Portfolio Optimisation as Trajectory Completion
A time-invariant framework for asset allocation under epistemic uncertainty, where the portfolio IS an oscillatory circuit graph and the optimal allocation IS the unique fixed point of a contraction mapping.
1. The Core Idea
Classical portfolio theory (Markowitz, 1952) treats assets as entries in a vector and optimises a quadratic objective. This discards network structure, ignores epistemic uncertainty, and produces allocations that are notoriously unstable.
We model the portfolio as an oscillatory circuit graph: each asset is a node occupying bounded phase space (prices are finite, volumes are finite, returns are bounded), and therefore exhibits oscillatory dynamics by the Poincare recurrence theorem. Each coupling between assets is an edge carrying a conductance derived from a universal transport formula that unifies correlation, capital flow, and information propagation.
Node states are represented as fuzzy membership functions encoding epistemic uncertainty. Kirchhoff's laws provide conservation (KCL: capital balance) and equilibrium (KVL: no-arbitrage) constraints, both lifted to fuzzy arithmetic via the Zadeh extension principle.
2. Fuzzy State Representation
In practice, the true valuation of an asset is never perfectly known. Bid-ask spreads, model uncertainty, and finite-sample correlation estimates all contribute epistemic uncertainty that point estimates discard.
We represent each asset's state as a trapezoidal fuzzy membership function μ̃i : R≥0 → [0,1], where μ̃i(x) = 1 means valuation x is fully consistent with all evidence and μ̃i(x) = 0 means x is excluded. The support width measures irreducible uncertainty; the core width measures the range of equally plausible values.
Fuzzy KCL (Capital Conservation)
At each node, the fuzzy flux balance constraint is enforced by intersecting the current membership function with the KCL-consistent set, restricting valuations to those simultaneously consistent with measurement and flow balance.
Fuzzy KVL (No-Arbitrage)
For each directed cycle, the fuzzy sum of potential differences must contain zero in its core. Any assignment violating this condition has its membership reduced.
3. Trajectory Completion
The backward trajectory of each asset node — its maximum-a-posteriori causal history given current state and network topology — is computed via the Viterbi algorithm. This backward trajectory is the asset's categorical address: a time-invariant geometric identity in the portfolio's state space.
Convergence Theorem (Banach Fixed-Point)
The trajectory completion operator T = T_Back ∘ T_KVL ∘ T_KCL is a contraction mapping on the Hausdorff product metric space of fuzzy state tuples. By the Banach fixed-point theorem, iteration converges geometrically to a unique fixed point X* — the optimal portfolio allocation.
Time-Invariance Theorem
The optimal allocation X* depends only on the current state and network topology, not on the absolute time of observation. The portfolio rebalances because the network structure has changed, not because the calendar has advanced.
4. Risk Analysis
Portfolio risk is measured by the total fuzzy support width of the fixed point, bounded by the spectral gap of the graph Laplacian:
Spectral Risk Bound
R ≤ R_0 / λ_2 · (N-M)/N, where λ_2 is the Fiedler value (algebraic connectivity), N is the total number of nodes, and M is the number of observed nodes.
Exponential Shock Decay
An external price shock at a boundary node propagates to internal node i with amplitude decaying exponentially: |Δφ_i| ≤ Δφ_0 · exp(-√λ_2 · d_G(b,i)), where d_G is the graph distance.
Diversification in this framework is not merely “holding many assets” — it is increasing the algebraic connectivity λ2 of the portfolio graph.
5. Classical Limit: Markowitz Recovery
Markowitz as Special Case
When all fuzzy states are crisp (zero epistemic uncertainty), the network is complete with uniform conductance, and backward trajectory constraints are vacuous, the fixed point reduces to the Markowitz mean-variance optimal portfolio.
The trajectory completion framework strictly generalises Markowitz in three independent directions: fuzzy states handle epistemic uncertainty, sparse non-uniform conductance replaces the full covariance matrix, and backward trajectory constraints enforce kinetic consistency.
6. Harmonic Coincidence and Regime Detection
FFT spectral decomposition of asset return series reveals harmonic coincidences between assets whose characteristic frequencies are rationally related. The shadow portfolio network built from these coincidences detects market regime transitions through topological changes in the spectral correlation graph.
7. Experimental Validation (7/7)
Convergence rate scales with λ₂
CONFIRMED — 69 iterations at λ₂=0.016, 46 at λ₂=121. Correlation = -0.87.
Time-invariance of optimal allocation
CONFIRMED — Zero drift across Δt ∈ {0, 10, 100, 1000, 10000}. Exact zero.
Fuzzy risk is a meaningful risk bound
CONFIRMED — Non-zero residual uncertainty preserved. Mean fuzzy risk = 0.008.
Risk scales as R ∝ 1/λ₂
CONFIRMED — Risk drops from 1000 to 8 as λ₂ increases from 0.016 to 229.
Shock decays exponentially with distance
CONFIRMED — R² > 0.99 exponential fit. Decay rate γ = 0.96 per hop.
Harmonic coincidence detects regime changes
CONFIRMED — Both spectral and correlation detect within 15 days of regime change.
Markowitz recovery as special case
CONFIRMED — Uniform conductance → exact 1/N weights (distance = 0.000000).