Pith. sign in

REVIEW 3 major objections 6 minor 24 references

Dynamic portfolios hedge the motion of a common-driver geometry, and switches in that geometry leave an unhedgeable residual.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Dynamic portfolio choice under causal/conditional separation decomposes into a projected myopic fund plus a first-order hedge of the rotating common-driver manifold, with geometry jumps Kunita–Watanabe-orthogonal to continuous trades.

T0 review reviewed 2026-07-10 challenge →

load-bearing objection Solid continuous-time portfolio theory that makes the conditioning geometry the priced state, with two genuinely sharp results (KW incompleteness of separator jumps; metric-strain transport) and honest synthetic checks; causal reading rests on a maintained hypothesis and companion identification. the 3 major comments →

arxiv 2607.06702 v1 pith:OKOT3VQE submitted 2026-07-07 q-fin.PM

Dynamic Causal Portfolio Choice: Hedging the Rotation of the Common-Driver Manifold

Alejandro Rodriguez Dominguez This is my paper
classification q-fin.PM MSC 91G1093E2062H22
keywords dynamic portfolio choicemanifold hedgingintertemporal hedging demandcausal separationKunita-Watanabe decompositioninformation geometryregime switching
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

When assets become mutually independent given a small set of observable drivers, the natural state of continuous-time portfolio choice is that conditioning geometry rather than the asset list. The geometry moves: sensitivities depend on the driver state, the separator can switch, and the induced information geometry rotates or jumps. The optimal policy then splits into a static projected allocation along the current geometry and an intertemporal hedge against that geometry's predictable motion, which the paper treats as a first-order demand rather than a refinement. Because the problem is organized by the drivers, the governing computation lives in the driver dimension, not the asset dimension. At a separator switch the value process jumps in a direction that continuous trading cannot span, so the market is incomplete along its own geometry. Synthetic economies isolate each piece: the hedge, the rotation, the dimension reduction, the unhedgeable jump, and the transport connection that forces geometric turnover.

Core claim

Under conditional factorization on a common-driver manifold, continuous-time optimal policy is the myopic projected Markowitz fund plus a manifold hedge against the predictable motion and rotation of that geometry; when the separator switches, the resulting value jump is Kunita–Watanabe orthogonal to every continuous traded strategy, so incompleteness lies in the market's own conditioning geometry, and the whole control problem reduces to the number of drivers rather than assets.

What carries the argument

The common-driver manifold as state: instantaneous residual orthogonality (the diffusion normal form) yields a projected myopic-plus-manifold-hedge policy whose priced state is the geometry's coordinates, together with the Kunita–Watanabe orthogonality of separator-switch jumps and a metric-compatible transport connection carrying a strain correction for geometric turnover.

Load-bearing premise

The drivers must truly separate the assets so residuals are independent and the conditional law of returns given the drivers stays stable under environment interventions; without that, the geometry being hedged is not interventionally meaningful.

What would settle it

In a synthetic economy with known state-dependent loadings and known switches, check whether the value jump's best-hedge R-squared by continuous assets stays near zero, and whether the manifold-rotation hedge adds a significant certainty-equivalent gain only when the geometry is estimated accurately; failure of either collapses the central claims.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. The paper develops continuous-time portfolio choice when assets are conditioned on a minimal driver set that induces mutual residual independence (a diagonal-plus-low-rank conditional covariance). In a controlled diffusion normal form, the conditioning geometry becomes the natural state: loadings may depend on the driver state, the separator may switch, and the induced information geometry rotates or jumps. The optimal CRRA policy decomposes into a myopic projected-Markowitz fund plus an intertemporal hedge against predictable motion of that geometry (Theorems 3.4, 3.6), with the HJB posed in driver dimension m rather than asset dimension n. Separator switches produce value jumps that are Kunita–Watanabe orthogonal to continuous self-financing strategies, so a continuous-asset market is incomplete along its own geometry (Theorem 3.11). A metric-compatible transport connection with a derived strain term governs exposure transport between dates (Theorem 4.5). Results are verified against synthetic DGPs that isolate each mechanism.

Significance. If the maintained conditional-factorization and continuous-asset hypotheses hold, the paper supplies a clean geometric organization of intertemporal hedging in which the priced state is the certified conditioning geometry rather than exogenous macro factors, together with a genuine incompleteness result (KW orthogonality of geometry jumps) and a derived metric-strain connection that has no close antecedent in the intertemporal portfolio literature. The dimension-m reduction and exact two-stage Woodbury assembly are practically relevant for large books. Strengths include carefully stated verification theorems (classical Riccati for constant B; viscosity under a priori bounds for state-dependent B), an explicit horizon–instantaneous bridge under H1–H3, honest scope language separating conditional-factor mathematics from the causal reading imported from the companion, synthetic experiments that falsify claims against a known DGP, and a full reproducibility package. The contribution is theoretical and mechanism-isolating rather than an empirical market test.

major comments (3)
  1. For state-dependent B(z)—the paper’s central “manifold-rotation” object—Theorem 3.4 and the discussion after (5) establish optimality only in the viscosity sense under an a priori bound that g, 1/g and ∇_z g remain bounded (also the hypothesis of Theorem 3.10). The grid-refinement study in §5.2 is numerical evidence of a Cauchy sequence, not an existence proof. Please either (i) prove existence of a bounded Lipschitz viscosity solution for the reduced viscous HJ under the paper’s ellipticity/mean-reversion assumptions, or (ii) restate the state-dependent optimality claim explicitly as conditional on that a priori bound, and separate more sharply the classical Riccati case (fully closed) from the viscosity case in the abstract and contribution list.
  2. Lemma 3.3 (horizon ⇔ instantaneous) and the dynamic theorems rest on H1–H3 and the structural separator of Definition A.2 / Lemma A.3. The paper is explicit that causal identification is deferred to companion [1] and that the dynamic results are conditional-factor statements. That is appropriate, but the title and abstract lead with “Dynamic Causal Portfolio Choice” and “common-driver manifold.” Please add a short, prominent scope box (or strengthen the existing §1 paragraph) stating that all dynamic control theorems are conditional-factor results given the normal form and residual orthogonality, and that the interventional-invariance (causal) reading is an additional claim that requires the companion’s identification. Without that, readers may treat the hedge as intervention-stable by default.
  3. Corollary 3.6 and §5.1 note that the finite-horizon Riccati has a bounded solution only for moderate risk aversion and premium relative to mean-reversion, and that beyond that region the value diverges. The manifold-hedge CE gains that support the “first-order effect” claim (§5.1–5.2: ~0.8–2.5%) are reported in the persistent, strongly-remunerated regime near that boundary. Please quantify the admissible (R, premium, κ) region more explicitly (e.g., a simple inequality or a figure of the blow-up locus) and report CE gains also well inside the comfortably bounded region, so that “first-order” is not read only near the singularity where compact W must bind.
minor comments (6)
  1. Figure 1 caption refers to “optimal policy (7)” while the displayed decomposition is (6); align equation numbers in captions with the main text.
  2. Proposition 3.7 is labeled “Proposition” in the text but “Theorem 3.7” in the contribution list and figure captions; unify numbering.
  3. In §3.5 / Theorem 3.8, the simulation fractions (“about one in ten,” “0/400,” “0/500”) are useful; please state the exact DGP parameters and seed protocol in the figure caption or a short appendix table so the monotonicity checks are fully reproducible from the text alone.
  4. Table 4’s “nearest antecedent” column is helpful; a one-line pointer to where the metric-strain term is derived (not posited) relative to Edelman–Arias–Smith would help non-geometry readers.
  5. Typos / notation: “L´ evy” and “Itˆ o” appear with broken accents in several places; “Gt-orthogonal” vs “G_t-orthogonal” is inconsistent; fix “sunspot-free” if it is not defined.
  6. The reproducibility package is cited (Zenodo DOI); please confirm that the pinned environment (Python 3.12, numpy 2.4.4, etc.) and fixed seeds regenerate every reported CE, R², and monotonicity fraction in §5.

Circularity Check

1 steps flagged

Mild self-citation for the causal framing; the HJB, KW, and transport derivations are self-contained conditional-factor mathematics, not forced by definition or by the companion.

specific steps
  1. self citation load bearing [§1 scope paragraph; Definition A.2 / Lemma A.3 (Appendix A.2); companion [1]]
    "Their identification with a causal separator, the guarantee that the conditioning set is intervention invariant rather than merely predictive, is established in the static companion [1]; the single consequence the dynamic theory uses, invariance of the priced geometry under environment interventions, is stated and proved here from an explicit structural hypothesis... A reader may take the development as a conditional-factor dynamic portfolio theory; the causal reading is the additional content the companion supplies."

    The dynamic theorems are proved under a maintained structural hypothesis (Definition A.2). That the declared drivers actually form such a separator—and thus that the hedge is against an intervention-stable object rather than a correlational artifact—is not established in this paper; it is imported from the same-author companion [1]. This makes the causal reading of the title and of the hedge load-bearing on self-citation, while leaving the conditional-factor mathematics intact. The paper is explicit about the split, so the circularity is limited to the interpretive layer, not the HJB/KW derivations.

full rationale

The paper’s load-bearing dynamic results (Theorems 3.4, 3.6, 3.10, 3.11, 4.5) are standard HJB verification, Kunita–Watanabe orthogonality of a purely discontinuous switch martingale to continuous asset integrals, and metric-compatible transport obtained by differentiating the frame normalization E⊤GE = I. Each is derived under explicitly stated conditional-factorization and market-structure hypotheses (normal form (4), H1–H3, continuous traded assets, a priori bounds on g). Synthetic experiments check those theorems against a known DGP; reported CE gains, residual R², and monotonicity fractions are therefore not fitted-input “predictions.” The only circularity-adjacent element is that the word “causal” and the interventional-stability reading of the hedge rest on Definition A.2 / Lemma A.3, whose identification is deferred to the same-author companion [1]. The paper itself flags this scope split (“a reader may take the development as a conditional-factor dynamic portfolio theory; the causal reading is the additional content the companion supplies”), so the self-citation is load-bearing for the causal title claim but not for the mathematical content of the strongest theorems. No self-definitional identity, no fitted parameter renamed as prediction, no uniqueness theorem imported from the authors, and no ansatz smuggled via self-citation were found. Score 2 reflects one non-central self-citation layer with independent central derivations.

Axiom & Free-Parameter Ledger

4 free parameters · 6 axioms · 4 invented entities

The central claims rest on a conditional-factor diffusion normal form plus a structural separator hypothesis imported from the companion, standard stochastic-control regularity, and continuous traded paths for incompleteness. Experimental free parameters calibrate regimes where the value stays bounded; they do not define the theorems but do select where hedge gains are measurable. Invented entities are mostly named geometric objects (manifold, configuration space, strain connection) rather than new physical forces; their independent handle is synthetic falsifiability against known DGPs, not external market measurement in this paper.

free parameters (4)
  • Relative risk aversion R and mean-reversion/premium calibrations (e.g. κ, premium levels) = Examples: R=4, κ=0.6, premium 0.35 (shape); R=8, κ=0.15, premium 0.55 (value)
    Chosen to exhibit bounded Riccati/value regimes and first-order hedge effects in §5; outside moderate R/premium relative to mean reversion the finite-horizon value can blow up (stated limitation).
  • Switch intensities λ_kj(z) and regime-loading configurations
    Bounded intensities enter the coupled HJB (Thm 3.10) and detector experiments; geometry of switches (rotation vs anisotropic) is dialed in central study I.
  • State-dependence amplitude of B(z) (manifold mobility dial) = Mobility sweep 0→0.6; geometric-turnover share up to ~66%
    Controls magnitude of rotation hedge, strain, curvature, and geometric turnover share in §5.2 and central study II.
  • Partial-Sharpe refinement margin (finite-sample)
    Practical decision rule for adding drivers scales a noise margin to estimation error (Thm 3.8 discussion); population sign is exact, sample rule is tuned.
axioms (6)
  • domain assumption Instantaneous separation: residual local martingales of assets are pairwise KW-orthogonal given the driver filtration, upgrading under ellipticity to the diffusion normal form (4).
    Definition 3.1, Lemma 3.2; organizes Q=BΛB⊤+diag(ς²) and all subsequent projectors.
  • domain assumption Horizon screening-off iff instantaneous residual orthogonality under nondegeneracy H1–H3 (recoverable W^Z, residual innovations independent of W^Z, state-dependent residual vols F^Z-measurable).
    Lemma 3.3; bridges static companion separation to continuous-time model.
  • ad hoc to paper Structural causal separator: r=f(Z,ε) with root-exogenous independent Z,ε and f invariant to environment interventions (Def A.2), implying invariance of μ(z),Q(z),B(z),M_C(z).
    Appendix A.2; identification deferred to companion [1]; needed for causal rather than merely conditional reading of the hedge.
  • domain assumption Traded assets have continuous paths; switch process is purely discontinuous, hence KW-orthogonal to continuous stable subspace.
    Theorem 3.11 hypothesis; incompleteness claim is conditional on this market structure.
  • standard math Admissible controls in compact convex W with linear tangent constraint Cw=0; CRRA utility; bounded Lipschitz uniformly elliptic coefficients for switching verification.
    Standing assumptions for HJB verification (Thms 3.4, 3.10) and to rule out unbounded-position pathologies.
  • domain assumption A priori boundedness of g,1/g,∇_z g (or non-blow-up of Riccati) for classical/viscosity verification in state-dependent and switching cases.
    Stated explicitly for Thms 3.4(ii), 3.6, 3.10; checked numerically via grid refinement rather than proved globally.
invented entities (4)
  • Common-driver manifold / conditioning geometry (D*, M_t, Δ_t) as priced state no independent evidence
    purpose: Replaces exogenous macro state variables in Merton ICAPM as the object whose predictable motion is hedged.
    Central modeling object of the framework; independent evidence in this paper is synthetic DGP isolation plus cited applied companions, not a new external measurement here.
  • Horizontal manifold-rotation hedge (vs fixed-span vertical hedge) independent evidence
    purpose: Isolates the novel intertemporal demand arising only when B(z) is state-dependent.
    Defined via Gt-orthogonal split in Thm 4.5 / §3.2; magnitude quantified synthetically (+~2% CE in persistent regime).
  • Configuration space C as disjoint union of Grassmannian bundles over the driver lattice independent evidence
    purpose: Classifies representation motions into flow, swap, and refinement/coarsening (Thm 3.7).
    Geometric taxonomy organizing switches and frontier motion; standard Grassmannian geometry applied to driver sets.
  • Metric-strain connection eω on pullback metric G(z)=B⊤Σ_ret⁻¹B independent evidence
    purpose: Metric-compatible transport of exposures inducing geometric turnover (rotation + breathing).
    Derived from differentiating frame orthonormality (Thm 4.5, App A.3); not posited ad hoc, but named as the paper’s transport law.

reviewed 2026-07-10 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Dynamic Causal Portfolio Choice: Hedging the Rotation of the Common-Driver Manifold." pith.science (2026). https://pith.science/paper/OKOT3VQE

@misc{pith2026260706702,
  author       = {Pith},
  title        = {Pith review of: Dynamic Causal Portfolio Choice: Hedging the Rotation of the Common-Driver Manifold},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKOT3VQE}},
  note         = {Machine review of arXiv:2607.06702}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

When a portfolio is conditioned on a minimal set of observable drivers under which its assets become mutually independent over the investment horizon, the dynamic investment problem acquires a distinctive geometric structure. We study continuous-time portfolio choice in this setting. The conditioning representation, rather than the asset vector, becomes the natural state of the problem, and it moves: the sensitivity of returns to the drivers depends on the state, the conditioning set may itself change over time, and the induced information geometry both rotates and, at discrete instants, jumps. The optimal policy separates into a static component that allocates along the conditioning geometry at each instant and an intertemporal component that hedges the predictable motion of that geometry, a first-order effect in the model rather than a refinement, placing the coordinates of the information geometry in the role played by exogenous state variables in classical intertemporal asset pricing. Because the problem is organized by the drivers, its computational cost is governed by their number rather than by the number of assets. Changes in the conditioning set generate a risk that continuous trading cannot span, so the market is incomplete in the direction of its own geometry. The analysis is carried out in a controlled diffusion model, and the resulting structure is illustrated on synthetic economies designed to isolate each mechanism.

Figures

Figures reproduced from arXiv: 2607.06702 by Alejandro Rodriguez Dominguez.

Figure 1
Figure 1. Figure 1: The optimal policy (7) as two projected funds. The myopic fund projects the conditional premium onto the tangent-admissible space; the manifold hedge passes the predicted state motion ψt through the driver noise ΛZ, the sensitivity B(z), and the same projector MC. The hedge is nonzero when the information geometry is predictably moving. when the loading map B(z) depends on the driver state; it is the hedge… view at source ↗
Figure 2
Figure 2. Figure 2: The moving manifold (Section 3.3). With state-dependent sensitivities B(z) the tangent space TxMt rotates as the driver state Z flows. The intertemporal term hedges this predictable rotation of the geometry; in the fixed-geometry case the span does not move and only the premium does. depend on z through B(z) and MC(z). When B is state-dependent the common-driver manifold rotates as the driver state moves (… view at source ↗
Figure 3
Figure 3. Figure 3: The configuration space C = F m Gr(m, n) of (9), stratified by the number of drivers m. A flow is a smooth path within one stratum; a swap is a jump within the same stratum (drivers change identity at fixed count); a refinement or coarsening is a jump between strata (the number of drivers changes). Each dot is a configuration TD(z), a point of a Grassmannian. 1. Flow (smooth rotation). D′ = D and z ′ → z c… view at source ↗
Figure 4
Figure 4. Figure 4: The three motions of the representation ( [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The time-dependent frontier potential ∆t = µ ⊤ DQ −1 D µD carried along the representation’s path (Section 3.5). Left: the concrete mean-variance frontiers of three configurations in the (σ, µ) plane; the maximal Sharpe ratio √ ∆ is the slope of the tangent ray, so a higher potential is a steeper frontier. A swap (A → B) changes the frontier at fixed dimension; a refinement (A → C) changes it and its dimen… view at source ↗
Figure 6
Figure 6. Figure 6: The configuration space and the time-dependent frontier ( [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Kunita–Watanabe orthogonality of a separator switch ( [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Geometric turnover (Theorem 4.4). Even with no change in the economic signal, the tangent frame carrying the portfolio’s systematic exposures moves between dates: the connection Γt→t+dt transports the frame (e1, e2) to (e ′ 1 , e′ 2 ). The motion splits into a horizontal rotation of the span (the skew part E⊤GE˙) and a vertical breathing of the metric (the symmetric strain 1 2E⊤GE˙ ). Rebalancing is forced… view at source ↗
Figure 9
Figure 9. Figure 9: Decomposition and hedging demand (Theorem 3.4). Left: the manifold-hedge component grows linearly with the distance of the driver state from its mean, while the myopic fund is flat; the hedge tracks predictable state motion. Right: out-of-sample certainty equivalent, myopic-only versus myopic-plus-hedge, in a persistent-state regime. precisely where a compact admissible set W must bind. The characterizatio… view at source ↗
Figure 10
Figure 10. Figure 10: Dimension reduction (Theorem 3.6). The dimension-m PDE recursion has cost independent of n (flat lower curve); all n-dependence lives in a one-time projection (upper curve). Log–log axes. 5.5 Switching verification The third experiment tests Theorem 3.10 without a regime oracle. A subtlety must be faced first: a pure rotation of the manifold leaves the return covariance invariant (an orthogonal change of … view at source ↗
Figure 11
Figure 11. Figure 11: Switching verification with an online regime detector ( [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Central study I (Theorem 3.11, Theorem 3.12). Each point is a switch geometry (rotation angle × anisotropic scale). Best-hedge R2 (vertical, log scale) stays at ∼ 10−4 for all switches, so unhedgeability is universal, while detectability KL (horizontal) ranges from zero (pure rotation) upward. Colour is the value-jump magnitude. The bottom-left region is Theorem 3.12: invisible, unhedgeable, costly. is pa… view at source ↗
Figure 13
Figure 13. Figure 13: Central study II (Theorem 4.5). Comparative statics in manifold mobility: strain norm, connection curvature (scaled), and geometric-turnover share all rise monotonically from exactly zero at a static manifold. One dial drives the entire geometric apparatus. Structural object Metric under 0% error Metric under 30% error Unhedgeability (residual best-hedge R2 ) 1.6 × 10−4 1.6 × 10−4 Metric compatibility (an… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages · 2 internal anchors

  1. [1]

    Rodr´ ıguez Dom´ ınguez

    A. Rodr´ ıguez Dom´ ınguez. Causal separation, conditional risk, and projected Markowitz portfolios. Working paper, 2025

    work page 2025

  2. [2]

    Markowitz

    H. Markowitz. Portfolio selection.Journal of Finance7(1):77–91, 1952. 30

    work page 1952

  3. [3]

    R.C. Merton. Optimum consumption and portfolio rules in a continuous-time model.Journal of Economic Theory3(4):373–413, 1971

    work page 1971

  4. [4]

    R.C. Merton. An intertemporal capital asset pricing model.Econometrica41(5):867–887, 1973

    work page 1973

  5. [5]

    J. A. Wachter. Portfolio and consumption decisions under mean-reverting returns: an exact solution for complete markets.Journal of Financial and Quantitative Analysis, 37(1):63–91, 2002

    work page 2002

  6. [6]

    T. S. Kim and E. Omberg. Dynamic nonmyopic portfolio behavior.Review of Financial Studies, 9(1):141–161, 1996

    work page 1996

  7. [7]

    Karatzas, S.E

    I. Karatzas, S.E. Shreve.Brownian Motion and Stochastic Calculus, 2nd ed. Springer, 1991

    work page 1991

  8. [8]

    Fleming, H.M

    W.H. Fleming, H.M. Soner.Controlled Markov Processes and Viscosity Solutions, 2nd ed. Springer, 2006

    work page 2006

  9. [9]

    X. Mao, C. Yuan.Stochastic Differential Equations with Markovian Switching. Imperial College Press, 2006

    work page 2006

  10. [10]

    L. R. Sotomayor and A. Cadenillas. Explicit solutions of consumption–investment problems in financial markets with regime switching.Mathematical Finance, 19(2):251–279, 2009

    work page 2009

  11. [11]

    Pham.Continuous-time Stochastic Control and Optimization with Financial Applications

    H. Pham.Continuous-time Stochastic Control and Optimization with Financial Applications. Springer, 2009

    work page 2009

  12. [12]

    Barles.Solutions de viscosit´ e des ´ equations de Hamilton–Jacobi

    G. Barles.Solutions de viscosit´ e des ´ equations de Hamilton–Jacobi. Springer, 1994

    work page 1994

  13. [13]

    M. G. Crandall, H. Ishii, and P.-L. Lions. User’s guide to viscosity solutions of second order partial differential equations.Bull. Amer. Math. Soc., 27(1):1–67, 1992

    work page 1992

  14. [14]

    H. Ishii. Perron’s method for Hamilton–Jacobi equations.Duke Math. J., 55(2):369–384, 1987

    work page 1987

  15. [15]

    Edelman, T

    A. Edelman, T. A. Arias, and S. T. Smith. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl., 20(2):303–353, 1998

    work page 1998

  16. [16]

    Jacod, A.N

    J. Jacod, A.N. Shiryaev.Limit Theorems for Stochastic Processes, 2nd ed. Springer, 2003

    work page 2003

  17. [17]

    Revuz, M

    D. Revuz, M. Yor.Continuous Martingales and Brownian Motion, 3rd ed. Springer, 1999

    work page 1999

  18. [18]

    S. Boyd, L. Vandenberghe.Convex Optimization. Cambridge University Press, 2004

    work page 2004

  19. [19]

    Gˆ arleanu, L.H

    N. Gˆ arleanu, L.H. Pedersen. Dynamic trading with predictable returns and transaction costs.Journal of Finance68(6):2309–2340, 2013

    work page 2013

  20. [20]

    Peters, P

    J. Peters, P. B¨ uhlmann, N. Meinshausen. Causal inference using invariant prediction: identification and confidence intervals.Journal of the Royal Statistical Society, Series B78(5):947–1012, 2016

    work page 2016

  21. [21]

    Rodr´ ıguez Dom´ ınguez

    A. Rodr´ ıguez Dom´ ınguez. Portfolio optimization based on neural networks sensitivities from assets dynamics respect common drivers.Machine Learning with Applications11:100447, 2023

    work page 2023

  22. [22]

    Rodr´ ıguez Dom´ ınguez

    A. Rodr´ ıguez Dom´ ınguez. A portfolio’s common causal conditional risk-neutral PDE. In M. Corazza, F. Gannon, F. Legros, C. Pizzi, V. Touz´ e (eds.),Mathematical and Statistical Methods for Actuarial Sciences and Finance (MAF 2024), pages 274–279. Springer Nature Switzerland, Cham, 2024

    work page 2024

  23. [23]

    Causal Portfolio Optimization: Principles and Sensitivity-Based Solutions

    A. Rodr´ ıguez Dom´ ınguez.Causal Portfolio Optimization: Principles and Sensitivity-Based Solutions. Monograph, forthcoming; arXiv:2504.05743, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  24. [24]

    Causal PDE-Control Models for Dynamic Portfolio Optimization with Latent Drivers

    A. Rodr´ ıguez Dom´ ınguez. Causal PDE-control for adaptive portfolio optimization under partial information. In A. Itkin, O. Kondratyev (eds.),Reviews in Modern Quantitative Finance, vol. 2. World Scientific, forthcoming; arXiv:2509.09585, 2025. 31

    work page internal anchor Pith review Pith/arXiv arXiv 2025

This paper was first reviewed by grok-4.5 on July 10, 2026.