Category Reports

Edouard Jaeck, Delphine Lautier This article proposes an empirical study of the Samuelson effect in electricity markets. Our motivations are twofold. First, although the literature largely assesses the decreasing pattern in the volatilities along the price curve in commodity markets, it has not extensively tested the presence of such a dynamic feature in electricity prices. Second, the analysis of a non-storable commodity enriches the literature on the behavior of commodity prices. Indeed, Read more [...]

Anthony LE CAVIL, Nadia OUDJANE and Francesco RUSSO We propose a nonlinear forward Feynman-Kac type equation, which represents the solution of a nonconservative semilinear parabolic Partial Differential Equations (PDE). We show in particular existence and uniqueness in the first part of the article. The second part is devoted to the construction of a probabilistic particle algorithm and the proof of its convergence. Illustrations of the efficiency of the algorithm are provided by numerical experiments. Read more [...]

Pierre Del Moral, Peng Hu, Nadia Oudjane, Bruno Remillard We analyze the robustness properties of the Snell envelope backward evolution equation for the discrete time optimal stopping problem. We consider a series of approximation schemes, including cut-off type approximations, Euler discretization schemes, interpolation models, quantization tree models, and the Stochastic Mesh method of Broadie-Glasserman. In each situation, we provide non asymptotic convergence estimates, including Lp-mean error Read more [...]

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 [...]