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摘要翻译：
在通常的随机波动模型中，驱动资产价格波动的过程是按照一个自治的一维随机微分方程演化的。我们假定这个方程的系数是光滑的。利用It\o公式，我们摆脱了资产价格动力学中关于驱动该SDE的布朗运动的随机积分。利用这一结构，我们提出了一个基于Milstein离散化的资产价格弱轨迹收敛阶格式和一个基于Ninomiya-Victoir离散化的资产价格弱轨迹收敛阶格式。当资产价格波动由一个Orstein-Uhlenbeck过程驱动时，我们还提出了一个具有改进收敛性的具体方案。我们通过数值实验验证了理论收敛速度，并表明我们的格式很好地适应了Giles[2008a,2008b]提出的多层Monte Carlo方法。
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英文标题：
《High order discretization schemes for stochastic volatility models》
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作者：
Benjamin Jourdain (CERMICS), Mohamed Sbai (CERMICS)
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最新提交年份：
2011
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分类信息：
一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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一级分类：Quantitative Finance 数量金融学
二级分类：Computational Finance 计算金融学
分类描述：Computational methods, including Monte Carlo, PDE, lattice and other numerical methods with applications to financial modeling
计算方法，包括蒙特卡罗，偏微分方程，格子和其他数值方法，并应用于金融建模
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英文摘要：
In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. We assume that the coefficients of this equation are smooth. Using It\^o's formula, we get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, we propose - a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, - a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence for the asset price. We also propose a specific scheme with improved convergence properties when the volatility of the asset price is driven by an Orstein-Uhlenbeck process. We confirm the theoretical rates of convergence by numerical experiments and show that our schemes are well adapted to the multilevel Monte Carlo method introduced by Giles [2008a, 2008b].
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PDF链接：
https://arxiv.org/pdf/0908.1926
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