带跳随机积分离散化的渐近误差分布研究

随机积分一直是随机分析和金融数学中的重要内容。为了在实际金融研究中使用随机积分，将理论上的随机积分离散化是必不可少的步骤。目前，研究随机积分离散化后的渐近误差分布对于掌控离散化风险有十分重要的意义。申请者在前期研究中发现，由于实际金融操作中，确定离散时间是不现实的。最近，很多学者将研究转向了随机离散时间设定下，随机积分离散化的渐近误差分布问题。但是现有文献往往将研究限制在连续轨道随机积分上 ，然而很多实证研究表明，金融数据常含有跳，所以有必要对带跳的随机积分离散化问题进行研究。本项目将在随机离散时间设定下，对以下问题开展研究:(1)带跳随机积分离散化的渐近误差分布问题;(2)纯跳随机积分离散化的渐近误差分布问题;(3)带跳随机二重积分离散化的渐近误差分布问题。这些研究不仅能够丰富随机过程的渐近理论，也为随机微分方程数值分析，对冲误差分析方面的研究提供理论基础。

Stochastic integral has played an important role in the fields of stochastic analysis as well as mathematical finance. However, discretization of the original integral sometimes is necessary before we put it into practical use. Research on the asymptotic error distribution for discretization of stochastic integral is meaningful, which can help us to control the risk caused by discretization. Preliminary studies have shown that it is unrealistic to quantify the discrete time in the real market. So recent papers about this distribution are mainly under the assumption of random discrete time and they basically restrained themselves on the continuous paths. But the real financial data normally have jumps, it is necessary to consider this problem for stochastic integral with jumps. In this project ,we will consider the following problems under the assumption of random discrete time ：1. the asymptotic error distributions for discretization of stochastic integral with jumps; 2 The asymptotic error distribution for discretization of pure jump stochastic integrals; 3 the asymptotic error distribution for discretization of stochastic iterated integrals with jumps. We believe that these discussions will not only enrich the asymptotic theory of stochastic processes but also provide some new theoretical results and tools for the numerical analysis of stochastic differential equations and hedging-error analysis.