Publications

2015

• Analysis and simulation of rare events for SPDEs
  
  • Bréhier Charles-Edouard
  • Gazeau Maxime
  • Goudenège Ludovic
  • Rousset Mathias
  ESAIM: Proceedings and Surveys, EDP Sciences, 2015, 48, pp.364-384. In this work, we consider the numerical estimation of the probability for a stochastic process to hit a set B before reaching another set A. This event is assumed to be rare. We consider reactive trajectories of the stochastic Allen-Cahn partial differential evolution equation (with double well potential) in dimension 1. Reactive trajectories are defined as the probability distribution of the trajectories of a stochastic process, conditioned by the event of hitting B before A. We investigate the use of the so-called Adaptive Multilevel Splitting algorithm in order to estimate the rare event and simulate reactive trajectories. This algorithm uses a reaction coordinate (a real valued function of state space defining level sets), and is based on (i) the selection, among several replicas of the system having hit A before B, of those with maximal reaction coordinate; (ii) iteration of the latter step. We choose for the reaction coordinate the average magnetization, and for B the minimum of the well opposite to the initial condition. We discuss the context, prove that the algorithm has a sense in the usual functional setting, and numerically test the method (estimation of rare event, and transition state sampling). (10.1051/proc/201448017)
  DOI : 10.1051/proc/201448017
• Quadratic BSDEs with jumps: a fixed-point approach
  
  • Kazi-Tani Mohamed Nabil
  • Possamaï Dylan
  • Zhou Chao
  Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2015, 20 (66), pp.1-28. (10.1214/EJP.v20-3363)
  DOI : 10.1214/EJP.v20-3363
• Time-Optimal Synthesis for Three Relevant Problems: The Brockett Integrator, the Grushin Plane and the Martinet Distribution
  
  • Barilari Davide
  • Boscain Ugo
  • Le Donne Enrico
  • Sigalotti Mario
  , 2015.
• Robust utility maximization in nondominated models with 2BSDE: the uncertain volatility model
  
  • Matoussi Anis
  • Possamaï Dylan
  • Zhou Chao
  Mathematical Finance, Wiley, 2015, 25 (2), pp.258-287. The problem of robust utility maximization in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose to study this problem in the framework of second-order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models of [2]. (10.1111/mafi.12031)
  DOI : 10.1111/mafi.12031
• Accelerated Share Repurchase: pricing and execution strategy
  
  • Guéant Olivier
  • Pu Jiang
  • Guillaume Royer
  International Journal of Theoretical and Applied Finance, World Scientific Publishing, 2015, 18 (3). In this article, we consider a specific optimal execution problem associated to accelerated share repurchase contracts. When firms want to repurchase their own shares, they often enter such a contract with a bank. The bank buys the shares for the firm and is paid the average market price over the execution period, the length of the period being decided upon by the bank during the buying process. Mathematically, the problem is new and related to both option pricing (Asian and Bermudan options) and optimal execution. We provide a model, along with associated numerical methods, to determine the optimal stopping time and the optimal buying strategy of the bank. (10.1142/S0219024915500193)
  DOI : 10.1142/S0219024915500193
• Zubov's equation for state-constrained perturbed nonlinear systems
  
  • Grüne Lars
  • Zidani Hasnaa
  Mathematical Control and Related Fields, AIMS, 2015, 5 (1), pp.55-71. The paper gives a characterization of the uniform robust domain of attraction for a nite non-linear controlled system subject to perturbations and state constraints. We extend the Zubov approach to characterize this domain by means of the value function of a suitable in nite horizon state-constrained control problem which at the same time is a Lyapunov function for the system. We provide associated Hamilton-Jacobi-Bellman equations and prove existence and uniqueness of the solutions of these generalized Zubov equations. (10.3934/mcrf.2015.5.55)
  DOI : 10.3934/mcrf.2015.5.55
• Derivation of nonlinear shell models combining shear and flexure: application to biological membranes
  
  • Pantz Olivier
  • Trabelsi Karim
  Mathematics and Mechanics of Complex Systems, International Research Center for Mathematics & Mechanics of Complex Systems (M&MoCS),University of L’Aquila in Italy, 2015, 3 (2), pp.101--138. Biological membranes are often idealized as incompressible elastic surfaces whose strain energy only depends on their mean curvature and pos-sibly on their shear. We show that this type of model can be derived using a formal asymptotic method by considering biological membranes to be thin, strongly anisotropic, elastic, locally homogeneous bodies.