International audienceWe study the approximation of Ef(X-T) by a Monte Carlo algorithm, where X is the solution of a stochastic differential equation and f is a given function. We introduce a new variance reduction method, which can be viewed as a statistical analogue of Romberg extrapolation method. Namely, we use two Euler schemes with steps delta and delta(beta), 0 < beta < 1. This leads to an algorithm which, for a given level of the statistical error, has a complexity significantly lower than the complexity of the standard Monte Carlo method. We analyze the asymptotic error of this algorithm in the context of general (possibly degenerate) diffusions. In order to find the optimal beta (which turns out to be beta = 1/2), we establish a central limit: type theorem, based on a result of Jacod and Protter for the asymptotic distribution of the error in the Euler scheme. We test our method on various examples. In particular, we adapt it to Asian options. In this setting, we have a CLT and, as a by-product, an explicit expansion of the discretization error

This paper deals with the problem of global parameter estimation in the Cox-Ingersoll-Ross (CIR) model (X t) t≥0. This model is frequently used in finance for example to model the evolution of short-term interest rates or as a dynamic of the volatility in the Heston model. In continuity with a recent work by Ben Alaya and Kebaier [1], we establish new asymptotic results on the maximum likelihood estimator (MLE) associated to the global estimation of the drift parameters of (X t) t≥0. To do so, we need to study first the asymptotic behavior of the quadruplet (log X t , X t , t 0 X s ds, t 0 ds Xs). This allows us to obtain various and original limit theorems on our MLE, with different rates and different types of limit distributions. Our results are obtained for both cases : ergodic and nonergodic diffusion.