英文标题：
《Geometrically Convergent Simulation of the Extrema of L\\\'{e}vy Processes》
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作者：
Jorge Ignacio Gonz\\\'alez C\\\'azares, Aleksandar Mijatovi\\\'c, Ger\\\'onimo
Uribe Bravo
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最新提交年份：
2021
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英文摘要：
We develop a novel approximate simulation algorithm for the joint law of the position, the running supremum and the time of the supremum of a general L\\\'evy process at an arbitrary finite time. We identify the law of the error in simple terms. We prove that the error decays geometrically in $L^p$ (for any $p\\geq 1$) as a function of the computational cost, in contrast with the polynomial decay for the approximations available in the literature. We establish a central limit theorem and construct non-asymptotic and asymptotic confidence intervals for the corresponding Monte Carlo estimator. We prove that the multilevel Monte Carlo estimator has optimal computational complexity (i.e. of order $\\epsilon^{-2}$ if the mean squared error is at most $\\epsilon^2$) for locally Lipschitz and barrier-type functionals of the triplet and develop an unbiased version of the estimator. We illustrate the performance of the algorithm with numerical examples.
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中文摘要：
我们开发了一种新的近似模拟算法，用于在任意有限时间内求解一般L趵evy过程的位置、运行上确界和上确界时间的联合规律。我们用简单的术语来确定误差定律。我们证明，与文献中可用近似的多项式衰减相比，误差在几何上以$L^p$（对于任何$p\\geq 1$）作为计算成本的函数衰减。我们建立了一个中心极限定理，并为相应的蒙特卡罗估计量构造了非渐近和渐近置信区间。我们证明了对于三元组的局部Lipschitz和势垒型泛函，多层蒙特卡罗估计具有最佳的计算复杂度（即，如果均方误差最大为$\\ε^{-2}$），并开发了该估计的无偏版本。我们用数值例子说明了算法的性能。
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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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一级分类：Statistics 统计学
二级分类：Methodology 方法论
分类描述：Design, Surveys, Model Selection, Multiple Testing, Multivariate Methods, Signal and Image Processing, Time Series, Smoothing, Spatial Statistics, Survival Analysis, Nonparametric and Semiparametric Methods
设计，调查，模型选择，多重检验，多元方法，信号和图像处理，时间序列，平滑，空间统计，生存分析，非参数和半参数方法
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