Insurance as an ergodicity problem

Abstract

In November 2014, the economist Ken Arrow and I had one of our long conversations about my efforts to re-imagine economic science from the perspective of the ergodicity problem (Peters, 2019). Ergodicity, as it pertains to economics, is about two different ways of averaging to deal with randomness. Let us say we measure some quantity at regularly spaced times t = 1 . . . T and model it as a stochastic process, x(t,ω), where ω denotes the realization of the process. If we want to reduce the process to a single informative number, we can average across the ensemble of realizations, yielding the expected value E [x] (t)= limN→∞ 1 N ∑N i x(t,ωi), or we can average across time, yielding the time average T [x] (ω)= limT→∞ 1 T ∑T t x(t,ωi), Fig. 1. If the process is ergodic, then the two ways of averaging will give the same result. We are interested in cases where this is not true.