## Team : mathfi

## Section: New Results

## Discretization of stochastic differential equations

**Participants:**E. Clément, A. Kohatsu Higa, D. Lamberton, J. Guyon, A. Alfonsi, B. Jourdain.

E.Clément, A. Kohatsu Higa and D.Lamberton develop a new approach for the error analysis of weak convergence of the Euler scheme, which enables them to obtain new results on the approximation of stochastic differential equations with memory. Their approach uses the properties of the linear equation satisfied by the error process instead of the partial differential equation derived from the Markov property of the process. It seems to be more general than the usual approach and gives results for the weak approximation of stochastic delay equations. A paper has been submitted and extensions are studied.

In his thesis, A. Kbaier develops a "statistic Romberg method" for weak approximation of stochastic differential equations. This method is especially efficient for the computation of Asian options.

J. Guyon, PhD student of B. Lapeyre and J.F. Delmas has studied how fast the Euler scheme X_{T}^{n}
with time-step t = T/n converges in law to the original random variable X_{T}. More precisely, he has
looked for which class of functions f the approximate expectation
converges to
with speed t. So far, (X_{t}, t0) has been a smooth
-valued diffusion.
When f is smooth, it is known from D. Talay and L. Tubaro that

Using Malliavin calculus, V. Bally and D. Talay
have shown that this development remains true when f is only
mesurable and bounded, in the case when the diffusion X is uniformly hypoelliptic.
When X is uniformly elliptic, J. Guyon has extended this result to
the general class of tempered distributions. When f is a
tempered distribution,
(resp. ) has to be understood as
(resp. ) where p
(resp. p_{n}) is the density of X_{T} (resp. X_{T}^{n}). In
particular, (3) is valid when f is a measurable function
with polynomial growth, a Dirac distribution or a derivative of a
Dirac distribution. The proof consists in controlling the
linear mapping and the remainder. It can be used to
show that (3) remains valid when f is a measurable
function with exponential growth, or when the tempered distribution
f acts on the deterministic initial value x of the diffusion X. An article is
being written, underlying applications to option pricing and hedging.

Under the supervision of Benjamin Jourdain, Aurélien Alfonsi is studying the weak and strong rates of convergence of various explicit and implicit discretization schemes for Cox-Ingersoll-Ross processes, both from a theoretical point of view and by numerical experiments.