Inria / Raweb 2004
Team: Mathfi

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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 XTn with time-step $ \Delta$t = T/n converges in law to the original random variable XT. More precisely, he has looked for which class of functions f the approximate expectation Im14 ${\#120124 \mfenced o=[ c=] {f(}X_T^n{)}}$ converges to Im15 ${\#120124 \mfenced o=[ c=] {f(}X_T{)}}$ with speed $ \Delta$t. So far, (Xt, t$ \ge$0) has been a smooth Im16 $\#8477 ^d$-valued diffusion. When f is smooth, it is known from D. Talay and L. Tubaro that

Im17 ${\#120124 \mfenced o=[ c=] {f(}X_T^n{)}-\#120124 \mfenced o=[ c=] {f(}X_T{)}={C(f)\#916 t}+O\mfenced o=( c=) \#916 t^2.}$(3)

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, Im15 ${\#120124 \mfenced o=[ c=] {f(}X_T{)}}$ (resp. Im14 ${\#120124 \mfenced o=[ c=] {f(}X_T^n{)}}$) has to be understood as Im18 ${\#9001 f,p\#9002 }$ (resp. Im19 ${\#9001 f,p_n\#9002 }$) where p (resp. pn) is the density of XT (resp. XTn). 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 Im20 ${f\#8614 C(f)}$ 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.