带跳随机积分的加权BMO逼近
Approximation of stochastic integrals with jumps via weighted BMO approach
摘要：
利用加权有界平均振荡（BMO）方法研究了带跳半鞅驱动的随机积分的离散时间逼近。该方法使得能够对取决于权重的近似误差进行$L_p$ -估计，$p \in (2, \infty)$，并且它允许改变基本测量，这使得误差估计不变。为了利用这种方法，我们提出了一个新的近似方案，从校正的基础上跟踪跳跃的基础半鞅的黎曼近似。我们还讨论了一种方法，以优化近似速度，通过调整离散时间的设置。当半鞅的小跳活动表现得像具有$α\in (1, 2)$的$α$稳定过程时，我们的方案在正则区域下实现了与Rosenbaum和Tankov [ \textit{Ann. Appl. Probab.} \textbf{24}（2014）1002- 1048]中相同的误差收敛速度。此外，我们的方法扩展到情况下$α\in (0, 1]$和$L_p$ -设置没有处理。作为应用，我们将半鞅为指数Lévy过程的特殊情形下的方法应用于欧式期权的均值-方差套期保值。
英文摘要：
This article investigates discrete-time approximations of stochastic integrals driven by semimartingales with jumps via weighted bounded mean oscillation (BMO) approach. This approach enables $L_p$-estimates, $p \in (2, \infty)$, for the approximation error depending on the weight, and it allows a change of the underlying measure which leaves the error estimates unchanged. To take advantage of this approach, we propose a new approximation scheme obtained from a correction for the Riemann approximation based on tracking jumps of the underlying semimartingale. We also discuss a way to optimize the approximation rate by adapting the discretization times to the setting. When the small jump activity of the semimartingale behaves like an $α$-stable process with $α\in (1, 2)$, our scheme achieves under a regular regime the same convergence rate for the error as in Rosenbaum and Tankov [\textit{Ann. Appl. Probab.} \textbf{24} (2014) 1002--1048]. Moreover, our approach extends to the case $α\in (0, 1]$ and to the $L_p$-setting which are not treated there. As an application, we apply the methods in the special case where the semimartingale is an exponential Lévy process to mean-variance hedging of European type options.
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