arXiv 1408.08982014-08-05math.PR

Lévy过程的重要抽样和统计Romberg方法

Importance Sampling and Statistical Romberg Method for Lévy Processes

M. Ben Alaya, K. Hajji, A. Kebaier

摘要：

在应用概率的许多领域中出现的一个重要的随机过程族是Lévy过程类。一般来说，这样的过程是不可模拟的，特别是对于那些具有无限活动的过程。在实践中，通常通过在某个截止大小$\varepsilon$（$\varepsilon\searrow 0$）处截断跳跃来近似它们。这个过程使我们考虑一个可模拟的复合泊松过程。本文首先介绍了，在这种情况下，统计Romberg方法，以提高经典的蒙特卡罗方法的复杂性。粗略地说，我们使用许多具有粗略截止$\varepsilon^β,$ $β\in(0,1)$的采样路径，以及一些具有精细截止$\varepsilon$的附加采样路径。证明了Monte Carlo方法和统计Romberg方法的Lindeberg-Feller型中心极限定理，其推断误差依赖于参数$\varepsilon$.这导致了一个准确的描述的最佳选择的参数与明确的极限方差。在此基础上，利用Monte Carlo和统计Romberg重要抽样方差约简方法，提出了一种用Esscher变换求Lévy过程最优测度变换的随机逼近方法.此外，我们开发了新的自适应Monte Carlo和统计Romberg算法，并证明了相关的中心极限定理。最后，数值模拟的处理，以说明自适应统计Romberg方法的效率，减少在同一时间的方差和计算工作量相关联的有效计算的期权价格时，标的资产的过程遵循一个指数纯跳CGMY模型。

英文摘要：

An important family of stochastic processes arising in many areas of applied probability is the class of Lévy processes. Generally, such processes are not simulatable especially for those with infinite activity. In practice, it is common to approximate them by truncating the jumps at some cut-off size $\varepsilon$ ($\varepsilon\searrow 0$). This procedure leads us to consider a simulatable compound Poisson process. This paper first introduces, for this setting, the statistical Romberg method to improve the complexity of the classical Monte Carlo one. Roughly speaking, we use many sample paths with a coarse cut-off $\varepsilon^β,$ $β\in(0,1)$, and few additional sample paths with a fine cut-off $\varepsilon$. Central limit theorems of Lindeberg-Feller type for both Monte Carlo and statistical Romberg method for the inferred errors depending on the parameter $\varepsilon$ are proved. This leads to an accurate description of the optimal choice of parameters with explicit limit variances. Afterwards, the authors propose a stochastic approximation method of finding the optimal measure change by Esscher transform for Lévy processes with Monte Carlo and statistical Romberg importance sampling variance reduction. Furthermore, we develop new adaptive Monte Carlo and statistical Romberg algorithms and prove the associated central limit theorems. Finally, numerical simulations are processed to illustrate the efficiency of the adaptive statistical Romberg method that reduces at the same time the variance and the computational effort associated to the effective computation of option prices when the underlying asset process follows an exponential pure jump CGMY model.

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