Weak approximation of killed diffusion using Euler schemes
Weak approximation of killed diffusion using Euler schemes
复制标题
使用欧拉方案的抑制扩散的弱近似
DOI：
10.1016/s0304-4149(99)00109-x
复制
发表时间：
2000
影响因子：
1.4
通讯作者：
E. Gobet
中科院分区：
文献类型：
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作者：
E. Gobet
我们研究了一个多维扩散（Xt）0≥t≥ttl在离开开集D时的弱逼近，当扩散由连续欧拉格式（X≥t）0≥t≤Tor逼近为离散欧拉格式（X≥ti）0≥i≤N，离散步长为t /N。如果我们将τ≔inf {t > 0: Xt∉D}和τ̃c≔inf {t > 0: X̃t∉D},我们证明了离散误差前[1 t <τ̃cf (X̃t)]−前[1 t <τf (Xt)]可以扩展到一阶N−1,提供支持或规律性条件f。离散方案,如果我们设置τ̃D≔inf {ti > 0: X̃ti∉D},错误例[1 t <τ̃df (X̃t)]−前[1 t <τf (Xt)]的N阶−1/2,类似的假设下f。这个收敛速度实际上是准确和内在问题的离散消磨时间。
We study the weak approximation of a multidimensional diffusion (Xt)0⩽t⩽Tkilled as it leaves an open set D, when the diffusion is approximated by its continuous Euler scheme ( X ̃t)0⩽t⩽Tor by its discrete one ( X ̃ti)0⩽i⩽N, with discretization step T/N. If we set τ ≔ inf {t>0 : Xt∉D} and τ ̃c≔ inf {t>0 : X ̃t∉D} , we prove that the discretization error Ex[ 1T<τ ̃cf( X ̃T)]− Ex[ 1T<τf(XT)] can be expanded to the first order in N−1, provided support or regularity conditions on f. For the discrete scheme, if we set τ ̃d≔ inf {ti>0 : X ̃ti∉D} , the error Ex[ 1T<τ ̃df( X ̃T)]− Ex[ 1T<τf(XT)] is of order N−1/2, under analogous assumptions on f. This rate of convergence is actually exact and intrinsic to the problem of discrete killing time.