% operators that are separated from the operand by a space

% operators that require brackets

% operators that require parentheses

% Paper specific

We generalize the formalism of dominant markets to account for stochastic “deductive processes,” and prove a theorem regarding the asymptotic behavior of such markets. In a following post, we will show how to use these tools to formalize the ideas outlined here.

Appendix A contains the key proofs. Appendix B contains the proofs of technical propositions used in Appendix A, which are mostly straightforward. Appendix C contains the statement of a version of the optional stopping theorem from Durret.

##Notation

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.

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

The way we previously laid out the dominant market formalism, the sequence of observations (represented by the sets {Xi}) was fixed. To study forecasting, we instead need to assume this sequence is sampled from some probability measure (the true environment).

For each n∈N, let On be a compact Polish space.On represents the space of possible observations at time n. Denote

Yn:=∏m<nOm

Given n≤m, πnm:Ym→Yn denotes the projection mapping and πn:=πn,n+1. Denote

X=∞∏n=0On

X is a compact Polish space. For each n∈N we denote πnω:X→Yn the projection mapping. Given y∈Yn, we denote Xy:=π−1nω(y), a closed subspace of X. Given n∈Z and x∈X, we denote x(n):=πmax(n,0)ω(x).

#Definition 1

A market is a sequence of mappings {Mn:Yn→P(X)}n∈N s.t.

Each Mn is measurable w.r.t.U(Yn) and B(P(X)).

For any y∈Yn, suppMn(y)⊆Xy.

As before, we define the space of trading strategies T(X):=C(P(X)×X), but this time we regard it as a Banach space.

#Definition 2

A trader is a sequence of mappings {Tn:Yn×P(X)n→T(X)}n∈N which are measurable w.r.t.U(Yn)⊗B(P(X)n) and B(T(X)).

Given a trader T and a market M, we define the mappings {TMn:Yn→T(X)}n∈N (measurable w.r.t.U(Yn) and B(T(X))) and {¯TMn:Yn→C(X)}n∈N (measurable w.r.t.U(Yn) and B(C(X))) as follows:

The “market maker” lemma now requires some additional work due to the measurability requirement:

#Lemma

Consider any trader T. Then, there is a market M s.t. for all n∈N and y∈Yn

suppMn(y)⊆argmaxXy¯TMn(y)

As before, we have the operator W:T(X)→T(X) defined by

Wτ(μ,x):=τ(μ,x)−Ez∼μ[τ(μ,z)]

We also introduce the notation {W¯TMn:Yn→C(X)}n∈N and {ΣWTMn:Ymax(n−1,0)→C(X)}n∈N which are measurable mappings defined by

W¯TMn(y)=WTMn(y,Mn(y))

ΣWTMn(y)=∑m<nW¯TMm(πmn(y))

#Definition 3

A market M is said to dominate a trader T when for any x∈X, if

infn∈Nminz∈Xx(n)ΣWTMn+1(x(n),z)>−∞

then

supn∈Nmaxz∈Xx(n)ΣWTMn+1(x(n),z)<+∞

#Theorem 1

Given any countable set of traders R, there is a market M s.t.M dominates all T∈R.

Theorem 1 is proved exactly as before (modulo Lemma), and we omit the details.

We now describe a class of traders associated with a fixed environment μ∗∈P(X) s.t. if a market dominates a trader from this class, a certain function of the pricing converges to 0 with μ∗-probability 1. In a future post, we will apply this result to a trader associated with an incomplete models Φ⊆P(X) by observing that the trader is in the class for any μ∗∈Φ.

#Definition 4

A trading metastrategy is a uniformly bounded family of measurable mappings {υn:Yn→T(X)}n∈N. Given μ∗∈P(X), υ is said said to be profitable for μ∗, when there are β>0 and {Kn∈μ∗∣πnω}n∈N s.t. for any n∈N, πnω∗μ∗-almost any y∈Yn and any μ∈P(Xy):

Even if a metastrategy is profitable, it doesn’t mean that a smart trader should use this metastrategy all the time: in order to avoid running out of budget, a trader shouldn’t place too many bets simultaneously. The following construction defines a trader that employs a metastrategy only when all previous bets are closed to being resolved.

#Definition 5

Fix a metastrategy υ. We define the trader Tυ and the measurable mappings {Uυn:Yn×P(X)n→C(X)}n∈N recursively as follows:

We can view Z as the graph of a multivalued mapping from Yn×T(X) to P(X). We will now show this multivalued mapping has a selection, i.e. a single-valued measurable mapping whose graph is a subset. Obviously, the selection is the desired M.

Z1 is closed by Proposition B.7.Z2 is closed by Proposition B.5.Z=Z1∩Z2 by Proposition B.6 and hence closed. In particular, the fiber Zyτ of Z over any (y,τ)∈Y×T(X) is also closed.

For any y∈Y, τ∈T(X), define iy:Y′→X by iy(y′):=(y,y′) and τy∈T(Y′) by τy(ν,y′):=τ(iy∗ν,y,y′). Applying Proposition A.1 to τy we get ν∈P(Y′) s.t.

suppτy(ν)⊆argmaxτy

It follows that (y,τ,iy∗ν)∈Z and hence Zyτ is non-empty.

Consider any U⊆P(X) open. Then, AU:=(Y×T(X)×U)∩Z is locally closed and in particular Fσ. Therefore, the image of AU under the projection to Y×T(X) is also Fσ and in particular Borel.

Applying the Kuratowski-Ryll-Nardzewski measurable selection theorem, we get the desired result.

#Proof of Lemma

For any n∈N, let Mn:Yn×T(X)→P(X) be as in Proposition A.2. We define Mn recursively by:

Mn(y):=Mn(y,TMn(y))

#Proposition A.3

Consider X a probability space, {Fn⊆2X}n∈N a filtration of X, t,α,β>0, {Sn:X→R}n∈N and {Δn:X→[0,t]}n∈N stochastic processes adapted to F. Assume that:

E[|S0|]<∞

∀n′≥n:|Sn′−Sn|≤n′−1∑m=nΔm+α

E[Sn+1−Sn∣Fn]≥βΔn

Then, infnSn>−∞ with probability 1.

The proof will use the following definition:

#Definition A

Consider a sequence {tn∈[0,1]}n∈N. The accumulation times of t are {nk∈N⊔{∞}}k∈N defined recursively by

n0:=0

nk+1={inf{n∈N∣∑n−1m=nktm≥1} if nk<∞∞ if nk=∞

Consider X a probability space and {Δn:X→[0,1]}n∈N a stochastic process. The accumulation times of Δ are {Nk:X→N⊔{∞}}k∈N defined pointwise as above. Clearly, they are stochastic processes and whenever Δ is adapted to a filtration {Fn⊆2X}n∈N, they are stopping times w.r.t. F.

#Proof of Proposition A.3

Without loss of generality, we can assume t=1 (otherwise we can renormalize S, Δ and α by a factor of t−1). Define {S0n:X→R}n∈N by

S0n:=Sn−βn−1∑m=0Δm

By Proposition B.8, S0 is a submartingale. Let {Nn:X→N⊔{∞}}n∈N be the accumulation times of Δ. By proposition N23, {S0min(n,Nk)}n∈N are submartingales for all k. By Proposition\ N24, each of them is uniformly integrable. Using the fact that Nk≤Nk+1 to apply Theorem C, we get

E[S0Nk+1∣FNk]≥S0Nk

Clearly, {S0Nk}k∈N is adapted to {FNk}k∈N. Doob’s second martingale convergence theorem implies that E[|S0Nk|]<∞ (S0Nk is the limit of the uniformly integrable submartingale {S0min(n,Nk)}n∈N). We conclude that {S0Nk}k∈N is a submartingale.

By Proposition B.12, |S0Nk+1−S0Nk|≤α+2. Applying the Azuma-Hoeffding inequality, we conclude that for any positive integer k:

It remains to show that if x∈X is s.t.infnSn(x)=−∞ then the condition above fails. Consider any such x∈X. |Sn(x)−S0(x)|≤∑n−1m=0Δn(x)+α, therefore ∑∞n=0Δn(x)=∞. On the other hand, by Proposition B.14, infkSNk(x)(x)=−∞.

#Proposition A.4

Consider X a probability space, {Fn⊆2X}n∈N a filtration of X, t,α,β>0, {S′n:X→R}n∈N and {Δn:X→[0,t]}n∈N stochastic processes adapted to F and {Sn:X→R}n∈N an arbitrary stochastic process. Assume that:

|Sn−S′n|≤α4

E[|S0|]<∞

|Sn+1−Sn|≤Δn

E[Sn+1−Sn∣Fn]≥βΔn

Then, infnSn>−∞ (equivalently infnS′n>−∞) with probability 1.

By Proposition A.3, infnS′′n>−∞ with probability 1. Since |S′′n−Sn|≤α2, we get the desired result.

#Proposition A.5

Consider {On}n∈N, {Yn:=∏m<nOm}n∈N and X:=∏nOn as before. Consider μ∗∈P(X), {Kn∈μ∗∣πnω}n∈N, υ a metastrategy profitable for μ∗ and M a market. Then, for μ∗-almost any x∈X:

infn∈Nminz∈Xx(n)ΣWTMυ,n+1(x(n),z)>−∞

#Proof of Proposition A.5

We regard X as a probability space using the σ-algebra U(X) and the probability measure μ∗. For any n∈N, we define Fn⊆U(X) and Sn,S′n,Δn:X→R by

Clearly, F is a filtration of X, S,S′,Δ are stochastic processes and S′,Δ are adapted to F. υ is uniformly bounded, therefore Tυ is uniformly bounded and so is Δ. Obviously, Δ is also non-negative.

By Proposition B.15, |Sn−S′n| are uniformly bounded.S0 is bounded and in particular E[|S0|]<∞. We have

Comparing with the inequality from before, we reach the desired conclusion.

#Proposition A.7

Consider the setting of Proposition A.4. Then, for almost all x∈X:

supn∈NSn(x)<∞⟹∞∑n=0Δn(x)<∞

#Proof of Proposition A.7

Define S′′n:=E[Sn∣Fn]. As in the proof of Proposition A.4, S′′ meets the conditions of Proposition A.3 and thus of Proposition A.6 also. By Proposition A.6, for almost all x∈X:

supn∈NS′′n(x)<∞⟹∞∑n=0Δn(x)<∞

As in the proof of Proposition A.4, |S′′n−Sn| is uniformly bounded, giving the desired result.

#Proof of Theorem 2

Let F, S, S′ and Δ be as in the proof of Proposition A.5. Using Proposition A.5 and the assumption that M dominates Tυ, we conclude that for μ∗-almost any x∈X, supnSn(x)<∞. As in the proof of Proposition A.5, the conditions of Proposition A.4 are satisfied, and therefore the conditions of Proposition A.7 are also satisfied. Applying Proposition A.7, we conclude that ∑nΔn(x)<∞. By Proposition B.17, it follows that for any n≫0, TMυn(x(n))=υ(x(n)). We get

If X,Y are compact Polish spaces and f:X×Y→R is continuous, then F:X→C(Y) defined by F(x)(y):=f(x,y) is continuous.

We omit the proof of Proposition B.1, since it appeared as “Proposition A.2” before.

#Proposition B.2

Fix X,Y compact Polish spaces. Define e:C(Y×X)×Y→C(X) by e(f,y)(x):=f(y,x). Then, e is continuous. In particular, we can apply this to Y=P(X) in which case e:T(X)×P(X)→C(X).

#Proof of Proposition B.2

Consider fk→f and yk→y. We have

maxx∈X|fk(yk,x)−f(yk,x)|≤∥fk−f∥→0

By Proposition B.1

maxx∈X|f(yk,x)−f(y,x)|→0

Combining, we get

maxx∈X|fk(yk,x)−f(y,x)|→0

#Proposition B.3

Fix Y,Y′ compact Polish spaces and denote X:=Y×Y′. Define F:Y×C(X)→R by

F(y,f):=maxy′∈Y′f(y,y′)

Then, F is continuous.

#Proof of Proposition B.3

Consider yk→y, fk→f. By Proposition B.1, yk→y implies that

limk→∞maxy′∈Y′f(yk,y′)=maxy′∈Y′f(y,y′)

Since fk→f, we get

limk→∞maxy′∈Y′fk(yk,y′)=maxy′∈Y′f(y,y′)

#Proposition B.4

Fix Y,Y′ compact Polish spaces. Denote X:=Y×Y′. Define Z⊆Y×P(X)×C(X) by

Z:={(y,μ,f)∈Y×P(X)×C(X)∣Eμ[f]=maxy′∈Y′f(y,y′)}

Then, Z is closed.

#Proof of Proposition B.4

Consider yk→y, μk→μ, fk→f, (yk,μk,fk)∈Z. By Proposition B.3, we get

Consider yk→y, μk→μ, (yk,μk)∈Z. We have Eμk[dyk]=0. By Proposition B.1, dyk→dy, therefore Eμ[dy]=0. By Proposition B.6, suppμ⊆y×Y′ and hence (y,μ)∈Z.

#Proposition B.8

Consider X a probability space, {Fn⊆2X}n∈N a filtration of X, {Sn:X→R}n∈N and {Δn:X→[0,1]}n∈N stochastic processes adapted to F. Assume that there are α,β>0 s.t.:

## Dominant stochastic markets

% operators that are separated from the operand by a space

% operators that require brackets

% operators that require parentheses

% Paper specific

We generalize the formalism of dominant markets to account for stochastic “deductive processes,” and prove a theorem regarding the asymptotic behavior of such markets. In a following post, we will show how to use these tools to formalize the ideas outlined here.

Appendix A contains the key proofs. Appendix B contains the proofs of technical propositions used in Appendix A, which are mostly straightforward. Appendix C contains the statement of a version of the optional stopping theorem from Durret.

##Notation

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.

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

The way we previously laid out the dominant market formalism, the sequence of observations (represented by the sets {Xi}) was fixed. To study forecasting, we instead need to assume this sequence is sampled from some probability measure (the true environment).

For each n∈N, let On be a compact Polish space.On represents the space of possible observations at time n. Denote

Yn:=∏m<nOm

Given n≤m, πnm:Ym→Yn denotes the projection mapping and πn:=πn,n+1. Denote

X=∞∏n=0On

X is a compact Polish space. For each n∈N we denote πnω:X→Yn the projection mapping. Given y∈Yn, we denote Xy:=π−1nω(y), a closed subspace of X. Given n∈Z and x∈X, we denote x(n):=πmax(n,0)ω(x).

#Definition 1

A

marketis a sequence of mappings {Mn:Yn→P(X)}n∈N s.t.Each Mn is measurable w.r.t.U(Yn) and B(P(X)).

For any y∈Yn, suppMn(y)⊆Xy.

As before, we define the space of trading strategies T(X):=C(P(X)×X), but this time we regard it as a Banach space.

#Definition 2

A

traderis a sequence of mappings {Tn:Yn×P(X)n→T(X)}n∈N which are measurable w.r.t.U(Yn)⊗B(P(X)n) and B(T(X)).Given a trader T and a market M, we define the mappings {TMn:Yn→T(X)}n∈N (measurable w.r.t.U(Yn) and B(T(X))) and {¯TMn:Yn→C(X)}n∈N (measurable w.r.t.U(Yn) and B(C(X))) as follows:

TMn(y):=Tn(y,M0(π0n(y)),M1(π1n(y))…Mn−1(πn−1,n(y)))

¯TMn(y):=TMn(y,Mn(y))

The “market maker” lemma now requires some additional work due to the measurability requirement:

#Lemma

Consider any trader T. Then, there is a market M s.t. for all n∈N and y∈Yn

suppMn(y)⊆argmaxXy¯TMn(y)

As before, we have the operator W:T(X)→T(X) defined by

Wτ(μ,x):=τ(μ,x)−Ez∼μ[τ(μ,z)]

We also introduce the notation {W¯TMn:Yn→C(X)}n∈N and {ΣWTMn:Ymax(n−1,0) →C(X)}n∈N which are measurable mappings defined by

W¯TMn(y)=WTMn(y,Mn(y))

ΣWTMn(y)=∑m<nW¯TMm(πmn(y))

#Definition 3

A market M is said to dominate a trader T when for any x∈X, if

infn∈Nminz∈Xx(n)ΣWTMn+1(x(n),z)>−∞

then

supn∈Nmaxz∈Xx(n)ΣWTMn+1(x(n),z)<+∞

#Theorem 1

Given any countable set of traders R, there is a market M s.t.M dominates all T∈R.

Theorem 1 is proved exactly as before (modulo Lemma), and we omit the details.

We now describe a class of traders associated with a fixed environment μ∗∈P(X) s.t. if a market dominates a trader from this class, a certain function of the pricing converges to 0 with μ∗-probability 1. In a future post, we will apply this result to a trader associated with an incomplete models Φ⊆P(X) by observing that the trader is in the class for any μ∗∈Φ.

#Definition 4

A

trading metastrategyis a uniformly bounded family of measurable mappings {υn:Yn→T(X)}n∈N. Given μ∗∈P(X), υ is said said to beprofitable for μ∗, when there are β>0 and {Kn∈μ∗∣πnω}n∈N s.t. for any n∈N, πnω∗μ∗-almost any y∈Yn and any μ∈P(Xy):EKn(y)[υ(y,μ)]−Eμ[υ(y,μ)]≥β(maxXyυ(y,μ)−minXyυ(y,μ))

Even if a metastrategy is profitable, it doesn’t mean that a smart trader should use this metastrategy all the time: in order to avoid running out of budget, a trader shouldn’t place too many bets simultaneously. The following construction defines a trader that employs a metastrategy only when all previous bets are closed to being resolved.

#Definition 5

Fix a metastrategy υ. We define the trader Tυ and the measurable mappings {Uυn:Yn×P(X)n→C(X)}n∈N recursively as follows:

U0:=0

Tυ0:=υ0

Uυ,n+1(y,{μm}m≤n):=Uυn(πn(y),{μm}m<n)+Tυn(y,{μm}m≤n)

Tυ,n+1(y,{μm}m≤n):={υn+1(y) if maxXyUυ,n+1(y,{μm}m≤n)−minXyUυ,n+1(y,{μm}m≤n)≤10 otherwise

#Theorem 2

Consider μ∗∈P(X), {Kn∈μ∗∣πnω}n∈N, υ a metastrategy profitable for μ∗ and M a market. Assume M dominates Tυ. Then, for μ∗-almost any x∈X:

limn→∞(EKn(x(n))[υ(x(n),Mn(x(n)))]−EMn(x(n))[υ(x(n),Mn(x(n)))])=0

That is, the market price of the “stock portfolio” traded by υ converges to its true μ∗-expected value.

##Appendix A

#Proposition A.1

Fix X a compact Polish space and τ∈T(X). Then, there exists μ∈P(X) s.t.

suppτ(μ)⊆argmaxτ

#Proof of Proposition A.1

Follows immediately from “Proposition 1” from before and Proposition B.6.

#Proposition A.2

Fix Y,Y′ compact Polish spaces. Denote X:=Y×Y′. Then, there exists M:Y×T(X)→P(X) measurable w.r.t.B(Y×T(X)) and B(P(X)) s.t. for any y∈Y and τ∈T(X):

suppM(y,τ)⊆argmaxy×Y′τ(M(y,τ))

#Proof of Proposition A.2

Define Z1,Z2,Z⊆Y×T(X)×P(X) by

Z1:={(y,τ,μ)∈Y×T(X)×P(X)∣suppμ⊆Xy}

Z2:={(y,τ,μ)∈Y×T(X)×P(X)∣Eμ[τ(μ)]=maxy∈Y′τ(μ,y,y′)}

Z:={(y,τ,μ)∈Y×T(X)×P(X)∣suppμ⊆argmaxy×Y′τ(μ)}

We can view Z as the graph of a

multivaluedmapping from Yn×T(X) to P(X). We will now show this multivalued mapping has aselection, i.e. a single-valued measurable mapping whose graph is a subset. Obviously, the selection is the desired M.Z1 is closed by Proposition B.7.Z2 is closed by Proposition B.5.Z=Z1∩Z2 by Proposition B.6 and hence closed. In particular, the fiber Zyτ of Z over any (y,τ)∈Y×T(X) is also closed.

For any y∈Y, τ∈T(X), define iy:Y′→X by iy(y′):=(y,y′) and τy∈T(Y′) by τy(ν,y′):=τ(iy∗ν,y,y′). Applying Proposition A.1 to τy we get ν∈P(Y′) s.t.

suppτy(ν)⊆argmaxτy

It follows that (y,τ,iy∗ν)∈Z and hence Zyτ is non-empty.

Consider any U⊆P(X) open. Then, AU:=(Y×T(X)×U)∩Z is locally closed and in particular Fσ. Therefore, the image of AU under the projection to Y×T(X) is also Fσ and in particular Borel.

Applying the Kuratowski-Ryll-Nardzewski measurable selection theorem, we get the desired result.

#Proof of Lemma

For any n∈N, let Mn:Yn×T(X)→P(X) be as in Proposition A.2. We define Mn recursively by:

Mn(y):=Mn(y,TMn(y))

#Proposition A.3

Consider X a probability space, {Fn⊆2X}n∈N a filtration of X, t,α,β>0, {Sn:X→R}n∈N and {Δn:X→[0,t]}n∈N stochastic processes adapted to F. Assume that:

E[|S0|]<∞

∀n′≥n:|Sn′−Sn|≤n′−1∑m=nΔm+α

E[Sn+1−Sn∣Fn]≥βΔn

Then, infnSn>−∞ with probability 1.

The proof will use the following definition:

#Definition A

Consider a sequence {tn∈[0,1]}n∈N. The

accumulation times of tare {nk∈N⊔{∞}}k∈N defined recursively byn0:=0

nk+1={inf{n∈N∣∑n−1m=nktm≥1} if nk<∞∞ if nk=∞

Consider X a probability space and {Δn:X→[0,1]}n∈N a stochastic process. The

accumulation times of Δare {Nk:X→N⊔{∞}}k∈N defined pointwise as above. Clearly, they are stochastic processes and whenever Δ is adapted to a filtration {Fn⊆2X}n∈N, they are stopping times w.r.t. F.#Proof of Proposition A.3

Without loss of generality, we can assume t=1 (otherwise we can renormalize S, Δ and α by a factor of t−1). Define {S0n:X→R}n∈N by

S0n:=Sn−βn−1∑m=0Δm

By Proposition B.8, S0 is a submartingale. Let {Nn:X→N⊔{∞}}n∈N be the accumulation times of Δ. By proposition N23, {S0min(n,Nk)}n∈N are submartingales for all k. By Proposition\ N24, each of them is uniformly integrable. Using the fact that Nk≤Nk+1 to apply Theorem C, we get

E[S0Nk+1∣FNk]≥S0Nk

Clearly, {S0Nk}k∈N is adapted to {FNk}k∈N. Doob’s second martingale convergence theorem implies that E[|S0Nk|]<∞ (S0Nk is the limit of the uniformly integrable submartingale {S0min(n,Nk)}n∈N). We conclude that {S0Nk}k∈N is a submartingale.

By Proposition B.12, |S0Nk+1−S0Nk|≤α+2. Applying the Azuma-Hoeffding inequality, we conclude that for any positive integer k:

Pr[S0Nk−S0<−βk]≤exp(−(βk)22(α+2)2k)=exp(−β2k2(α+2)2)

Since ∑kexp(−β2k2(α+2)2)<∞, it follows that

Pr[∃k∈N∀l>k:S0Nl−S0≥−βl]=1

Pr[∃k∈N∀l>k:SNl−S0≥β(Nl−1∑n=0Δn−l)]=1

By Proposition B.13

Pr[∞∑n=0Δn=∞⟹∃k∈N∀l>k:SNl−S0≥0]=1

It remains to show that if x∈X is s.t.infnSn(x)=−∞ then the condition above fails. Consider any such x∈X. |Sn(x)−S0(x)|≤∑n−1m=0Δn(x)+α, therefore ∑∞n=0Δn(x)=∞. On the other hand, by Proposition B.14, infkSNk(x)(x)=−∞.

#Proposition A.4

Consider X a probability space, {Fn⊆2X}n∈N a filtration of X, t,α,β>0, {S′n:X→R}n∈N and {Δn:X→[0,t]}n∈N stochastic processes adapted to F and {Sn:X→R}n∈N an arbitrary stochastic process. Assume that:

|Sn−S′n|≤α4

E[|S0|]<∞

|Sn+1−Sn|≤Δn

E[Sn+1−Sn∣Fn]≥βΔn

Then, infnSn>−∞ (equivalently infnS′n>−∞) with probability 1.

#Proof of Proposition A.4

Define S′′n:=E[Sn∣Fn]. We have

E[|S′′0|]=E[|E[S0∣F0]|]≤E[E[|S0|∣F0]]=E[|S0|]<∞

|S′′n−S′n|=|E[Sn∣Fn]−S′n|=|E[Sn−S′n∣Fn]|≤α4

|S′′n−Sn|≤|S′′n−S′n|+|S′n−Sn|≤α2

∀n′≥n:|S′′n′−S′′n|≤|Sn′−Sn|+α≤n′−1∑m=nΔm+α

E[S′′n+1−S′′n∣Fn]=E[E[Sn+1∣Fn+1]−E[Sn∣Fn]∣Fn]=E[Sn+1−Sn∣Fn]≥βΔn

By Proposition A.3, infnS′′n>−∞ with probability 1. Since |S′′n−Sn|≤α2, we get the desired result.

#Proposition A.5

Consider {On}n∈N, {Yn:=∏m<nOm}n∈N and X:=∏nOn as before. Consider μ∗∈P(X), {Kn∈μ∗∣πnω}n∈N, υ a metastrategy profitable for μ∗ and M a market. Then, for μ∗-almost any x∈X:

infn∈Nminz∈Xx(n)ΣWTMυ,n+1(x(n),z)>−∞

#Proof of Proposition A.5

We regard X as a probability space using the σ-algebra U(X) and the probability measure μ∗. For any n∈N, we define Fn⊆U(X) and Sn,S′n,Δn:X→R by

Fn:=π−1nω(U(Yn))

Sn(x):=ΣWTMυn(x(n−1),x)

S′n(x):=minz∈Xx(n−1)ΣWTMυn(x(n−1),z)

Δn(x):=maxz∈Xx(n)¯TMυn(x(n),z)−minz∈Xx(n)¯TMυn(x(n),z)

Clearly, F is a filtration of X, S,S′,Δ are stochastic processes and S′,Δ are adapted to F. υ is uniformly bounded, therefore Tυ is uniformly bounded and so is Δ. Obviously, Δ is also non-negative.

By Proposition B.15, |Sn−S′n| are uniformly bounded.S0 is bounded and in particular E[|S0|]<∞. We have

|Sn+1(x)−Sn(x)|=|W¯TMυn(x(n),x)|≤Δn(x)

Let β>0 and Kn∈μ∗∣πnω be as in Definition 4.

E[Sn+1−Sn∣Fn]=Ez∼Kn(x(n))[W¯TMυn(x(n),z)]

E[Sn+1−Sn∣Fn]=Ez∼Kn(x(n))[¯TMυn(x(n),z)]−Ez∼Mn(x(n))[¯TMυn(x(n),z)]

By definition of Tυ, ¯TMυn(x(n)) is equal to either υn(x(n),Mn(x(n))) or 0. In either case, we get (almost everywhere)

E[Sn+1−Sn∣Fn]≥β(maxXx(n)¯TMυn(x(n))−minXx(n)¯TMυn(x(n)))=βΔn

Applying Proposition A.4, we get the desired result.

#Proposition A.6

Consider the setting of Proposition A.3. Then, for almost all x∈X:

supn∈NSn(x)<∞⟹∞∑n=0Δn(x)<∞

#Proof of Proposition A.6

Define {S0n:X→R}n∈N by

S0n:=Sn−βn−1∑m=0Δm

Let {Nn}n∈N be the accumulation times of Δ. Consider any x∈X s.t.supnSn(x)=s(x)<∞ but ∑nΔn(x)=∞. Proposition B.13 implies that

S0Nk(x)(x)≤SNk(x)(x)−βk≤s(x)−βk

As in the proof of Proposition A.3, we can apply the Azuma-Hoeffding inequality to S0N and get that for any positive integer k

Pr[S0Nk−S0<−k34]≤exp(−k322(α+2)2k)=exp(−k122(α+2)2)

It follows that

∞∑k=1Pr[S0Nk−S0<−k34]<∞

Pr[∃k∈N∀l>k:S0Nl−S0<−l34]=0

Pr[∃m∈N∀k∈N:S0Nk≤m−βk]=0

Comparing with the inequality from before, we reach the desired conclusion.

#Proposition A.7

Consider the setting of Proposition A.4. Then, for almost all x∈X:

supn∈NSn(x)<∞⟹∞∑n=0Δn(x)<∞

#Proof of Proposition A.7

Define S′′n:=E[Sn∣Fn]. As in the proof of Proposition A.4, S′′ meets the conditions of Proposition A.3 and thus of Proposition A.6 also. By Proposition A.6, for almost all x∈X:

supn∈NS′′n(x)<∞⟹∞∑n=0Δn(x)<∞

As in the proof of Proposition A.4, |S′′n−Sn| is uniformly bounded, giving the desired result.

#Proof of Theorem 2

Let F, S, S′ and Δ be as in the proof of Proposition A.5. Using Proposition A.5 and the assumption that M dominates Tυ, we conclude that for μ∗-almost any x∈X, supnSn(x)<∞. As in the proof of Proposition A.5, the conditions of Proposition A.4 are satisfied, and therefore the conditions of Proposition A.7 are also satisfied. Applying Proposition A.7, we conclude that ∑nΔn(x)<∞. By Proposition B.17, it follows that for any n≫0, TMυn(x(n))=υ(x(n)). We get

EKn(x(n))[υ(x(n),Mn(x(n)))]−EMn(x(n))[υ(x(n),Mn(x(n)))]=EKn(x(n))[¯TMυn(x(n))]−EMn(x(n))[¯TMυn(x(n))])

EKn(x(n))[υ(x(n),Mn(x(n)))]−EMn(x(n))[υ(x(n),Mn(x(n)))]≤Δn(x)

limn→∞(EKn(x(n))[υ(x(n),Mn(x(n)))]−EMn(x(n))[υ(x(n),Mn(x(n)))])=0

##Appendix B

#Proposition B.1

If X,Y are compact Polish spaces and f:X×Y→R is continuous, then F:X→C(Y) defined by F(x)(y):=f(x,y) is continuous.

We omit the proof of Proposition B.1, since it appeared as “Proposition A.2” before.

#Proposition B.2

Fix X,Y compact Polish spaces. Define e:C(Y×X)×Y→C(X) by e(f,y)(x):=f(y,x). Then, e is continuous. In particular, we can apply this to Y=P(X) in which case e:T(X)×P(X)→C(X).

#Proof of Proposition B.2

Consider fk→f and yk→y. We have

maxx∈X|fk(yk,x)−f(yk,x)|≤∥fk−f∥→0

By Proposition B.1

maxx∈X|f(yk,x)−f(y,x)|→0

Combining, we get

maxx∈X|fk(yk,x)−f(y,x)|→0

#Proposition B.3

Fix Y,Y′ compact Polish spaces and denote X:=Y×Y′. Define F:Y×C(X)→R by

F(y,f):=maxy′∈Y′f(y,y′)

Then, F is continuous.

#Proof of Proposition B.3

Consider yk→y, fk→f. By Proposition B.1, yk→y implies that

limk→∞maxy′∈Y′f(yk,y′)=maxy′∈Y′f(y,y′)

Since fk→f, we get

limk→∞maxy′∈Y′fk(yk,y′)=maxy′∈Y′f(y,y′)

#Proposition B.4

Fix Y,Y′ compact Polish spaces. Denote X:=Y×Y′. Define Z⊆Y×P(X)×C(X) by

Z:={(y,μ,f)∈Y×P(X)×C(X)∣Eμ[f]=maxy′∈Y′f(y,y′)}

Then, Z is closed.

#Proof of Proposition B.4

Consider yk→y, μk→μ, fk→f, (yk,μk,fk)∈Z. By Proposition B.3, we get

maxy′∈Y′f(y,y′)=limk→∞maxy′∈Y′fk(yk,y′)=limk→∞Eμk[fk]=Eμ[f]

Hence, (y,μ,f)∈Z.

#Proposition B.5

Fix Y,Y′ compact Polish spaces. Denote X:=Y×Y′. Define Z⊆Y×P(X)×T(X) by

Z:={(y,μ,τ)∈Y×P(X)×T(X)∣Eμ[τ(μ)]=maxy′∈Y′τ(μ,y,y′)}

Then, Z is closed.

#Proof of Proposition B.5

By Proposition B.2, Z is the continuous inverse image of a subset of Y×P(X)×C(X) which is closed by Proposition B.4.

#Proposition B.6

Fix X a compact Polish space. Consider f∈C(X) and μ∈P(X) and denote M:=maxf. Then, suppμ⊆f−1(M) iff Eμ[f]=M.

#Proof of Proposition B.6

If suppμ⊆f−1(M) then Prx∼μ[f(x)≠M]=0 and therefore Eμ[f]=M.

Now, assume Eμ[f]=M. For any k∈N, Markov’s inequality yields

Prx∼μ[M−f(x)≥1k]≤kEx∼μ[M−f(x)]=0

Taking k→∞, we get Prx∼μ[M>f(x)]=0 and hence suppμ⊆f−1(M).

#Proposition B.7

Consider Y,Y′ compact Polish spaces. Denote X:=Y×Y′. Define Z⊆Y×P(X) by

Z:={(y,μ)∈Y×P(X)∣suppμ⊆y×Y′}

Then, Z is closed.

#Proof of Proposition B.7

We fix metrizations for Y and Y′ and metrize X by

dX((y1,y′1),(y2,y′2)):=max(dY(y1,y2),dY′(y′1,y′2))

For each y∈Y, denote dy:=dy×Y′.

Consider yk→y, μk→μ, (yk,μk)∈Z. We have Eμk[dyk]=0. By Proposition B.1, dyk→dy, therefore Eμ[dy]=0. By Proposition B.6, suppμ⊆y×Y′ and hence (y,μ)∈Z.

#Proposition B.8

Consider X a probability space, {Fn⊆2X}n∈N a filtration of X, {Sn:X→R}n∈N and {Δn:X→[0,1]}n∈N stochastic processes adapted to F. Assume that there are α,β>0 s.t.:

E[|S0|]<∞

|Sn−S0|≤n−1∑m=0Δm+α

E[Sn+1−Sn∣Fn]≥βΔn

Define {S0n:X→R}n∈N by

S0n:=Sn−βn−1∑m=0Δm

Then, S0 is a submartingale.

#Proof of Proposition B.8

Obviously, S0 is adapted to F. We have

E[|S0n|]≤E[|Sn|]+βn≤E[|S0|]+E[|Sn−S0|]+βn≤E[|S0|]+2βn+α<∞

E[S0n+1∣Fn]=E[Sn+1∣Fn]−βn∑m=0Δm

E[S0n+1∣Fn]≥E[Sn∣Fn]+βΔn−βn∑m=0Δm

E[S0n+1∣Fn]≥E[Sn∣Fn]−βn−1∑m=0Δm

E[S0n+1∣Fn]≥E[S0n∣Fn]

#Proposition B.9

Let X be a probability space, {Fn⊆2X}n∈N a filtration on