Category Reports

RR-Fime-16-05 Rene Aïd, Matteo Basei, Giorgia Callegaro, Luciano Campi, Tiziano Vargiolu We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Moreover, we give regularity properties of the value function as well as a description of the shape of the control regions. Based on these theoretical Read more [...]

RR-Fime-16-04 Erwan Pierre, Stéphane Villeneuve, Xavier Warin We consider a singular control problem with regime switching that arises in problems of optimal investment decisions of cash-constrained firms. The value function is proved to be the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. Moreover, we give regularity properties of the value function as well as a description of the shape of the control regions. Based on these theoretical results, a numerical deterministic Read more [...]

RR-FiME-16-03 Pierre Henry-Labordère, Nadia Oudjane, Xiaolu Tan, Nizar Touzi, Xavier Warin We provide a representation result of parabolic semi-linear PD-Es, with polynomial nonlinearity, by branching diffusion processes. We extend the classical representation for KPP equations, introduced by Skorokhod [23], Watanabe [27] and McKean [18], by allowing for polynomial nonlinearity in the pair (u;Du), where u is the solution of the PDE with space gradient Du. Similar to the previous literature, our Read more [...]

RR-FiME-16-02 Thomas DESCHATRE We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions Read more [...]

RR-FIME-16-01 Mahamadou Doumbia, Nadia Oudjane, Xavier Warin We develop a pure Monte Carlo method to compute E(g(XT )) where g is a bounded and Lipschitz function and Xt an Ito process. This approach extends the method proposed in [7] to the general multidimensional case with a SDE with varying coefficients. A variance reduction method relying on interacting particle systems is also developped.

RR-FiME-15-04 E. GOBET, J. G. LOPEZ-SALAS, P. TURKEDJIEV, AND C. VAZQUEZ In this paper, we design a novel algorithm based on Least-Squares Monte Carlo (LSMC) in order to approximate the solution of discrete time Backward Stochastic Differential Equations (BSDEs). Our algorithm allows massive parallelization of the computations on multicore devices such as graphics processing units (GPUs). Our approach consists of a novel method of stratification which appears to be crucial for large scale parallelization. Read more [...]

RR-FiME-15-02 Anthony LECAVIL, Nadia OUDJANE and Francesco RUSSO We introduce a new class of nonlinear Stochastic Differential Equations in the sense ofMcKean, related to non conservative nonlinear Partial Differential equations (PDEs). We discuss existence and uniqueness pathwise and in law under various assumptions. We propose an original interacting particle system for which we discuss the propagation of chaos. To this system, we associate a random function which is proved to converge to a solution Read more [...]

RR-FiME-15-05 Laure Durand-Viel and Bertrand Villeneuve We analyze the impact of the length of incomplete contracts on investment and surplus sharing. In the bilateral relationship explored, the seller controls the input and the buyer invests. With two-part tariffs, the length of the contract is irrelevant: the surplus is maximal and goes to the seller. In linear contracts, the seller prefers the shortest contract and the buyer the longest one. Further, the commitment period concentrates the incentives, Read more [...]

RR-FiME-15-06 Céline Grislain-Letrémy and Bertrand Villeneuve Urbanization in areas prone to natural hazards is massive and will grow. Economic analysis offers several tools to contain this phenomenon: insurance pricing in relation to risk, and zoning and building standards in exposed areas. Both approaches are theoretically equivalent, but their applications pose different challenges, and financial incentives were exaggerately reduced in France. In both cases, a more rigorous policy will meet Read more [...]

RR-FiME-15-03 Ivar Ekeland and Maurice Queyranne  In open pit mining, one must dig a pit, that is, excavate the upper layers of ground before reaching the ore. The walls of the pit must satisfy some mechanical constraints, in order not to collapse. The question then arises how to mine the ore optimally, that is, how to …nd the optimal pit. We set up the problem in a continuous (as opposed to discrete) framework, and we show, under weak assumptions, the existence of an optimum pit. For this, we Read more [...]