Total variation bound for Milstein scheme without iterated integrals

IF 0.8 Q3 STATISTICS & PROBABILITY
Toshihiro Yamada
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引用次数: 0

Abstract

Abstract The paper gives new results for the Milstein scheme of stochastic differential equations. We show that (i) the Milstein scheme holds as a weak approximation in total variation sense and is given by second-order polynomials of Brownian motion without using iterated integrals under non-commutative vector fields; (ii) the accuracy of the Milstein scheme is better than that of the Euler–Maruyama scheme in an asymptotic sense. In particular, we prove d TV ⁢ ( X T ε , X ¯ T ε , Mil , ( n ) ) ≤ C ⁢ ε 2 / n d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{Mil},(n)})\leq C\varepsilon^{2}/n and d TV ⁢ ( X T ε , X ¯ T ε , EM , ( n ) ) ≤ C ⁢ ε / n d_{\mathrm{TV}}(X_{T}^{\varepsilon},\bar{X}_{T}^{\varepsilon,\mathrm{EM},(n)})\leq C\varepsilon/n , where d TV d_{\mathrm{TV}} is the total variation distance, X ε X^{\varepsilon} is a solution of a stochastic differential equation with a small parameter 𝜀, and X ¯ ε , Mil , ( n ) \bar{X}^{\varepsilon,\mathrm{Mil},(n)} and X ¯ ε , EM , ( n ) \bar{X}^{\varepsilon,\mathrm{EM},(n)} are the Milstein scheme without iterated integrals and the Euler–Maruyama scheme, respectively. In computational aspect, the scheme is useful to estimate probability distribution functions by a simple simulation without Lévy area computation. Numerical examples demonstrate the validity of the method.
无迭代积分的Milstein格式的全变分界
摘要本文给出了随机微分方程米尔斯坦格式的新结果。我们证明(i) Milstein格式在全变分意义上是弱逼近,并且在非交换向量场下由布朗运动的二阶多项式给出,而不使用迭代积分;(ii)在渐近意义上,Milstein格式的精度优于Euler-Maruyama格式。特别地,我们证明了d TV减去(X T ε, X¯T ε, Mil,(n))≤C减去ε 2/n d_ {\mathrm{TV}} ({X_T}^ {\varepsilon}, \bar{X} _T{^ }{\varepsilon, \mathrm{Mil},(n}))\leq C \varepsilon ^{2}/n和d TV减去(X T ε, X¯T ε, EM,(n))≤C减去ε /n d_ {\mathrm{TV}} ({X_T}^ {\varepsilon}, \bar{X} _T{^ }{\varepsilon, \mathrm{EM},(n)})\leq C \varepsilon /n,其中d TV减去d_ {\mathrm{TV}}为总变异距离,X ε X^ {\varepsilon}是一个具有小参数的随机微分方程的解,X¯ε, Mil,(n) \bar{X} ^ {\varepsilon, \mathrm{Mil},(n)}和X¯ε, EM,(n)\bar{X} ^ {\varepsilon, \mathrm{EM},(n)}分别是无迭代积分的Milstein格式和Euler-Maruyama格式。在计算方面,该方案可以通过简单的模拟来估计概率分布函数,而无需计算lsamvy面积。数值算例验证了该方法的有效性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Monte Carlo Methods and Applications
Monte Carlo Methods and Applications STATISTICS & PROBABILITY-
CiteScore
1.20
自引率
22.20%
发文量
31
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