Stochasticity in Feedback Loops; Great Expectations and Guaranteed Ruin

TL;DR

This paper explores stochastic feedback systems, revealing how they can stabilize unstable systems or lead to paradoxical outcomes like guaranteed ruin despite unbounded expected profit.

Contribution

It provides explicit stability conditions for scalar stochastic feedback systems with arbitrary distributions and highlights counterintuitive phenomena such as stabilization without sign knowledge.

Findings

States evolve towards heavy-tailed distributions

Stochastic feedback can stabilize unknown unstable systems

Investment schemes can yield profit and guaranteed bankruptcy

Abstract

Stochastic feedback systems give rise to a variety of notions of stability. The conditions for the stability of the median, mean, and variance stability conditions differ. These conditions can be stated explicitly for scalar discrete-time systems with (almost) arbitrary distributions of the stochastic feedback gain. The state variable in such systems evolves towards a heavy-tailed distribution and exhibits some non-intuitive characteristics. For example, one can use stochastic feedback to stabilise unstable systems where one does not know the sign of the unstable pole or the sign of the system gain. A more dramatic example is an investment scheme which simultaneously yields unbounded expected profit and almost certain bankruptcy to every investor.