Ergodicity and Ruin: The State Space–Time Inequality

A simplified general case

Consider the extremely simplified example, the sequence of independent random variables (X_i) with support in the positive real numbers (\mathbb{R}^+). The convergence theorems of classical probability theory address the behavior of the sum or average: (\frac{1}{n}\sum_i X_i) by the (weak) law of large numbers (convergence in probability). As shown in the story of the casino in Chapter 19, n going to infinity produces convergence in probability to the true mean return m. Although the law of large numbers applies to draws i that can be strictly separated by time, it assumes (some) independence, and certainly path independence.

Now consider (X_{i,t}) where every state variable (X_i) is indexed by a unit of time t: 0 < t < T. Assume that the “time events” are drawn from the exact same probability distribution: (P(X_i) = P(X_{i,t})).

We define a time probability the evolution over time for a single agent i.

In the presence of terminal, that is irreversible, ruin, every observation is now conditional on some attribute of the preceding one, and what happens at period t depends on t − 1, what happens at t − 1 depends on t − 2, etc. We now have path dependence.

Next what we call failure of ergodicity:

Theorem 1 (state space-time inequality): Assume that (\mathbb{E}{space}[X] = m) and (\mathbb{E}{time}[X] = m), the state space expectation for a static initial period t, and the time expectation for any agent i, both obtained through the weak law of large numbers. We have

(\mathbb{E}{time}[X] < \mathbb{E}{space}[X])

Proof:

(\mathbb{E}{time}[X] = \mathbb{E}[X \cdot \mathbb{I}{survival}]) where (\mathbb{I}{survival}) is the indicator function requiring survival at the previous period. Hence the limits of n for t show a decreasing temporal expectation: (\lim{n \to \infty} \mathbb{E}_{time}[X] = 0).

We can actually prove divergence.

(\mathbb{E}_{time}[X] \to 0) as (T \to \infty) if ( \mathbb{P}(\text{ruin}) > 0).

As we can see by making T < ∞, by recursing the law of iterated expectations, we get the inequality for all T.

We can see the ensemble of risk takers expecting a return m in any period t, while every single risk taker is guaranteed to eventually go bust.

Other approaches: we can also approach the proof more formally in a measure-theoretic way by showing that while space sets for “nonruin” A are disjoint, time sets are not. The approach relies on the fact that for a measure V:

(V(\bigcup_{t} A_t) \neq \sum_t V(A_t))

Almost all papers discussing the actuarial “overestimation” of tail risk via options (see review in Barberis 2003) are void by the inequality in Theorem 1. Clearly they assume that an agent only exists for a single decision or exposure. Simply, the original papers documenting the “bias” assume that the agents will never ever again make another decision in their remaining lives.

The usual solution to this path dependence—if it depends on only ruin—is done by introducing a function of X to allow the ensemble (path independent) average to have the same properties as the time (path dependent) average—or survival conditioned mean. The natural logarithm seems a good candidate. Hence (\ln X) and (X) belong to the same probabilistic class; hence a probability measure on one is invariant with respect to the other—what is called ergodicity. In that sense, when analyzing performance and risk, under conditions of ruin, it is necessary to use a logarithmic transformation of the variable (Peters 2011), or boundedness of the left tail (Kelly 1956), while maximizing opportunity in the right tail (Gell-Mann 2016), or boundedness of the left tail (Geman et al. 2015).

What we showed here is that unless one takes a logarithmic transformation (or a similar—smooth—function producing −∞ with ruin set at X = 0), both expectations diverge. The entire point of the precautionary principle is to avoid having to rely on logarithms or transformations by reducing the probability of ruin.

In their magisterial paper, Peters and Gell-Mann (2014) showed that the Bernoulli use of the logarithm wasn’t for a concave “utility” function, but, as with the Kelly criterion, to restore ergodicity. A bit of history:

• Bernoulli discovers logarithmic risk taking under the illusion of “utility.”
• Kelly and Thorp recovered the logarithm for maximal growth criterion as an optimal gambling strategy. Nothing to do with utility.
• Samuelson disses logarithm as aggressive, not realizing that semi-logarithm (or partial logarithm), i.e., on partial of wealth, can be done. From Menger to Arrow, via Chernoff and Samuelson, many in decision theory are shown to be making the mistake of ergodicity.
• Pitman in 1975 shows that a Brownian motion subjected to an absorbing barrier at 0, with censored absorbing paths, becomes a three-dimensional Bessel process. The drift of the surviving paths is 1/(2t) which integrates to a logarithm.
• Peters and Gell-Mann recover the logarithm for ergodicity and, in addition, put the Kelly-Thorpe result on rigorous physical grounds.
• With Cirillo, this author (Taleb and Cirillo 2015) discovers the log as unique smooth transformation to create a dual of the distribution in order to remove one-tail compact support to allow the use of extreme value theory.
• We can show (Briys and Taleb, in progress and private communication) the necessity of logarithmic transformation as simple ruin avoidance, which happens to be a special case of the HARA utility class.