Euler scheme and tempered distributions
Euler scheme and tempered distributions
复制标题
DOI：
10.1016/j.spa.2005.11.011
复制
发表时间：
2006-06
影响因子：
1.4
通讯作者：
Julien Guyon
中科院分区：
文献类型：
--
作者：
Julien Guyon
Given a smooth Rd-valued diffusion (Xtx,t∈[0,1]) starting at point x, we study how fast the Euler scheme X1n,xwith time step 1/n converges in law to the random variable X1x. To be precise, we look for the class of test functions f for which the approximate expectation E[f(X1n,x)] converges with speed 1/n to E[f(X1x)]. 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, E[f(X1x)] (resp. E[f(X1n,x)]) has to be understood as 〈f,p(1,x,⋅)〉 (resp. 〈f,pn(1,x,⋅)〉) where p(t,x,⋅) (resp. pn(t,x,⋅)) is the density of Xtx(resp. Xtn,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 for a function ∂xα∂yβπ and an O(1/n2) remainder rnwhich 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.
