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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.
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.
The way we previously laid out the dominant market formalism, the sequence of observations (represented by the sets ) was fixed. To study forecasting, we instead need to assume this sequence is sampled from some probability measure (the true environment).
For each , let be a compact Polish space. represents the space of possible observations at time . Denote
Given , denotes the projection mapping and . Denote
is a compact Polish space. For each we denote the projection mapping. Given , we denote , a closed subspace of . Given and , we denote .
A market is a sequence of mappings s.t.
Each is measurable w.r.t. and .
For any , .
As before, we define the space of trading strategies , but this time we regard it as a Banach space.
A trader is a sequence of mappings which are measurable w.r.t. and .
Given a trader and a market , we define the mappings (measurable w.r.t. and ) and (measurable w.r.t. and ) as follows:
The "market maker" lemma now requires some additional work due to the measurability requirement:
Consider any trader . Then, there is a market s.t. for all and
As before, we have the operator defined by
We also introduce the notation and which are measurable mappings defined by
A market is said to dominate a trader when for any , if
Given any countable set of traders , there is a market s.t. dominates all .
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 s.t. if a market dominates a trader from this class, a certain function of the pricing converges to 0 with -probability 1. In a future post, we will apply this result to a trader associated with an incomplete models by observing that the trader is in the class for any .
A trading metastrategy is a uniformly bounded family of measurable mappings . Given , is said said to be profitable for , when there are and s.t. for any , -almost any and any :
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.
Fix a metastrategy . We define the trader and the measurable mappings recursively as follows:
Consider , , a metastrategy profitable for and a market. Assume dominates . Then, for -almost any :
That is, the market price of the "stock portfolio" traded by converges to its true -expected value.
Fix a compact Polish space and . Then, there exists s.t.
#Proof of Proposition A.1
Follows immediately from "Proposition 1" from before and Proposition B.6.
Fix compact Polish spaces. Denote . Then, there exists measurable w.r.t. and s.t. for any and :
#Proof of Proposition A.2
We can view as the graph of a multivalued mapping from to . 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 .
is closed by Proposition B.7. is closed by Proposition B.5. by Proposition B.6 and hence closed. In particular, the fiber of over any is also closed.
For any , , define by and by . Applying Proposition A.1 to we get s.t.
It follows that and hence is non-empty.
Consider any open. Then, is locally closed and in particular . Therefore, the image of under the projection to is also and in particular Borel.
Applying the Kuratowski-Ryll-Nardzewski measurable selection theorem, we get the desired result.
#Proof of Lemma
For any , let be as in Proposition A.2. We define recursively by:
Consider a probability space, a filtration of , , and stochastic processes adapted to . Assume that:
Then, with probability 1.
The proof will use the following definition:
Consider a sequence . The accumulation times of are defined recursively by
Consider a probability space and a stochastic process. The accumulation times of are defined pointwise as above. Clearly, they are stochastic processes and whenever is adapted to a filtration , they are stopping times w.r.t. .
#Proof of Proposition A.3
Without loss of generality, we can assume (otherwise we can renormalize , and by a factor of ). Define by
By Proposition B.8, is a submartingale. Let be the accumulation times of . By proposition N23, are submartingales for all . By Proposition\ N24, each of them is uniformly integrable. Using the fact that to apply Theorem C, we get
Clearly, is adapted to . Doob's second martingale convergence theorem implies that ( is the limit of the uniformly integrable submartingale ). We conclude that is a submartingale.
By Proposition B.12, . Applying the Azuma-Hoeffding inequality, we conclude that for any positive integer :
Since , it follows that
By Proposition B.13
It remains to show that if is s.t. then the condition above fails. Consider any such . , therefore . On the other hand, by Proposition B.14, .
Consider a probability space, a filtration of , , and stochastic processes adapted to and an arbitrary stochastic process. Assume that:
Then, (equivalently ) with probability 1.
#Proof of Proposition A.4
Define . We have
By Proposition A.3, with probability 1. Since , we get the desired result.
Consider , and as before. Consider , , a metastrategy profitable for and a market. Then, for -almost any :
#Proof of Proposition A.5
We regard as a probability space using the -algebra and the probability measure . For any , we define and by
Clearly, is a filtration of , are stochastic processes and are adapted to . is uniformly bounded, therefore is uniformly bounded and so is . Obviously, is also non-negative.
By Proposition B.15, are uniformly bounded. is bounded and in particular . We have
Let and be as in Definition 4.
By definition of , is equal to either or 0. In either case, we get (almost everywhere)
Applying Proposition A.4, we get the desired result.
Consider the setting of Proposition A.3. Then, for almost all :
#Proof of Proposition A.6
Let be the accumulation times of . Consider any s.t. but . Proposition B.13 implies that
As in the proof of Proposition A.3, we can apply the Azuma-Hoeffding inequality to and get that for any positive integer
It follows that
Comparing with the inequality from before, we reach the desired conclusion.
Consider the setting of Proposition A.4. Then, for almost all :
#Proof of Proposition A.7
Define . As in the proof of Proposition A.4, meets the conditions of Proposition A.3 and thus of Proposition A.6 also. By Proposition A.6, for almost all :
As in the proof of Proposition A.4, is uniformly bounded, giving the desired result.
#Proof of Theorem 2
Let , , and be as in the proof of Proposition A.5. Using Proposition A.5 and the assumption that dominates , we conclude that for -almost any , . 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 . By Proposition B.17, it follows that for any , . We get
If are compact Polish spaces and is continuous, then defined by is continuous.
We omit the proof of Proposition B.1, since it appeared as "Proposition A.2" before.
Fix compact Polish spaces. Define by . Then, is continuous. In particular, we can apply this to in which case .
#Proof of Proposition B.2
Consider and . We have
By Proposition B.1
Combining, we get
Fix compact Polish spaces and denote . Define by
Then, is continuous.
#Proof of Proposition B.3
Consider , . By Proposition B.1, implies that
Since , we get
Fix compact Polish spaces. Denote . Define by
Then, is closed.
#Proof of Proposition B.4
Consider , , , . By Proposition B.3, we get
Fix compact Polish spaces. Denote . Define by
Then, is closed.
#Proof of Proposition B.5
By Proposition B.2, is the continuous inverse image of a subset of which is closed by Proposition B.4.
Fix a compact Polish space. Consider and and denote . Then, iff .
#Proof of Proposition B.6
If then and therefore .
Now, assume . For any , Markov's inequality yields
Taking , we get and hence .
Consider compact Polish spaces. Denote . Define by
Then, is closed.
#Proof of Proposition B.7
We fix metrizations for and and metrize by
For each , denote .
Consider , , . We have . By Proposition B.1, , therefore . By Proposition B.6, and hence .
Consider a probability space, a filtration of , and stochastic processes adapted to . Assume that there are s.t.:
Then, is a submartingale.
#Proof of Proposition B.8
Obviously, is adapted to . We have