Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process,Physical Review E

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Geometric Brownian motion under stochastic resetting: A stationary yet nonergodic process
Physical Review E ( IF 2.2 ) Pub Date : 2021-07-19 , DOI: 10.1103/physreve.104.014121
Viktor Stojkoski _{1,

2} , Trifce Sandev _{2,

3,

4} , Ljupco Kocarev _{2,

5} , Arnab Pal ₆

Affiliation

We study the effects of stochastic resetting on geometric Brownian motion with drift (GBM), a canonical stochastic multiplicative process for nonstationary and nonergodic dynamics. Resetting is a sudden interruption of a process, which consecutively renews its dynamics. We show that, although resetting renders GBM stationary, the resulting process remains nonergodic. Quite surprisingly, the effect of resetting is pivotal in manifesting the nonergodic behavior. In particular, we observe three different long-time regimes: a quenched state, an unstable state, and a stable annealed state depending on the resetting strength. Notably, in the last regime, the system is self-averaging and thus the sample average will always mimic ergodic behavior establishing a stand-alone feature for GBM under resetting. Crucially, the above-mentioned regimes are well separated by a self-averaging time period which can be minimized by an optimal resetting rate. Our results can be useful to interpret data emanating from stock market collapse or reconstitution of investment portfolios.

中文翻译：

随机重置下的几何布朗运动：一个静止但非遍历的过程

我们研究随机重置对带漂移的几何布朗运动 (GBM) 的影响，这是一种用于非平稳和非遍历动力学的规范随机乘法过程。重置是一个过程的突然中断，它连续更新其动态。我们表明，虽然重置使 GBM 静止，但由此产生的过程仍然是非遍历的。相当令人惊讶的是，重置的效果对于表现非遍历行为至关重要。特别是，我们观察到三种不同的长时间状态：淬火状态、不稳定状态和稳定退火状态，具体取决于复位强度。值得注意的是，在最后一个机制中，系统是自我平均的，因此样本平均值将始终模仿遍历行为，为重置下的 GBM 建立独立的特征。至关重要的是，上述状态被一个自平均时间段很好地分开，该时间段可以通过最佳复位率最小化。我们的结果可用于解释源自股市崩盘或投资组合重组的数据。

更新日期：2021-07-19

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