Simulation and approximation of Lévy-driven stochastic differential equations
Simulation and approximation of Lévy-driven stochastic differential equations
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DOI：
10.1051/ps/2009017
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发表时间：
2009-01
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通讯作者：
N. Fournier
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作者：
N. Fournier
我们考虑了levy驱动随机微分方程的近似欧拉格式。我们研究了路径的收敛速度。我们表明，当用高斯变量近似小跳跃时，收敛速度比简单地忽略它们要快得多。例如，当驱动过程的Levy测度表现为|z |−1−α dz接近0时，对于某些α∈(1,2)，我们得到1阶/√n的误差，计算代价为n阶α。关于忽略小跳跃时的类似错误，请参见[S]。由Levy过程驱动的随机微分方程解的数值模拟。随机过程。应用学报，103(2003)311-349]，计算成本为n α /(2−α)阶，当α接近2时，计算成本巨大。在同样的精神下，我们研究了Levy过程没有大的跳跃时Levy驱动的S.D.E.用brown S.D.E.逼近的问题。我们的结果依赖于[E。中心极限定理中最小距离的上界。安。亨利·庞加莱研究所。[J] .数学学报（自然科学版），2004,(3):1 - 2。梅杰和塔斯纳迪，独立rv部分和的近似和样本。verw。科学通报32(1975):111-131。
We consider the approximate Euler scheme for Levy-driven stochastic differential equations. We study the rate of convergence in law of the paths. We show that when approximating the small jumps by Gaussian variables, the convergence is much faster than when simply neglecting them. For example, when the Levy measure of the driving process behaves like |z |−1−α dz near 0 , for some α ∈ (1,2), we obtain an error of order 1/√n with a computational cost of order nα . For a similar error when neglecting the small jumps, see [S. Rubenthaler, Numerical simulation of the solution of a stochastic differential equation driven by a Levy process. Stochastic Process. Appl. 103 (2003) 311–349], the computational cost is of order n α /(2−α ) , which is huge when α is close to 2. In the same spirit, we study the problem of the approximation of a Levy-driven S.D.E. by a Brownian S.D.E. when the Levy process has no large jumps. Our results rely on some results of [E. Rio, Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincare Probab. Stat. 45 (2009) 802–817] about the central limit theorem, in the spirit of the famous paper by Komlos-Major-Tsunady [J. Komlos, P. Major and G. Tusnady, An approximation of partial sums of independent rvs and the sample df I. Z. Wahrsch. verw. Gebiete 32 (1975) 111–131].