论文标题
通过加权BMO接近跳跃的随机积分的近似
Approximation of stochastic integrals with jumps via weighted BMO approach
论文作者
论文摘要
本文调查了由半明星驱动的随机积分的离散时间近似，并通过加权有界平均振荡（BMO）方法跳跃。此方法可启用$ l_p $ - 估计，$ p \ in（2，\ infty）$，根据重量的不同，近似错误，它允许更改基础措施，该措施使错误估计不变。为了利用这种方法，我们提出了一种新的近似方案，该方案是根据基于基础半明星的跟踪跳跃的riemann近似校正获得的。我们还讨论了一种通过将离散时间调整为设置来优化近似率的方法。当Semimartingale的小型跳跃活动的行为就像$α$稳定的过程，$α\ in（1，2）$ in（1，2）$时，我们的方案在常规状态下达到了与Rosenbaum和Tankov [\ textit {ann {ann {ann的误差的常规融合率一样。应用。 probab。} \ textbf {24}（2014）1002--1048]。此外，我们的方法扩展到了（0，1] $的情况下，以及$ l_p $ -setting，这些$ l_p $ setting在那里未对其进行处理。作为一种应用，我们在特殊情况下应用了Semimartingale是指数级别的Lévy过程的方法，以将欧洲类型的均值偏差套在一起。
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.
