Category Rapports

Oct
2016

Contracting Theory with Competitive Interacting Agents

Romuald Elie, Dylan Possamaï In a framework close to the one developed by Holmstrom and Milgrom [44], we study the optimal contracting scheme between a Principal and several Agents. Each hired Agent is in charge of one project, and can make efforts towards managing his own project, as well as impact (positively or negatively) the projects of the other Agents. Considering economic agents in competition with relative performance concerns, we derive the optimal contracts in both first best and moral Read more [...]

Sep
2016

A Non-Intrusive Stratified Resampler for Regression Monte Carlo: Application to Solving Non-Linear Equations

EMMANUEL GOBET, GANG LIU, AND JORGE P. ZUBELLI Abstract. Our goal is to solve certain dynamic programming equations associated to a given Markov chain X, using a regression-based Monte Carlo algorithm. More speci cally, we assume that the model for X is not known in full detail and only a root sample X1; : : : ;XM of such process is available. By a strati cation of the space and a suitable choice of a probability measure , we design a new resampling scheme that allows to compute local regressions Read more [...]

Sep
2016

Pricing American options using martingale bases

J. Lelong In this work, we propose an algorithm to price American options by directly solving the dual minimization problem introduced by Rogers [2002]. Our approach relies on approximating the set of uniformly square integrable martingales by a finite dimensional Wiener chaos expansion. Then, we use a sample average approximation technique to efficiently solve the optimization problem. Unlike all the regression based methods, our method can transparently deal with path dependent options Read more [...]

Sep
2016

Volatility in electricity derivative markets: the Samuelson effect revisited

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 [...]

Sep
2016

Forward Feynman-Kac Type Representation for Semilinear Nonconservative Partial Differential Equations

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 Read more [...]

Juil
2016

On the Robustness of the Snell envelope

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 [...]

Mai
2016

Nonzero-sum Stochastic Differential Games with Impulse Controls and Applications to Retail Energy Markets

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 [...]

Avr
2016

Numerical Approximation of a Cash-Constrained Firm Value with Investment Opportunities

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 [...]

Mar
2016

Branching Diffusion Representation of Semilinear PDEs and Monte Carlo Approximation

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 [...]

Jan
2016

On the Control of the Difference between two Brownian Motions: A Dynamic Copula Approach

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 [...]

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