Euler scheme and tempered distributions

Given a smooth R^d-valued diffusion ( X_t^x, t ∈ [ 0, 1 ] ) starting at point x, we study how fast the Euler scheme X₁^{n, x} with time step 1 / n converges in law to the random variable X₁^x. To be precise, we look for the class of test functions f for which the approximate expectation E [ f ( X₁^{n, x} ) ] converges with speed 1 / n to E [ f ( X₁^x ) ]. 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 C₁ f ( x ) such that {A formula is presented}. If X is uniformly elliptic, we expand this result to the case when f is a tempered distribution. In such a case, E [ f ( X₁^x ) ] (resp. E [ f ( X₁^{n, x} ) ]) has to be understood as { f, p ( 1, x, . ) } (resp.  < f, p_n ( 1, x, · ) > ) where p ( t, x, · ) (resp. p_n ( t, x,· )) is the density of X_t^x (resp. X_t^{n, x}). 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 {A formula is presented} for a function δ_x^α δ_y^ßπ and an O ( 1 / n² ) remainder r_n 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.