Long time behaviour of a stochastic nano particle

In this article, we are interested in the behaviour of a single ferromagnetic mono-domain particle submitted to an external field with a stochastic perturbation. This model is the first step toward the mathematical understanding of thermal effects on a ferromagnet. In a first part, we present the stochastic model and prove that the associated stochastic differential equation is well defined. The second part is dedicated to the study of the long time behaviour of the magnetic moment and in the third part we prove that the stochastic perturbation induces a non reversibility phenomenon. Last, we illustrate these results through numerical simulations of our stochastic model. The main results presented in this article are the rate of convergence of the magnetization toward the unique stable equilibrium of the deterministic model. The second result is a sharp estimate of the hysteresis phenomenon induced by the stochastic perturbation (remember that with no perturbation, the magnetic moment remains constant)

Adaptive optimal allocation in stratified sampling methods

International audienceIn this paper, we propose a stratified sampling algorithm in which the random drawings made in the strata to compute the expectation of interest are also used to adaptively modify the proportion of further drawings in each stratum. These proportions converge to the optimal allocation in terms of variance reduction. And our stratified estimator is asymptotically normal with asymptotic variance equal to the minimal one. Numerical experiments confirm the efficiency of our algorithm

On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients

In this paper, we provide a scheme for simulating one-dimensional processes generated by divergence or non-divergence form operators with discontinuous coefficients. We use a space bijection to transform such a process in another one that behaves locally like a Skew Brownian motion. Indeed the behavior of the Skew Brownian motion can easily be approached by an asymmetric random walk

Asymptotic equivalence of jumps Lévy processes and their discrete counterpart

Shorter version focusing on the statistical analysis of the Lévy measure. A new example has been added.We establish the global asymptotic equivalence between a pure jumps Lévy process $\{X_t\}$ on the time interval $[0,T]$ with unknown Lévy measure $\nu$ belonging to a non-parametric class and the observation of $2m^2$ Poisson independent random variables with parameters linked with the Lévy measure $\nu$. The equivalence result is asymptotic as $m$ tends to infinity. The time $T$ is kept fixed and the sample path is continuously observed. This result justifies the idea that, from a statistical point of view, knowing how many jumps fall into a grid of intervals gives asymptotically the same amount of information as observing $\{X_t\}$

Stochastic expansion for the pricing of call options with discrete dividends

International audienceIn the context of an asset paying affine-type discrete dividends, we present closed analytical approximations for the pricing of European vanilla options in the Black-Scholes model with time-dependent parameters. They are obtained using a stochastic Taylor expansion around a shifted lognormal proxy model. The final formulae are respectively first, second and third order approximations w.r.t. the fixed part of the dividends. Using Cameron-Martin transformations, we provide explicit representations of the correction terms as Greeks in the Black-Scholes model. The use of Malliavin calculus enables us to provide tight error estimates for our approximations. Numerical experiments show that the current approach yields very accurate results, in particular compared to known approximations of [BGS03,VW09], and quicker than the iterated integration procedure of [HHL03] or than the binomial tree method of [VN06]