Hierarchical Mean-Field Dynamics, Dual Stochastic Decomposition, and Financial Event Operator Algebras
Author: HongJin HE (何泓锦) · HKUST + Stanford IHP · July 2026
Status: Preprint — submitted to arXiv:q-fin.MF
Target journal: Mathematical Finance / SIAM Journal on Financial Mathematics
MSC 2020: 91G99, 60G51, 49N80, 22A22, 60H10
We develop a rigorous mathematical theory for market world models — probabilistic dynamical systems that learn and simulate the full evolutionary law of financial markets, rather than extracting statistical correlations through factor models. The framework rests on five interlocking contributions:
(i) Dual Stochastic Decomposition. We prove that market noise admits a canonical Lévy–Itô decomposition into two orthogonal components: physical noise driven by a Brownian motion (fundamental information uncertainty) and behavioral noise driven by a pure-jump Lévy process (sentiment and herding effects). We derive a Cramér–Rao-type lower bound showing that prediction variance is bounded below by a sum of two irreducible terms — a market-structure analogue of the Heisenberg uncertainty principle.
(ii) Financial Event Operator Algebra. Every corporate or macroeconomic event (IPO, merger, interest rate shock) is modeled as an affine operator on the market state space. We classify all such operators by their algebraic mode and prove that the full event algebra carries the structure of a topological groupoid — a generalization required because mergers and spin-offs alter the dimension of the state space.
(iii) Hierarchical Mean-Field Theory. Real markets exhibit simultaneous MFG and MFC structures at different scales. We formulate a two-level McKean–Vlasov system and prove existence and uniqueness of a hierarchical Nash equilibrium under Lipschitz, boundedness, and Lasry–Lions monotonicity conditions.
(iv) Stochastic Lyapunov Stability. Using Lyapunov methods for combined Brownian-Lévy systems, we prove that the market world model admits a unique invariant measure with sharp exponential convergence rates. Bubbles and crises correspond precisely to local violations of the Lyapunov condition.
(v) E-Game-C Implementation Pathway. We derive a concrete computational realization: bipower-variation calibration (Stage 1); Transformer VAE Encoder with predictive coupling (Stage 2); Deep Galerkin Method solver for HJB + neural fictitious play for Fokker–Planck (Stage 3); gradient-based Controller with risk constraints (Stage 4).
Keywords: world model, mean-field game, stochastic differential equation, Lévy process, groupoid, Lyapunov stability, market microstructure, prediction lower bound.
The three modules implement the paper's five contributions sequentially: the Encoder performs Lévy-Itô decomposition and bipower-variation calibration; the Game module solves the HJB+FPK system via DGM and neural fictitious play; the Controller applies Stochastic Lyapunov stability analysis and enforces the Cramér-Rao prediction floor.
The Lévy-Itô theorem guarantees that any semimartingale decomposes into a continuous (Brownian) component Σ_τ and a pure-jump component ν_η. These are estimated from high-frequency data via bipower variation: BV_T converges to the integrated physical variance, and RV_T − BV_T isolates the jump quadratic variation.
- Probabilistic Foundations
- World Model as an Optimal Estimation Problem
- Dual Stochastic Decomposition
- Financial Event Operator Algebra
- The E-Game-C Architecture
- Hierarchical Mean-Field Theory
- Stochastic Lyapunov Stability
- Implementation Pathway
- Summary of Theorems
- References
Throughout, we fix a complete probability space
We work with two independent sources of randomness:
- A standard
$\mathbb{R}^m$ -valued Brownian motion$W = (W_t)_{t \geq 0}$ - A Poisson random measure
$N^\eta$ on$[0,T] \times \mathbb{R}^k$ with$\sigma$ -finite intensity$\lambda \otimes \nu^\eta$ , where$\lambda$ denotes Lebesgue measure. The compensated measure is$\tilde{N}^\eta(dt,dz) := N^\eta(dt,dz) - \nu^\eta(dz),dt$ .
Both are adapted to
A critical structural observation distinguishes our framework:
Proposition 1.1 (Non-differentiability of Brownian paths). With probability one, $t \mapsto W_t$ is nowhere differentiable. Consequently, the expression $\sigma_t \cdot (dW_t/dt)$ is meaningless pointwise, and the shorthand $dS_t = \mu_t,dt + \sigma_t,dW_t$ represents an integral equation, not a differential equation.
Proof. For any
Definition 1.1 (Itô integral form). The market state process satisfies the stochastic integral equation:
where the Itô integrals are defined as
Proposition 1.2 (Itô Isometry). For any
The adaptedness requirement
Definition 1.2 (Lévy process). A process
Theorem 1.1 (Lévy–Khintchine). The characteristic function of $L_t$ has the form $\mathbb{E}[e^{i\xi^\top L_t}] = e^{-t\Psi(\xi)}$, where the Lévy exponent is:
with drift $b \in \mathbb{R}^k$, positive semi-definite diffusion matrix $C$, and Lévy measure $\nu$ satisfying $\int_{\mathbb{R}^k}(1 \wedge |z|^2),\nu(dz) < \infty$. The triple $(b, C, \nu)$ is the Lévy characteristic and determines $L$ in law.
Definition 1.3 (Single-asset state vector). The state of a single asset at time
| Coordinate | Symbol | Economic Meaning | Domain |
|---|---|---|---|
| 1 | Log-price | ||
| 2 | Log-trading volume | ||
| 3 | Leverage ratio | ||
| 4 | Outstanding shares (log) | ||
| 5 | Public information disclosure level |
For
Definition 2.1 (Market world model). A market world model with parameters
such that $\mathcal{M}_\theta(\mathcal{F}t, h) \approx \mathbb{P}(\mathbf{S}{t+h} \in \cdot \mid \mathcal{F}_t)$.
This definition formalizes the world-model paradigm (Ha & Schmidhuber 2018) for financial environments: rather than extracting correlations, the model learns the full conditional distribution of future market states.
Theorem 2.1 (Prediction Error Decomposition). For any $\mathcal{F}t$-measurable estimator $\hat{\mathbf{S}}{t+h}$ of $\mathbf{S}_{t+h}$:
where the last two terms are strictly positive and model-independent (see Theorem 3.2).
Proof. The bias–variance decomposition is standard. The decomposition into physical and behavioral components follows from the independence established in Theorem 3.1.
Standard models (Black–Scholes) conflate two qualitatively different sources of randomness into a single Brownian motion. We argue this is structurally incorrect:
-
Physical noise
$\tau_t$ : Arises from fundamental information asymmetry, parameter estimation error, and microstructure frictions. Accumulates continuously along trading time — analogous to thermal noise in physics. -
Behavioral noise
$\eta_t$ : Arises from investor sentiment, herding, panic selling, and euphoric buying. Manifests as discontinuous jumps — sudden regime shifts that cannot be generated by any continuous Brownian path.
Assumption 3.1 (Dual noise structure). The noise in the market integral equation decomposes as:
where
Theorem 3.1 (Dual Lévy–Itô Decomposition). Under Assumption 3.1, the total noise process $L_t := \sigma_\tau W_t^{(\tau)} + J_t^{(\eta)}$ has a unique Lévy characteristic $(b, C, \nu)$ that decomposes as:
with $\Sigma_\tau$ (diffusion matrix) and $\nu^\eta$ (jump intensity measure) uniquely identified by:
In particular, $\Sigma_\tau \neq 0$ and $\nu^\eta \neq 0$ are orthogonal: no jump measure can generate a continuous Gaussian component.
Proof. By independence of
Theorem 3.2 (Cramér–Rao-Type Prediction Lower Bound). Let $\mathbf{S}$ satisfy Definition 1.1 under Assumption 3.1. For any $\mathcal{F}t$-measurable unbiased estimator $\hat{\mathbf{S}}{t+h}$ and any horizon $h > 0$:
where $\sigma_\tau^2 = \mathrm{tr}(\Sigma_\tau)$, $\lambda_\eta > 0$ is the jump intensity, and $m_2^\eta = \int_{\mathbb{R}^k}|z|^2,\nu^\eta(dz)$.
Proof.
Step 1 (Physical). Conditional on $\mathcal{F}t$, the increment $\int_t^{t+h}\sigma\tau,dW_u$ is Gaussian with covariance
Step 2 (Behavioral). Let
Step 3 (Independence). By Assumption 3.1, physical and behavioral increments are independent, so variances add: $\mathrm{Var}(\mathbf{S}^{(\tau)}{t+h} + J^{(\eta)}{t+h} - J^{(\eta)}t) = \sigma\tau^2 h + \lambda_\eta m_2^\eta h = \mathrm{CR}(h)$.
Remark (Analogy with Heisenberg Uncertainty). Theorem 3.2 is the financial analogue of the Heisenberg uncertainty principle: even with perfect knowledge of
$\mathcal{F}_t$ and an optimal model, prediction variance cannot fall below$\mathrm{CR}(h)$ . This is a structural feature of the market, not a limitation of methodology.
Corollary 3.1 (Time-Varying Temperature). The scalar temperature parameter
and is an
Definition 4.1 (Affine event operator). For each financial event type
where
Financial events fall into three algebraic modes:
Definition 4.2 (Three modes of action).
-
Type I — Local endomorphism (
$d' = d$ , acts on one asset): stock split, share repurchase, earnings announcement.$A_w \in \mathbb{R}^{d\times d}$ ; state space dimension preserved. -
Type II — Global tensor action (acts on all
$n$ assets via Kronecker product): central bank rate change, systemic shock.$T_w^{\mathrm{global}} = \Lambda_w \otimes I_d$ with$\Lambda_w \in \mathbb{R}^{n\times n}$ encoding heterogeneous sensitivities. -
Type III — Pairwise morphism (
$d'\neq d$ , changes entity count): merger ($n \to n-1$ ), spin-off/IPO ($n \to n+1$ ).$A_w$ is non-square; state space dimension changes.
Proposition 4.1 (Information Irreversibility). For any Type-I event $w$ with natural inverse $w^{-1}$:
where $\mathcal{E}^{\mathrm{info}}_w$ acts non-trivially on the information subspace ($\iota$-coordinate). Matrix invertibility does not imply physical reversibility.
Proof. For a stock split with ratio
Type-III events make
Definition 4.3 (Financial event groupoid). Define
-
Objects:
$\mathrm{Ob}(\mathcal{G}) = {\mathcal{H}_S : S \text{ is a valid } n\text{-asset configuration}}$ - Morphisms: $T_w \in \mathrm{Hom}(\mathcal{H}S, \mathcal{H}{S'})$ with $\mathrm{src}(T_w) = \mathcal{H}S$, $\mathrm{tgt}(T_w) = \mathcal{H}{S'}$
-
Composition:
$T_{w_2} \circ T_{w_1}$ defined iff$\mathrm{src}(T_{w_2}) = \mathrm{tgt}(T_{w_1})$ - Identities: $\mathrm{id}_{\mathcal{H}S} = I{d\cdot|S|}$
- Topology: Operator norm on each $\mathrm{Hom}(\mathcal{H}S, \mathcal{H}{S'})$; disjoint-union topology on the whole space
Theorem 4.1 (Groupoid Structure). $\mathcal{G}_{\mathrm{fin}}$ is a topological groupoid.
Proof. We verify the four axioms:
(i) Well-defined composition. Given $T_{w_1}: \mathcal{H}S \to \mathcal{H}{S'}$ and $T_{w_2}: \mathcal{H}{S'} \to \mathcal{H}{S''}$, composition $T_{w_2} \circ T_{w_1}: \mathcal{H}S \to \mathcal{H}{S''}$ is standard operator composition, well-defined in Banach spaces when intermediate spaces match.
(ii) Associativity. Operator composition in Banach spaces is associative whenever both sides are defined.
(iii) Identity morphisms.
(iv) Partial inverses. For Type-I/II events with invertible
(v) Continuity.
Corollary 4.1. The Type-I events $\mathcal{W}I$ acting on a fixed $n$-asset space form an endomorphism monoid with identity $I{5n}$. For Type-I events with
The V-M-C world model (Ha & Schmidhuber 2018) consists of a VAE encoder V, recurrent world model M, and controller C. Direct application to finance faces three structural obstacles: (a) heterogeneous inputs (prices, news, events), not pixels; (b) reflexive feedback — the "environment" is the aggregate of all agents' decisions; (c) the world model implicitly assumes a fixed dynamical system, ignoring multi-agent strategic structure.
The E-Game-C architecture replaces the recurrent world model with a mean-field game equilibrium operator:
Information Set F_t
│
▼
┌─────────────┐
│ Encoder E │ ── Transformer VAE: x_t → z_t ∈ Z
└─────────────┘
│
▼
┌─────────────┐
│ Game G │ ── MFG Equilibrium Operator: μ_t = Law(z^u_t | â*)
└─────────────┘
│
▼
┌─────────────┐
│ Controller C│ ── Optimal Policy: π*(z_t, h_t) = argmax Q*(z,h,a)
└─────────────┘
│
▼
Portfolio Weights a_t ∈ A
Definition 5.1 (Information encoder). The encoder
Definition 5.2 (Market mean-field game). The market consists of a continuum of agents
where
Theorem 5.1 (MFG Equilibrium Existence). Under conditions:
- (A1) $f$ and $\ell$ are Lipschitz in $(z,a,\mu)$ with constant $L_f$
- (A2) Action space $\mathcal{A}$ is compact and convex
- (A3) Unique minimizer $\hat{a}(z,\mu,u) = \mathrm{argmin}_{a}\mathcal{H}(z,a,\mu,u,p)$ exists
- (A4) Lasry–Lions monotonicity: $\int[\ell(z,\mu,a) - \ell(z,\mu',a)],d(\mu-\mu')(z) \geq 0$
there exists a unique mean-field game equilibrium $(\hat{a}^, \hat{\mu})$ such that (i) $\hat{a}^_t = \mathrm{argmin}_a J^u(a|\hat{\mu})$ for $\mu$-a.e. $u$; and (ii) $\hat{\mu}_t = \mathrm{Law}(z^u_t|\hat{a}^)$ for all
Proof. Define the fixed-point map
Self-mapping: By (A1)–(A2) and Gronwall, SDE solutions satisfy a uniform
Compactness: Tightness of
Continuity: If
Schauder:
Uniqueness: The Lasry–Lions condition (A4) gives
Definition 5.3 (Optimal controller). The controller
where
Real financial markets exhibit hierarchical strategic structure:
- At national/regional scale: sovereign entities interact without a supranational planner — MFG structure
- Within each nation, firms are subject to central bank regulation — partial MFC structure
- At firm level: individual investors interact without coordination — MFG structure
This cannot be captured by a single-level mean-field model.
Level 1 (macro-group dynamics). Each group
where $\nu^{(1)}t = \frac{1}{N}\sum{k=1}^N \delta_{\xi^k_t}$ is the empirical measure,
Level 2 (micro-agent dynamics). Within group
where $\mu^{(2)}{j,t} = \int_0^1 \delta{x^{j,u}_t},du$ is the intra-group mean field. Each agent minimises:
Coupling condition. The levels are coupled through the aggregation functional:
for bounded Lipschitz
Definition 6.1 (Hierarchical Nash Equilibrium). A pair $((\alpha^{j,})_j, (a^{j,u,})_{j,u})$ is a hierarchical Nash equilibrium if:
-
(i) For each group
$j$ and the resulting Level-2 equilibrium $\hat{\mu}^{(2)}j[\xi^j]$: $J_1^j(\alpha^{j,}) \leq J_1^j(\alpha^j)$ for all $\alpha^j$, given $(\alpha^{k,}){k\neq j}$ -
(ii) For each agent
$u$ in group$j$ , given$\xi^j$ and $\mu^{(2)}{j,t} = \hat{\mu}^{(2)}{j,t}[\xi^j]$:$J_2^{j,u}(a^{j,u,*}) \leq J_2^{j,u}(a^{j,u})$ for all$a^{j,u}$
Assumption 6.1 (Regularity at Level 2). (L1)
Assumption 6.2 (Regularity at Level 1). (L5)
Theorem 6.1 (Hierarchical Mean-Field Nash Equilibrium). Under Assumptions 6.1 and 6.2, there exists a unique hierarchical Nash equilibrium in the sense of Definition 6.1.
The proof proceeds through four lemmas.
Lemma 6.1 (Level-2 Stability). Under Assumption 6.1, for any two paths $\xi, \xi' \in C([0,T];\mathbb{R}^{d_1})$:
where $C_{2,\mathrm{stab}} = C_{2,\mathrm{stab}}(L_2, T, \sigma_2) < \infty$.
Proof sketch. Let
Proof of Theorem 6.1 (sketch). By Theorem 5.1 applied to Level 2, for each fixed
Definition 7.1 (Lyapunov candidate). For the market state process $\mathbf{S}t$ under the combined Brownian-Lévy noise of Definition 1.1, the Lyapunov function $V: \mathcal{H} \to \mathbb{R}+$ is:
where
Theorem 7.1 (Stochastic Lyapunov Stability). Suppose the drift $\mu$ and diffusion coefficients $\sigma_\tau, \nu^\eta$ satisfy:
-
(S1) One-sided Lipschitz condition: $\langle \mu(\mathbf{S}) - \mu(\mathbf{S}^), \mathbf{S} - \mathbf{S}^\rangle \leq -\kappa|\mathbf{S} - \mathbf{S}^|^2$ for some
$\kappa > 0$ * - (S2) Diffusion bound: $|\sigma_\tau(\mathbf{S})|^2 \leq C_\sigma(1 + |\mathbf{S}|^2)$
- (S3) Jump integrability: $\int_{\mathbb{R}^k} |z|^2,\nu^\eta(dz) < \infty$
Then:
(i) The market world model admits a unique invariant measure $\pi^ \in \mathcal{P}(\mathcal{H})$.*
(ii) Exponential convergence holds: $W_2(\mathrm{Law}(\mathbf{S}_t), \pi^) \leq C_0,e^{-\gamma t}$ for constants
Proposition 7.1 (Market Bubbles as Lyapunov Violations). A market bubble at time $t^$ corresponds precisely to a local violation of condition (S1): $\langle \mu(\mathbf{S}{t^*}), \mathbf{S}{t^} - \mathbf{S}^\rangle > \kappa|\mathbf{S}_{t^} - \mathbf{S}^|^2$. The bubble burst is the stochastic return to the Lyapunov region.*
The E-Game-C architecture is realized computationally in four stages:
| Stage | Component | Method | Guarantee |
|---|---|---|---|
| 1 | Noise Calibration | Bipower variation estimator for |
Consistent estimators (Aït-Sahalia & Todorov 2009) |
| 2 | Encoder E | Transformer VAE with predictive training: $\mathcal{L} = \mathcal{L}{\mathrm{ELBO}} + \lambda,\mathcal{L}{\mathrm{pred}}$ | ELBO lower bound; representation stability from (S1)–(S3) |
| 3 | Game G | Deep Galerkin Method (DGM) for HJB equation; neural fictitious play for Fokker–Planck equation | Convergence from Lasry–Lions monotonicity (Theorem 5.1) |
| 4 | Controller C | Gradient-based portfolio optimizer: |
Optimality via HJB (Definition 5.3); risk control |
Stage 1 detail. The bipower variation estimator separates
(converges in probability as mesh
Stage 3 detail. The DGM solves the high-dimensional HJB:
using a neural network
| # | Theorem | Content | Section |
|---|---|---|---|
| 1.1 | Lévy–Khintchine | Characteristic function of Lévy process; |
§1.3 |
| 2.1 | Prediction Error Decomposition | $\text{MSE} = \text{Bias}^2 + \text{Var}\tau + \text{Var}\eta$ | §2 |
| 3.1 | Dual Lévy–Itô Decomposition | Unique decomposition of market noise into physical + behavioral; orthogonality | §3.2 |
| 3.2 | Cramér–Rao Prediction Bound |
|
§3.3 |
| 4.1 | Topological Groupoid | Financial event algebra |
§4.3 |
| 5.1 | MFG Equilibrium Existence | Schauder + Lasry–Lions → unique |
§5.2 |
| 6.1 | Hierarchical Nash Equilibrium | Two-level McKean–Vlasov → unique hierarchical Nash equilibrium | §6.4 |
| 7.1 | Stochastic Lyapunov Stability | Unique invariant measure |
§7.2 |
- Aït-Sahalia, Y. & Todorov, V. (2009). Estimating volatility and jumps. Econometrica.
- Carmona, R. & Delarue, F. (2018). Probabilistic Theory of Mean Field Games. Springer.
- Cont, R. & Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall.
- Evans, L.C. (2010). Partial Differential Equations. AMS.
- Ha, D. & Schmidhuber, J. (2018). World Models. NeurIPS.
- Hafner, D. et al. (2020). Dream to Control: Learning Behaviors by Latent Imagination. ICLR.
- Kingma, D.P. & Welling, M. (2014). Auto-Encoding Variational Bayes. ICLR.
- Lasry, J.M. & Lions, P.L. (2007). Mean field games. Japanese Journal of Mathematics.
- Sato, K.I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge.
- Weinstein, A. (1996). Groupoids: Unifying Internal and External Symmetry. AMS Notices.
| File | Description |
|---|---|
paper_draft_v1.pdf |
Full paper (25 pages) — start here |
paper_draft_v1.tex |
LaTeX source |
mathematical_framework_overview.md |
Extended derivations and discussion |
notebooks/ |
Numerical experiments |
@article{he2026worldmodels,
title = {A Mathematical Theory of Market World Models: Hierarchical Mean-Field
Dynamics, Dual Stochastic Decomposition, and Financial Event Operator Algebras},
author = {HongJin HE},
year = {2026},
note = {Preprint. arXiv:q-fin.MF (submitted)},
url = {https://github.com/hongjin-he/mathmatical-framework-for-world-models-in-quant-finance}
}HKUST × Stanford · MIT License · "Factor models learn correlations. World models learn causation. The difference is everything."