Stochastic Process is Topology

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OBSERVATIONS

• branch pattern of infinite length is stochastic process
• so stochastic process is topology
• topology have its shape, order type
• order types are ordinals
• ordinals made metamathematical (with sufficiently strong reflection principle) are forcing notions
• Lebesgue used order types to define integration
• set theorists related enumeration of integration pattern and determinacy

WHAT WE NEED TO BELIEVE IN COMPLETENESS OF PROBABILITY?

• just an assumption saying unmeasurable set would not appear
• regularity assumptions on time (transversality, no ponzi, ..)
• require smoothness of time and economic spacetime hypermanifold
• set-theoretic axioms (stronger reflection principle)
• meta set-theoretic axiom sufficient to proof Con(ZF+AD)

Continuity of Equilibrium Hypermanifold in Economic Spacetime (memorandum)

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MOTIVATION

• Existence of Walrasian equilibrium manifold depends on continuous contraction mapping in n-space of goods with non-negativity of price and quantity
• Think in n+1-"spacetime" and see evolution along time or convergence to above manifold at a point of Ramsey's "bliss"

CONTINUITY OF TIME

• I mean dt/dx in n+1-spacetime, or in adequately blown-ups of such spacetime
• In other words, regularity of stochastic process
• In other words, completeness of expectation
• (added) dynamic constraints like transversality and no Ponzi game must be related to continuity of time

PROBLEM OF JUMP VARIABLES AND DIVERGENT PATH

• "Walrasian equilibrium" in spacetime needs continuity of time
• If no continuity of time is assured, generally divergent paths exist
• ex-ante vs ex-post (realized state of economy is always continuous evolution of the past in ex-post)
• how we treat ex-ante irregular changes as extra dimension added to continuous economic spacetime?

STRATEGIC FOUNDATION OF ECONOMIC DYNAMICS

• existence of pay-off in infinitely repeated games follows from axiom of determinacy (Mycielski-Steinhaus(1962))
• such axiomatics support is really needed? one serious problem is AD contradicts AC so it hurts Walrasian equilibrium thru Uzawa equivalence

So many quantizations (memorandum)

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• glücklichneujahr!!!

QUANTIZATION

• mathematical form which represent infinite dimensional space
• must be separable (and summable) under certain restriction

INFINITELY MANY WAYS OF QUANTIZATION

• wave functions
• linear operators (hamiltonian, probability density, observations, ..)
• paths (in path integral)
• homotopies (might be used in generalization of path integral)
• wavelets
• triangulations of relativistic universe (simplicial decomposition)
• chain complexes (in homology algebra, as generalization of triangulation and observation, and these might be used in future categorical formulations)
• quantum sets (a la Takeuti and Ozawa, as parametrized boolean algebra and generalization of von Neumann's quantum logic)
• formulae valid in various systems of non-standard analysis allowing infinitely many infinitesimals
• (anything that forms topology and can be completed when bounded, given adequate interpretation)

What would be the "22nd century maths"?

• it looks just a matter of encoding (only after when throughly carried out, it should likely be realized that also old maths can be fully employed to formulate and solve)
• similarly in definition of Borel sets, it might include somewhat disguised (and likely finitized in properties of prime numbers) form of transfinite ordinals (order types) to order topologies
• it means: we might need meta-topology, which is just a topology itself, over topologies

Why?

• Since it must be a first-order approximation of "framework subsuming any possible mathematically-definable compatible and incompatible views as reflecting the only reality"
• "views" might be considered as topological spaces
• And it must eliminate infinities (poles) with finitization and interpretation

Completeness of probability measure and abstract 3-body problem (memorandum)

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multi-body system and probability

• what is probabilistic system?
• deriving virtual one-body approximation of multi-body system with probability
• or collapsing into one wave function
• complete measure turns many-body into one-body "falling" into incomparably large "ground"
• Lagrangian of the system make definite description for virtual one-body

future and incomplete probability measure

• sum of contributions from possible Feynman diagrams (no a priori completeness assured)
• estimating contributions from indistinguishable states as zero
• ignoring experimentally unobserved states
• completeness of cases might change over history and environment
• (added) e.g. solar neutrino density and probabilities of beta-decay or other phenomena involving weak forth
• expectation or Knightian uncertainty (tail risk that should be hedged only with longer-than-the-longest bond)
• abstract 3-body problem or game with indefinite sum

gravity or entropic force

• uncertain quantum entropy, indefinite radius of the universe
• Baysian gravity?
• is gravitational constant really constant?

TODO

• understand path integral, S-matrix and Wheeler-DeWitt theory
• relate them with gravity model of trade and transportation

unprepared for dinner yet (my wife just messaged)

unprepared for dinner yet:
lacking one thing for each menu to cook
no garlic for pasta
no celery for minestrone
no flour for chowder
no meat for stewed meat
and, only cabbage for casserole

Who's behind this silly site? (egocentric advertisement)

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• Just Kidding
• Who's behind this silly site? The uneducated blogger answers. Ask me in English (Japanese question might be answered). Anonymous question is allowed, feel free to visit http://ask.fm/TuvianNavy

On Personal Matters and Forthcoming Posts (update November 2013)

For my and my family's health concern, I left Tokyo and moved to Kyoto from September 2013. For courtesy of my employer, my labor contract is kept unchanged. I have some heartache, headache, slightly high blood pressure and random pains but feel somewhat eased. My wife's throat turned well. We enjoy bigger rooms and now I have bookshelf!

I am preparing (at least) two essays, one on history of extensive thoughts for recent four centuries and the other on some biographical facts of ergodic theorist Shizuo Kakutani. Other subjects which have appeared in this blog would be rewritten in paper form.

There is no actual plan for forthcoming posts, but dimension theory (for cases of discrete and continuous dimensions) must be treated first.

(added) I also have a long-awaited plan for simple explaination of various versions of fixed point theorems in economical context, but I have no through understanding on this subject yet.