Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while time averages suffer collapse . A common tactic for dealing with this difference is diversification. In this letter we show that diversification will not solve the problem but only delay the collapse .
Geometric Brownian motion (GBM) is a model for systems as varied as financial instruments and populations. The statistical properties of GBM are complicated by non-ergodicity, which can lead to ensemble averages exhibiting exponential growth while any individual trajectory collapses according to its time-average . A common tactic for bringing time averages closer to ensemble averages is diversification. In this letter we study the effects of diversification using the concept of ergodicity breaking .
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time averages suffer collapse
any individual trajectory collapses according to its time-average
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dealing with this difference
bringing time averages closer to ensemble averages
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show that diversification will not solve the problem but only delay the collapse
study the effects of diversification using the concept of ergodicity breaking