Benjamin Jourdain

TL;DR: In this article, a probabilistic numerical method for solving a partial differential equation with fractional diffusion and nonlinear drift is presented. But it is based on the Euler model.

TL;DR: In this paper, the convergence properties of the Wang-Landau algorithm were analyzed and an associated central limit theorem was proved for Markov Chain Monte Carlo (MCMC) algorithms.

TL;DR: A modified Ninomiya–Victoir scheme is proposed, which may be strongly coupled with order 1 to the Giles–Szpruch scheme at the finest level of a multilevel Monte Carlo estimator.

Abstract: By Gyongy's theorem, a local and stochastic volatility model is calibrated to the market prices of all call options with positive maturities and strikes if its local volatility function is equal to the ratio of the Dupire local volatility function over the root conditional mean square of the stochastic volatility factor given the spot value. This leads to a SDE nonlinear in the sense of McKean. Particle methods based on a kernel approximation of the conditional expectation, as presented by Guyon and Henry-Labord\`ere (2011), provide an efficient calibration procedure even if some calibration errors may appear when the range of the stochastic volatility factor is very large. But so far, no existence result is available for the SDE nonlinear in the sense of McKean. In the particular case where the local volatility function is equal to the inverse of the root conditional mean square of the stochastic volatility factor multiplied by the spot value given this value and the interest rate is zero, the solution to the SDE is a fake Brownian motion. When the stochastic volatility factor is a constant (over time) random variable taking finitely many values and the range of its square is not too large, we prove existence to the associated Fokker-Planck equation. Thanks to Figalli (2008), we then deduce existence of a new class of fake Brownian motions. We then extend these results to the special case of the LSV model called Regime Switching Local Volatility, where the stochastic volatility factor is a jump process taking finitely many values and with jump intensities depending on the spot level. Under the same condition on the range of its square, we prove existence to the associated Fokker-Planck PDE. We then deduce existence of the calibrated model by extending the results in Figalli (2008).

TL;DR: In this paper, the authors present des modeles aleatoires elementaires and certaines de leurs applications courantes: algorithmes d'optimisation, gestion des approvisionnements, dimensionnement de files d'attente, fiabilite et dimensionnements d'ouvrages.