Euler scheme and tempered distributions

Given a smooth RdRd-valued diffusion (Xtx,t∈[0,1]) starting at point xx, we study how fast the Euler scheme X1n,x with time step 1/n1/n converges in law to the random variable X1x. To be precise, we look for the class of test functions ff for which the approximate expectation E[f(X1n,x)] converges with speed 1/n1/n to E[f(X1x)].When ff is smooth with polynomially growing derivatives or, under a uniform hypoellipticity condition for XX, when ff is only measurable and bounded, it is known that there exists a constant C1f(x)C1f(x) such that equation(1)E[f(X1n,x)]−E[f(X1x)]=C1f(x)/n+O(1/n2).If XX is uniformly elliptic, we expand this result to the case when ff is a tempered distribution. In such a case, E[f(X1x)] (resp. E[f(X1n,x)]) has to be understood as 〈f,p(1,x,⋅)〉〈f,p(1,x,⋅)〉 (resp. 〈f,pn(1,x,⋅)〉〈f,pn(1,x,⋅)〉) where p(t,x,⋅)p(t,x,⋅) (resp. pn(t,x,⋅)pn(t,x,⋅)) is the density of Xtx (resp. Xtn,x). In particular, (1) is valid when ff 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 ff is a measurable function with exponential growth. Actually our results are symmetric in the two space variables xx and yy of the transition density and we prove that ∂xα∂yβpn(t,x,y)−∂xα∂yβp(t,x,y)=∂xα∂yβπ(t,x,y)/n+rn(t,x,y) for a function ∂xα∂yβπ and an O(1/n2)O(1/n2) remainder rnrn which are shown to have gaussian tails and whose dependence on tt is made precise. We give applications to option pricing and hedging, proving numerical convergence rates for prices, deltas and gammas.