Given a smooth -valued diffusion starting at point x, we study how fast the Euler scheme with time step 1/n converges in law to the random variable . To be precise, we look for the class of test functions f for which the approximate expectation converges with speed 1/n to . When f is smooth with polynomially growing derivatives or, under a uniform hypoellipticity condition for X, when f is only measurable and bounded, it is known that there exists a constant C1f(x) such that If X is uniformly elliptic, we expand this result to the case when f is a tempered distribution. In such a case, (resp. ) has to be understood as <f,p(1,x,[dot operator])> (resp. <f,pn(1,x,[dot operator])>) where p(t,x,[dot operator]) (resp. pn(t,x,[dot operator])) is the density of (resp. ). In particular, (1) is valid when f is a measurable function with polynomial growth, a Dirac mass or any derivative of a Dirac mass. We even show that (1) remains valid when f is a measurable function with exponential growth. Actually our results are symmetric in the two space variables x and y of the transition density and we prove that for a function and an O(1/n2) remainder rn which are shown to have gaussian tails and whose dependence on t is made precise. We give applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas.
