Financial markets exhibit heterogeneous investment horizons and nonstationary dependence, so portfolio risk and comovement are intrinsically multiscale and time varying. Classical single-horizon covariance models and repeated static reoptimization can therefore be fragile in high-dimensional universes, especially considering transaction costs.

We proposed a reduced-order portfolio optimization method in which long-only allocations evolved dynamically through simplex-preserving mirror flow. The strategy was parameterized by a small number of time-varying decision variables, and exposures to a larger asset universe were obtained by combining a few investable basis portfolios. The basis was updated online from the principal components of a wavelet-based multiscale covariance estimator and mapped to admissible weights through an entropy-based softmax construction. A wavelet multiscale layer was integrated as a structural component of the model rather than as a preprocessing step, inducing both a multiscale quadratic risk geometry and a risk-normalized multiscale return signal that drove allocation dynamics.

We provided a self-contained formulation and established core properties of the scheme, including positive definiteness and stability of the multiscale covariance updates, stability of the principal component analysis (PCA)–softmax basis under perturbations, simplex invariance of the mirror update, and explicit turnover control that transferred from reduced allocations to traded allocations, including the effect of periodic basis refresh. Empirical experiments were then conducted on real equity data, including transaction costs, compared against equal-weight and classical mean–variance baselines, including sensitivity and ablation analyses. The results illustrated that the multiscale and reduced-order components produced materially different dynamic allocations and contributed to risk-adjusted performance and trading stability. We referred to the proposed method as WPROD-R, a shorthand for a wavelet-driven product (multiplicative mirror-flow) scheme with a reduced-order representation.

We introduced GATE-WPCA-PI—geometry-aware, tracking-error-controlled allocation with wavelet principle component analysis features and a proportional-integral controller—a practical portfolio construction framework that linked multi-scale market geometry to explicit, out-of-sample risk targeting. At each rebalance, the daily returns were embedded in a multi-resolution wavelet feature space and compressed via principal component analysis to form a similarity kernel. A simple discriminative-power score gated the optimizer: when the cross section was heterogeneous, the feature geometry was activated; when it was homogeneous, the method reverted to a correlation-only view. Allocations were obtained from an implementable mean–variance surrogate with (ⅰ) a geometry penalty that discouraged concentration in highly similar assets, (ⅱ) quadratic and absolute turnover costs, (ⅲ) an entropy floor, and (ⅳ) standard long-only, budget, and sleeve caps. A proportional-integral (PI) law treated the tracking error (TE) as a controllable state and steered realized TE toward a feasible band under trading frictions.

Exergy-dissipation portfolio optimization (EDPO) is a physics-inspired alternative to mean-variance allocation that treats portfolio formation as a nonequilibrium optimization problem. Rather than maximizing an estimated mean return subject to a variance penalty, EDPO maximizes an exergy rate $X_a(\mathbf{w})$, interpreted as a risk-sensitive certainty equivalent of log-growth, while penalizing an entropy production rate (EPR) $\mathcal{D}(\mathbf{w})$ that quantifies time-irreversibility in the portfolio return process. We propose a tractable long-only implementation with stabilizers including weight caps, Kullback–Leibler (KL), and turnover regularization along with a dissipation-budget thermostat: a primal–dual control law that updates $\lambda$ to keep the in-sample dissipation ${\mathcal{D}}_{\mathrm{I}\mathrm{S}}$ close to a target budget ${\sigma }_{\mathrm{b}\mathrm{u}\mathrm{d}}$ without explicit regime labels. The accompanying theory establishes (i) a variational identity $X_a={\mathbb{E}}_Q[g]+a^{-1}{D}_{\mathrm{K}\mathrm{L}}(Q\Vert P)$, linking exergy to a tilted expected log-growth and an information term, and (ii) a constrained efficient frontier in the $(X_a,\mathcal{D})$ plane, with a convex candidate-set dual used as an internal consistency diagnostic. An ablation study confirms that combining exergy and dissipation improves risk-adjusted performance relative to either term in isolation. Empirically, a rolling backtest over 2015–2025 on four exchange-traded fund (ETF) universes shows that EDPO is competitive with standard benchmarks. In the diversified ETF7 universe, EDPO attains the highest Sharpe ratio (0.955) with moderate quarterly turnover (0.251), while remains comparable in more correlated universes. These model-based thermodynamic diagnostics are economically interpretable: stress episodes coincide with elevated $\mathcal{D}$, and the thermostat raises $\lambda$ when dissipation pressure increases. Finally, $\mathcal{D}$ contains incremental predictive content for realized volatility in three of the four universes, as measured by mean squared error (MSE) ratios below one, although the associated forecast-gain tests remain exploratory once serial dependence and multiple comparisons are taken into account. A compact empirical-evidence package further shows that the conclusions are most sensitive to the coarse-graining level $K$, moderately sensitive to the lookback length $L$, and least sensitive to the smoothing parameter $\alpha$. The strategy remains viable under plausible transaction costs.