filtrationsigma-algebraprobability-spacemeasure-theoryfinancial-forecasting

Filtration-Indexed Probability Space (滤链索引概率空间)

A filtered probability space (\Omega, \mathcal{F}, \{\mathcal{F}_n\}_{n\ge1}, P) is a measurable space (\Omega, \mathcal{F}) with a probability measure P and a non-decreasing family \{\mathcal{F}_n\} of sub-σ-algebras — so \mathcal{F}_n \subseteq \mathcal{F}_{n+1} \subseteq \mathcal{F} and \mathcal{F}_n "sees more information" as n grows. In Achury Arredondo Sanchez Ascanio 2024 Filtration ARIMA Random Walk the filtration is the structural device that lets an ARIMA Forecast Model and a Random Walk Price Pathwise share the same sample space \Omega = \mathbb{R} while carrying different information.

Two-σ-algebra filtration used in the paper

Let \mathcal{F}_A be the σ-algebra generated by the h+1 intervals U(\eta_i, \delta/2) around the random-walk endpoints (Section 3 of the source); let \mathcal{F}_B be the Borel σ-algebra on \mathbb{R} that supports the ARIMA-implied normal density (Section 2). Then

\mathcal{F}_A \;\subseteq\; \mathcal{F}_B \;\subseteq\; \mathcal{F}.

  • \mathcal{F}_A is coarser: it only resolves "the price lands in this U(\eta_i, \delta/2) bin or that one."
  • \mathcal{F}_B is finer: it resolves any open interval on \mathbb{R} and supports a normal CDF.
  • \mathcal{F}_A \subseteq \mathcal{F}_B because every U(\eta_i, \delta/2) is a Borel set.

The conditional expectation \mathbb{E}[\,\cdot \mid \mathcal{F}_A\,] is the operation that takes the ARIMA normal density and collapses it to a mass per bin — that is the normal tail-difference term in Filtered Probability Trend Hypothesis Proposition 1.

Why a filtration and not a mixture

A naive approach would weight the ARIMA and the random walk with a fixed scalar \alpha \in (0,1) and call it a day. The filtration does something different: it lets the two distributions live on different resolutions of the same sample space and then enforces a product measure on the intersection. Concretely,

  • If the ARIMA density is flat across the bins (no information beyond "is it up or down"), the product reduces to the binomial mass on its own.
  • If the binomial is very peaked (a strongly up-trending or down-trending market where the walk is nearly deterministic), the product pulls the ARIMA's normal mass into the corresponding bin and multiplies it heavily.
  • If the two disagree — ARIMA says "bin i" with high normal mass but the random walk says "bin j \ne i" — the renormalising denominator will spread the mass back out, and the top-2-bin test may declare the trend undecidable.

This is the practical content of putting \mathcal{F}_A inside \mathcal{F}_B rather than just averaging two predictions.

Diagram

flowchart TB
    OMEGA[Sample space Ω = ℝ prices]
    FA[σ-algebra 𝓕_A: random-walk bins U(ηᵢ, δ/2)]
    FB[σ-algebra 𝓕_B: Borel σ-algebra from ARIMA normal]
    FFULL[σ-algebra 𝓕 = 𝓕_B]
    P[P: filtered probability measure]
    P -->|product of normal tail-diff × binomial mass, renormalised| FFULL

    OMEGA --> FA
    OMEGA --> FB
    FA -->|subset| FB
    FB -->|equals| FFULL
    FA -->|conditional expectation gives bin mass| P
    FB -->|normal density integrates on each bin| P

Conditional expectations in the construction

  • \mathbb{E}[\,1_{U(\eta_i, \delta/2)} \mid \mathcal{F}_B\,] = \Phi\!\bigl(\frac{\eta_i + \delta/2 - \hat Y_h}{\sigma \sqrt h}\bigr) - \Phi\!\bigl(\frac{\eta_i - \delta/2 - \hat Y_h}{\sigma \sqrt h}\bigr) — the ARIMA-implied normal mass on bin i.
  • \mathbb{E}[\,1_{U(\eta_i, \delta/2)} \mid \mathcal{F}_A\,] = P_i = \binom{h}{i} p^i (1-p)^{h-i} — the binomial mass on bin i.
  • The product / renormalised expression is the filtered probability P\bigl(Y_h \in U(\eta_i, \delta/2)\bigr) in Filtered Probability Trend Hypothesis Proposition 1.

This is one of the cleanest worked examples in the literature of using conditional-expectation notation in a forecasting problem rather than in pure stochastic-process theory.

Limitations as a measure-theoretic construction

  • Information set is fixed. The filtration is the same for all 25 names. A more refined construction would let \sigma and the random-walk step sizes drift with realised volatility, but that breaks the closed-form tractability of Proposition 1.
  • Product measure, not a mixture. A genuinely Bayesian update would be P(\text{bin} \mid \text{ARIMA}), not the product of the marginals. Achury et al. justify the product by appealing to the disjointness of the bins, but in a strict measure-theoretic sense it is a scoring rule rather than a posterior.
  • Two σ-algebras only. A natural extension is \{\mathcal{F}_n\}_{n=1}^{N} with N "levels of information" — e.g. add \mathcal{F}_C for volume, \mathcal{F}_D for sentiment, \mathcal{F}_E for macro. The paper lists this as future-work item (1).
  • No optional stopping / no martingale structure. Classical filtrations are usually built on martingales; this filtration is a static construction, so the Doob-Meyer decomposition does not apply. For trading purposes this is fine, but it is worth noting for anyone trying to extend it.

Cross-references