An adaptive scheme for the approximation of dissipative systems

We propose a new scheme for the long time approximation of a diffusion when the drift vector field is not globally Lipschitz. Under this assumption, regular explicit Euler scheme --with constant or decreasing step-- may explode and implicit Euler scheme are CPU-time expensive. The algorithm we introduce is explicit and we prove that any weak limit of the weighted empirical measures of this scheme is a stationary distribution of the stochastic differential equation. Several examples are presented including gradient dissipative systems and Hamiltonian dissipative systems

Connecting discrete and continuous lookback or hindsight options in exponential L\'evy models

Motivated by the pricing of lookback options in exponential L\'evy models, we study the difference between the continuous and discrete supremum of L\'evy processes. In particular, we extend the results of Broadie et al. (1999) to jump-diffusion models. We also derive bounds for general exponential L\'evy models.Comment: 31 p

Behavior of the Euler scheme with decreasing step in a degenerate situation

The aim of this paper is to study the behavior of the weighted empirical measures of the decreasing step Euler scheme of a one-dimensional diffusion process having multiple invariant measures. This situation can occur when the drift and the diffusion coefficient are vanish simultaneously. As a first step, we give a brief description of the Feller's classification of the one-dimensional process. We recall the concept of attractive and repulsive boundary point and introduce the concept of strongly repulsive point. That allows us to establish a classification of the ergodic behavior of the diffusion. We conclude this section by giving necessary and sufficient conditions on the nature of boundary points in terms of Lyapunov functions. In the second section we use this characterization to study the decreasing step Euler scheme. We give also an numerical example in higher dimension

Human ascariasis: diagnostics update

Soil-transmitted helminths (STHs) infect over one billion people worldwide. Ascariasis may mimic a number of conditions, and individual clinical diagnosis often requires a thorough work-up. Kato-Katz thick smears are the standard detection method for Ascaris and, despite low sensitivity, are often used for mapping and monitoring and evaluation of national control programmes. Although increased sampling (number of stools) and diagnostic (number of examinations per stool) efforts can improve sensitivity, Kato-Katz is less sensitive than other microscopy methods such as FLOTAC®. Antibody-based diagnostics may be a sensitive diagnostic tool; however, their usefulness is limited to assessing transmission in areas aiming for elimination. Molecular diagnostics are highly sensitive and specific, but high costs limit their use to individual diagnosis, drug - efficacy studies and identification of Ascaris suum. Increased investments in research on Ascaris and other STHs are urgently required for the development of diagnostic assays to support efforts to reduce human suffering caused by these infections

When can the two-armed bandit algorithm be trusted?

We investigate the asymptotic behavior of one version of the so-called two-armed bandit algorithm. It is an example of stochastic approximation procedure whose associated ODE has both a repulsive and an attractive equilibrium, at which the procedure is noiseless. We show that if the gain parameter is constant or goes to 0 not too fast, the algorithm does fall in the noiseless repulsive equilibrium with positive probability, whereas it always converges to its natural attractive target when the gain parameter goes to zero at some appropriate rates depending on the parameters of the model. We also elucidate the behavior of the constant step algorithm when the step goes to 0. Finally, we highlight the connection between the algorithm and the Polya urn. An application to asset allocation is briefly described

A duality approach for the weak approximation of stochastic differential equations

In this article we develop a new methodology to prove weak approximation results for general stochastic differential equations. Instead of using a partial differential equation approach as is usually done for diffusions, the approach considered here uses the properties of the linear equation satisfied by the error process. This methodology seems to apply to a large class of processes and we present as an example the weak approximation of stochastic delay equations.Comment: Published at http://dx.doi.org/10.1214/105051606000000060 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org