Stochastic components in a feedback loop introduce state behaviors that are fundamentally different from those observed in a deterministic system. The effect of injecting a stochastic signal additively in linear feedback systems can be viewed as the addition of filtered stochastic noise. If the stochastic signal enters the feedback loop in a multiplicative manner, a much richer set of state behaviors emerges. These phenomena are investigated for the simplest possible system: a multiplicative noise in a scalar, integrating feedback loop. The same dynamics arise when considering a first-order system in feedback with a stochastic gain. The dynamics of this form arise naturally in a number of domains, including compound investments in finance, chemical reaction dynamics, population dynamics, epidemiology, control over lossy communication channels, and adaptive control. Understanding the nature of such dynamics in a simple system is a precursor to recognizing them in more complex stochastic dynamical systems.

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反馈回路中的随机性：寄予厚望和保证的破产

反馈回路中的随机成分引入的状态行为与确定性系统中观察到的状态行为根本不同。在线性反馈系统中加性注入随机信号的效果可以看作是滤波后的随机噪声的添加。如果随机信号以乘法的方式进入反馈回路，则会出现更丰富的状态行为。对于最简单的系统，对这些现象进行了研究：标量中的乘法噪声，积分反馈环路。当考虑具有随机增益的反馈中的一阶系统时，会产生相同的动态变化。这种形式的动力自然而然地出现在许多领域，包括金融的复合投资，化学反应动力，人口动力，流行病学，对有损通讯渠道的控制，和自适应控制。在简单的系统中了解此类动力学的性质是在更复杂的随机动力学系统中识别它们的先决条件。