金融中的耦合连续时间随机游动
Coupled continuous time random walks in finance
摘要：
连续时间随机游走（CTRW）在物理学中用于模拟异常扩散，通过在粒子跳跃之间加入随机等待时间。在金融中，粒子跳跃是对数回报，等待时间衡量交易之间的延迟。这两个随机变量（日志返回和等待时间）通常不是独立的。对于这些耦合的CTRW模型，我们现在可以计算极限随机过程（就像布朗运动是简单随机游走的极限一样），即使是在重尾（幂律）价格跳跃和/或等待时间的情况下。这个极限过程的概率密度函数求解分数阶偏微分方程。在某些情况下，这些方程可以显式求解，以产生对长期价格变化的描述，基于单个交易的高分辨率模型，其中包括等待时间和随后的对数回报之间的统计相关性。在重尾情况下，这涉及到算子稳定的时空随机向量，推广了熟悉的稳定模型。在本文中，我们将回顾基本理论，并提出两个应用与tick-by-tick股票和期货数据。
英文摘要：
Continuous time random walks (CTRWs) are used in physics to model anomalous diffusion, by incorporating a random waiting time between particle jumps. In finance, the particle jumps are log-returns and the waiting times measure delay between transactions. These two random variables (log-return and waiting time) are typically not independent. For these coupled CTRW models, we can now compute the limiting stochastic process (just like Brownian motion is the limit of a simple random walk), even in the case of heavy tailed (power-law) price jumps and/or waiting times. The probability density functions for this limit process solve fractional partial differential equations. In some cases, these equations can be explicitly solved to yield descriptions of long-term price changes, based on a high-resolution model of individual trades that includes the statistical dependence between waiting times and the subsequent log-returns. In the heavy tailed case, this involves operator stable space-time random vectors that generalize the familiar stable models. In this paper, we will review the fundamental theory and present two applications with tick-by-tick stock and futures data.
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