forward-backward stochastic differential equations that can be solved efficiently by a least squares Monte Carlo algorithm. We illustrate the approach with a suitable numerical example and discuss the extension of the algorithm to high-dimensional systems

Influence of Sulphur and Phophorus on the Hot Deformation of Fe-Cr 13% High Purity Steel

A series of Fe-Cr13%-C high purity steels containing increasing volume fractions of Sulphur (30, 60 and 100ppm) and Phosphorus (30, 60 and 100ppm) were prepared in order to study their hot deformation properties by tensile tests at various strain rates (10-1s-1 to 10-4s-1) and at temperatures from 700°C to 1100°C. It is observed that the hot ductility is lowered at 1000°C with the addition of sulphur. However, this decrease is relatively small (about 30% for l00ppm of sulphur) and quite similar for all additions of sulphur. When phosphorus is added, the embrittlement is along the whole deformed specimen. The usual criteria of ductility by parameter Z do not seem to be sufficient to describe this embrittlement

EFFECT OF S AND C SEGREGATION ON GRAIN BOUNDARY PROPERTIES IN ULTRA HIGH PURITY IRON

L'effet de ségrégation de S et C sur les propriétés des joints de grains (JDG) dans le fer de très haute pureté a été examiné du point de vue de leur rôle dans la rupture à chaud. A basse teneur en S la fragilité interganulaire est controlée par la présence des precipités de AIN. L'addition de C améliore la ductilité détériorée par la ségrégation du S. Le carbon change le mécanisme de la rupture. La présence de C et de S décroit la diffusivité des JDG et empêche leur migration. Cependant l'addition de C attenue l'effet particulièrement fort du S sur la migration des JDG.Ultra high purity iron doped with S and C was investigated from the point of view of the influence of S and C segregation on grain boundary (GB) properties related to high temperature fracture. At low S contents intergranular fracture was found to be controlled by GB precipitates presence (AIN). C addition improves high temperature ductility deteriorated by S segregation. C and S effect on grain boundary properties seems to be complex. Both elements were found to decrease GB diffusivity and inhibit GB migration. S effect on migration is particularly strong, but it is healed by C presence. C addition seems to change fracture mechanism from mechanical decohesion on GB towards diffusional voids formation and growth. This effect combined with C influence on migration and cohesion are favourable for good high temperature ductility