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 X 1 n , x with time step 1 / n converges in law to the random variable X 1 x . To be precise, we look for the class of test functions f for which the approximate expectation E [ f ( X 1 n , x ) ] converges with speed 1 / n to E [ f ( X 1 x ) ] . 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 C 1 f ( x ) such that (1) E [ f ( X 1 n , x ) ] − E [ f ( X 1 x ) ] = C 1 f ( x ) / n + O ( 1 / n 2 ) . s uniformly elliptic, we expand this result to the case when f is a tempered distribution. In such a case, E [ f ( X 1 x ) ] (resp. E [ f ( X 1 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 ∂ x α ∂ y β p n ( t , x , y ) − ∂ x α ∂ y β p ( t , x , y ) = ∂ x α ∂ y β π ( t , x , y ) / n + r n ( t , x , y ) for a function ∂ x α ∂ y β π and an O ( 1 / n 2 ) 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.