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Le mardi 20 janvier à 9h00, nous nous retrouverons en salle A709 pour la Matinée du Ceremade. Elle sera suivie d’une réception en Espace 7.
Computing ratios of normalizing constants plays an important role in statistical modeling. Two important examples are hypothesis testing in latent variables models, and model comparison in Bayesian statistics. In both examples, the likelihood ratio and the Bayes factor are defined as the ratio of the normalizing constants of posterior distributions. We propose in this article a novel methodology that estimates this ratio using stochastic approximation principle. Our estimator is consistent and asymptotically Gaussian. Its asymptotic variance is smaller than the one of the popular optimal bridge sampling estimator. Furthermore, it is much more robust to little overlap between the two unnormalized distributions considered. Thanks to its online definition, our procedure can be integrated in an estimation process in latent variables model, and therefore reduce the computational effort. The performances of the estimator are illustrated through a simulation study and compared to two other estimators : the ratio importance sampling and the optimal bridge sampling estimators.
We consider a nonlinear Fokker-Planck equation derived from a Cucker-Smale model for flocking with noise. There is a known phase transition depending on the noise between a regime with a unique stationary solution which is isotropic (symmetry) and a regime with a continuum of polarized stationary solutions (symmetry breaking). If the value of the noise is larger than the threshold value, the solution of the evolution equation converges to the unique radial stationary solution. This solution is linearly unstable in the symmetry-breaking range, while polarized stationary solutions attract all solutions with sufficiently low entropy. We prove that the convergence measured in a weighted $L^2$ norm occurs with an exponential rate and that the average speed also converges with exponential rate to a unique limit which determines a single polarized stationary solution.
We introduce a joint microfoundation for volatility and order flow that is consistent with salient empirical facts. In particular, it identifies the so-called core order flow as the latent driver underlying the main statistical regularities of financial markets. We start from a two-layer Hawkes framework that separates core activity, representing non-reactive trading rooted in long-term information, from reaction activity, capturing endogenous responses to observed trades. In the natural continuous time limit of this model, volatility (and the unsigned order flow) is rough but the signed order flow is a mixture of a smooth process with long-range dependence and a martingale. In addition, under no-arbitrage constraints, the model endogenously connects the exponents governing order flow, volume, market impact and volatility.
The lifted TASEP is a variant of the totally asymmetric exclusion process where at each time-step, instead of trying to move forward a uniformly chosen particle, we try to move forward a marked particle which then may pass the marker to another particle. It was introduced by physicists as a toy model for non-reversible event-chain Monte-Carlo algorithms, which are expected to reach equilibrium faster than reversible dynamics. We will study the behaviour of this system on the integer line by evidencing a connexion with true self-avoiding walks, yielding timescales of the dynamics. This is based on joint work with Clément Erignoux, Werner Krauth, François Simenhaus and Cristina Toninelli.
In this talk, we revisit the resolution of continuous-time principal–agent problems with drift and volatility control, originally addressed by Cvitanić, Possamaï, and Touzi (2018) [1] through dynamic programming and second-order backward stochastic differential equations (2BSDEs), and develop new results in this framework. We begin by introducing an alternative problem in which the principal is allowed todirectly control the quadratic variation of the output process. On the one hand, the resolution of this contractible-volatility problem follows the classical methodology of Sannikov (2008) [2], thus relying on standard (first-order) BSDEs only. On the other hand, we demonstrate that the original contract form introduced in [1] allows the principal to achieve her contractible-volatility value, thereby ensuring both the optimality of this contract form and the equivalence between the original and the alternative problems. At the same time, this alternative approach reveals that the optimality of the original contract form implicitly relies on an additional duality assumption, which was not identified in [1]. This observation motivates the construction of new families of contracts that remain optimal even when the duality assumption fails. Altogether, this line of work both simplifies and strengthens the existing theory of continuous-time principal–agent problems with volatility control, and opens new directions for further extensions and applications in economics and finance. Talk based on joint works with Alessandro Chiusolo, Dylan Possama¨ı, and Nizar Touzi.
[1] J. Cvitanić, D. Possamaï, and N. Touzi. Dynamic programming approach to principal–agent problems. Finance and Stochastics, 22(1):1–37, 2018
[2] Y. Sannikov. A continuous–time version of the principal–agent problem. The Review of Economic Studies, 75(3):957–984, 2008
The planetary problem is an almost integrable newtonian approximation of the Solar System, where light, point planets move around a heavy Sun. In the first approximation, planets describe Keplerian ellipses as if they underwent the only attraction of the Sun. But what is the long term effect of the mutual attraction of planets? Have the slow variations of the elliptic elements ("constants") zero average, or can they accumulate? It turns out that, in the problem with 3 planets (or more, conjecturally), some initial conditions in the 18-dimensional phase space lead to a random dynamical behavior, letting for example a planet flip its orbital plane, or letting the major semi axis of another planet grow by an arbitrary ratio. This contradicts the conclusion of the first stability theorem of Lagrange-Laplace, and positively answers Arnold's 1964 conjecture.
