% operators that are separated from the operand by a space

% operators that require brackets

% operators that require parentheses

% Paper specific

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 X a metric space, x∈X and r∈R, Br(x):={y∈X∣d(x,y)<r}.

Given X a topological space:

P(X) is the space of Borel probability measures on X equipped with the weak* topology.

C(X) is the Banach space of continuous functions X→R with uniform norm.

T(X):=C(P(X)×X)

B(X) is the Borel σ-algebra on X.

U(X) is the σ-algebra of universally measurable sets on X.

Given μ∈P(X), suppμ denotes the support of μ.

Given X and Y measurable spaces, K:Xmk−→Y is a Markov kernel from X to Y. For any x∈X, we have K(x)∈P(Y). Given μ∈P(X), μ⋉K∈P(X×Y) is the semidirect product of μ and K and K∗μ∈P(Y) is the pushforward of μ by K.

Given X, Y Polish spaces, π:X→Y Borel measurable and μ∈P(X), we denote μ∣π the set of Markov kernels K:Ymk−→X s.t. π∗μ⋉K is supported on the graph of π and K∗π∗μ=μ. By the disintegration theorem, μ∣π is always non-empty and any two kernels in μ∣π coincide π∗μ-almost everywhere.

##Results

Consider X any compact Polish metric space and d:X×X→R its metric. We denote Lip(X) the Banach space of Lipschitz continuous functions X→R with the norm

∥f∥Lip:=maxx|f(x)|+supx≠y|f(x)−f(y)|d(x,y)

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

dKR(μ,ν):=sup∥f∥Lip≤1|Eμ[f]−Eν[f]|

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 Φ⊆P(X) convex. We will describe a class of trading strategies that are designed to exploit any μ∈Φ.

#Definition 1

Let α>0. τ∈T(X) is said to be an α-representative of Φ when for all μ∈P(X), ν∈Φ we have

∥τ(μ)∥≤max(dKR(μ,Φ)−α,0)

Eν[τ(μ)]≥Eμ[τ(μ)]+12(dKR(μ,Φ)−α)dKR(μ,Φ)

#Lemma 1

Consider Y another compact Polish space and Φ⊆Y×P(X) closed. For any y∈Y, define Φy⊆P(X) by

Φy:={μ∈P(X)∣(y,μ)∈Φ}

Assume that for any y∈Y, Φy is convex. Then, for any α>0, there exists υα:Y→T(X) measurable w.r.t. U(Y) and B(T(X)) s.t. for all y∈Y, υα(y) is an α-representative of Φy.

For each n∈N, let On be a compact subset of RD(n) (D:N→N is arbitrary). Denote

Yn:=∏m<nOm

Yn is a compact subset of R∑m<nD(m). We will regard it as equipped with the Euclidean metric. Denote

X=∞∏n=0On

X is a compact Polish space. For each n∈N we denote πn:X→Yn the projection mapping. Given n∈N and x∈X, we have x(n):=πn(x)∈Yn and xn∈On.

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

#Definition 2

Consider Φ⊆P(X). For any n∈N and y∈Yn, we define Φ′′y⊆P(X) by

Φ′′y:={limr→0(μ∣π−1n(Br(y)))∣μ∈Φ}

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

% operators that are separated from the operand by a space

% operators that require brackets

% operators that require parentheses

% Paper specific

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 X a metric space, x∈X and r∈R, Br(x):={y∈X∣d(x,y)<r}.

Given X a topological space:

P(X) is the space of Borel probability measures on X equipped with the weak* topology.

C(X) is the Banach space of continuous functions X→R with uniform norm.

T(X):=C(P(X)×X)

B(X) is the Borel σ-algebra on X.

U(X) is the σ-algebra of universally measurable sets on X.

Given μ∈P(X), suppμ denotes the support of μ.

Given X and Y measurable spaces, K:Xmk−→ Y is a Markov kernel from X to Y. For any x∈X, we have K(x)∈P(Y). Given μ∈P(X), μ⋉K∈P(X×Y) is the semidirect product of μ and K and K∗μ∈P(Y) is the pushforward of μ by K.

Given X, Y Polish spaces, π:X→Y Borel measurable and μ∈P(X), we denote μ∣π the set of Markov kernels K:Ymk−→X s.t. π∗μ⋉K is supported on the graph of π and K∗π∗μ=μ. By the disintegration theorem, μ∣π is always non-empty and any two kernels in μ∣π coincide π∗μ-almost everywhere.

##Results

Consider X any compact Polish metric space and d:X×X→R its metric. We denote Lip(X) the Banach space of Lipschitz continuous functions X→R with the norm

∥f∥Lip:=maxx|f(x)|+supx≠y|f(x)−f(y)|d(x,y)

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

dKR(μ,ν):=sup∥f∥Lip≤1|Eμ[f]−Eν[f]|

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 Φ⊆P(X) convex. We will describe a class of trading strategies that are designed to exploit any μ∈Φ.

#Definition 1

Let α>0. τ∈T(X) is said to be an

α-representative of Φwhen for all μ∈P(X), ν∈Φ we have∥τ(μ)∥≤max(dKR(μ,Φ)−α,0)

Eν[τ(μ)]≥Eμ[τ(μ)]+12(dKR(μ,Φ)−α)dKR(μ,Φ)

#Lemma 1

Consider Y another compact Polish space and Φ⊆Y×P(X) closed. For any y∈Y, define Φy⊆P(X) by

Φy:={μ∈P(X)∣(y,μ)∈Φ}

Assume that for any y∈Y, Φy is convex. Then, for any α>0, there exists υα:Y→T(X) measurable w.r.t. U(Y) and B(T(X)) s.t. for all y∈Y, υα(y) is an α-representative of Φy.

For each n∈N, let On be a compact subset of RD(n) (D:N→N is arbitrary). Denote

Yn:=∏m<nOm

Yn is a compact subset of R∑m<nD(m). We will regard it as equipped with the Euclidean metric. Denote

X=∞∏n=0On

X is a compact Polish space. For each n∈N we denote πn:X→Yn the projection mapping. Given n∈N and x∈X, we have x(n):=πn(x)∈Yn and xn∈On.

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

#Definition 2

Consider Φ⊆P(X). For any n∈N and y∈Yn, we define Φ′′y⊆P(X) by

Φ′′y:={limr→0(μ∣π−1n(Br(y)))∣μ∈Φ}

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

Denote Φ′y