Stojkoski Viktor, Sandev Trifce, Kocarev Ljupco, Pal Arnab
Faculty of Economics, Ss. Cyril and Methodius University, 1000 Skopje, Macedonia.
Research Center for Computer Science and Information Technologies, Macedonian Academy of Sciences and Arts, Bul. Krste Misirkov 2, 1000 Skopje, Macedonia.
Phys Rev E. 2021 Jul;104(1-1):014121. doi: 10.1103/PhysRevE.104.014121.
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.
我们研究了随机重置对带漂移的几何布朗运动（GBM）的影响，GBM是一种用于非平稳和非遍历动力学的典型随机乘法过程。重置是一个过程的突然中断，它会连续更新其动力学。我们表明，尽管重置使GBM变得平稳，但所得过程仍然是非遍历的。非常令人惊讶的是，重置的效果在表现非遍历行为方面起着关键作用。特别地，我们观察到三种不同的长期状态：一个淬火状态、一个不稳定状态和一个稳定的退火状态，这取决于重置强度。值得注意的是，在最后一种状态下，系统是自平均的，因此样本平均值将始终模拟遍历行为，为重置下的GBM建立一个独立的特征。至关重要的是，上述状态由一个自平均时间段很好地分隔开，该时间段可以通过最优重置率最小化。我们的结果对于解释源自股市崩溃或投资组合重组的数据可能是有用的。
