无迭代积分的Milstein格式的全变分界

IF 0.8 Q3 STATISTICS & PROBABILITY Monte Carlo Methods and Applications Pub Date : 2023-05-26 DOI:10.2139/ssrn.4333285
Toshihiro Yamada
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引用次数: 0

摘要

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。
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Total variation bound for Milstein scheme without iterated integrals
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.
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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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