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This post is formal treatment of the idea outlined here.

Given a countable set of incomplete models, we define a forecasting function that converges in the Kantorovich-Rubinstein metric with probability 1 to every one of the models which is satisfied by the true environment. This is analogous to Blackwell-Dubins merging of opinions for complete models, except that Kantorovich-Rubinstein convergence is weaker than convergence in total variation. The forecasting function is a dominant stochastic market for a suitably constructed set of traders.


Appendix A contains the proofs. Appendix B restates the theorems about dominant stochastic markets for ease of reference. Appendix C states a variant of the Michael selection theorem due to Yannelis and Prabhakar which is used in Appendix A.

##Notation

Given a metric space, and , .

Given a topological space:

  • is the space of Borel probability measures on equipped with the weak* topology.

  • is the Banach space of continuous functions with uniform norm.

  • is the Borel -algebra on .

  • is the -algebra of universally measurable sets on .

  • Given , denotes the support of .

Given and measurable spaces, is a Markov kernel from to . For any , we have . Given , is the semidirect product of and and is the pushforward of by .

Given , Polish spaces, Borel measurable and , we denote the set of Markov kernels s.t. is supported on the graph of and . By the disintegration theorem, is always non-empty and any two kernels in coincide -almost everywhere.

##Results

Consider any compact Polish metric space and its metric. We denote the Banach space of Lipschitz continuous functions with the norm

(as before, with the weak topology) can be regarded as a compact subset of (with the strong topology), yielding a metrization of which we will call the Kantorovich-Rubinstein metric :

In fact, the above differs from the standard definition of the Kantorovich-Rubinstein metric (a.k.a. 1st Wasserstein metric, a.k.a. earth mover's metric), but this abuse of terminology is mild since the two are strongly equivalent.

Now consider convex. We will describe a class of trading strategies that are designed to exploit any .

#Definition 1

Let . is said to be an -representative of when for all , we have

#Lemma 1

Consider another compact Polish space and closed. For any , define by

Assume that for any , is convex. Then, for any , there exists measurable w.r.t. and s.t. for all , is an -representative of .


For each , let be a compact subset of ( is arbitrary). Denote

is a compact subset of . We will regard it as equipped with the Euclidean metric. Denote

is a compact Polish space. For each we denote the projection mapping. Given and , we have and .

The following definition provides a notion of updating an incomplete model by observations:

#Definition 2

Consider . For any and , we define by

Note that the limit in the definition above need not exist for every .

Denote