Section: New Results
Discretization of stochastic differential equations
In order to compute options prices and hedges by Monte Carlo simulations, it is necessary to discretize the SDE giving the model with respect to time. Usually this is done by using the standard explicit Euler scheme since schemes with higher order of strong convergence involve multiple stochastic integrals which are difficult to simulate.
D. Lamberton and G. Pagès have studied the approximation of the invariant measure for SDEs with a return condition using the Euler scheme with decreasing time-steps  . Their student V. Lemaire has investigated the same scheme when the SDE admits several invariant measures  ,  . J. Guyon  has proved that for uniformly elliptic SDEs with smooth coefficients, a tempered distribution applied to the density of the Euler scheme converges to the same distribution applied to the density of the solution of the SDE at rate given by the size of the time-step. This allows him to give the exact rate of convergence of the Euler scheme for the deltas and gammas of a european option. E. Clément, A. Kohatsu Higa and D. Lamberton have developed a new approach for the error analysis of weak convergence of the Euler scheme based on the linear equation satisfied by the error process  . This method is more general than the usual approach introduced by Talay and Tubaro and provides the means of the weak approximation of stochastic delay equations. In their theses, A. Kbaier  has developed a "statistic Romberg method" for the weak approximation of SDEs and A. Alfonsi has studied various explicit and implicit schemes for the discretization of Cox-Ingersoll-Ross processes for which the standard Euler scheme is not well-posed  . V. Lemaire  has proposed a new explicit scheme with stochastic time-steps to deal with SDEs when the coefficients are locally Lipschitz continuous but fail to be globally Lipschitz continuous.