Error distributions for random grid approximations of multidimensional stochastic integrals
Error distributions for random grid approximations of multidimensional stochastic integrals
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DOI：
10.1214/12-aap858
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发表时间：
2011-08
影响因子：
1.8
通讯作者：
C. Lindberg;Holger Rootz'en
中科院分区：
文献类型：
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作者：
C. Lindberg;Holger Rootz'en
本文证明了在非等距随机网格下，多个随机积分关于局部布朗半鞅的逼近误差的联合收敛性质。收敛所需的条件是被积函数的勒贝格积分一致趋于零，且平方变分和协变分过程收敛。本文还提供了工具，简化了对这些条件的检查，并扩大了结果的范围。利用这些结果证明了基于多维离散微分方程解的积分随机网格逼近的一个显式极限定理，并找到了“设计”和优化逼近误差分布的方法。作为例子，我们简要讨论离散期权套期保值的策略。
This paper proves joint convergence of the approximation error for several stochastic integrals with respect to local Brownian semimartingales, for nonequidistant and random grids. The conditions needed for convergence are that the Lebesgue integrals of the integrands tend uniformly to zero and that the squared variation and covariation processes converge. The paper also provides tools which simplify checking these conditions and which extend the range for the results. These results are used to prove an explicit limit theorem for random grid approximations of integrals based on solutions of multidimensional SDEs, and to find ways to "design" and optimize the distribution of the approximation error. As examples we briefly discuss strategies for discrete option hedging.