Papers

We model market impact as the response to submitted order flow net of counterflow from latent traders, activated when price displacements from the level that would prevail without the order exceed individual thresholds. Order flow depletes this pool, and a generalized Langevin equation governs its recovery over several time scales. Its memory kernels are finite sums of exponentials, so its Markovian lift is exact rather than an approximation. For an undepleted pool, aggregation under explicit assumptions on individual trading responses yields an intermediate square-root regime between linear small- and large-order limits, without imposing a square-root impact law. Scaling thresholds and responses with price noise makes impact in this regime proportional to volatility, and thresholds that grow with the execution horizon make it independent of duration. With constant displayed depth, expected round-trip costs are nonnegative under the log-price convention, independently of the memory. Numerical experiments show that depletion narrows the square-root range and that memory spectra producing similar single-order impacts can respond differently after substantial prior trading. Calibration to market data is left to a companion paper.

We develop a non-perturbative framework for stochastic-volatility option pricing built on the two-particle-irreducible (2PI) effective action and the Dyson-Schwinger gap equations of quantum field theory. In log-price, log-volatility or Lamperti coordinates, the joint law of the state variables is approximated by a self-consistent Gaussian whose mean and effective diffusion follow from the 2PI stationarity conditions, with the exponential and CEV vertices evaluated through the exact Gaussian moment-generating function rather than a Taylor cut. For exponential vertices the skeletons beyond this level sum in closed form: in exp-OU the evaluated melon level cuts the Hartree error by 20-130x, to 0.01-0.11 bp. Whether the dressed inverse propagator is local in time separates Markovian models, where the gap equation collapses to a few ODEs, from rough (Volterra) models, where the full two-time propagator is retained. At nonzero correlation the field is integrated deterministically by conditioning on its two leading Gaussian modes, with exact conditional moments and a moment-matched law for the integrated variance and leverage integral; in the CEV models the correlated drift enters a Dyson equation around the exact absorbed-CEV propagator, resummed by jumps along the variance clock. These closures match Monte Carlo references for rough Bergomi approximately at their resolution (0.01-0.1 bp) down to H=0.07, rho=-0.9, and price SABR to 0.97 bp over a 72-cell grid with maturities to ten years (75 bp for Hagan's expansion) and rough SABR to 0.92 bp. Conditional on the volatility field, forward-start smiles and continuously-monitored barriers reduce to field-only quadratures on the native Volterra field, and the causal response block yields the impulse-vega curve in one adjoint contraction. Derivations, extended benchmarks and secondary applications are in an ancillary technical supplement.

We study the joint dynamics of the best bid and ask prices with a spread-gated Hawkes-flocking model. The model tracks four types of best-quote movements: spread-narrowing movements are switched off when the spread is at its one-tick minimum, and a cross-side excitation term, whose activation depends on the prevailing spread, links the two sides of the book. We show that the process is non-explosive on every finite horizon, give an $O(N)$ recursive likelihood, and validate the maximum likelihood estimator by simulation. On real intraday limit order book data for two large-tick stocks, INTC and MSFT, the restriction that removes the cross-side term is rejected, and the full model improves fit substantially by AIC and BIC; the likelihood is multimodal on a single day, so estimation uses a multi-start search. As an application, we derive the closed-form optimal size of a single-period limit order placed at the best or second-best quote, given the model's next-event probabilities and externally supplied execution probabilities.

We study a finite-horizon reversible investment problem in which a risk-neutral firm adjusts capacity at a proportional purchase cost and a lower salvage value under multi-factor geometric Brownian motion. Via the singular control–optimal switching correspondence, the marginal value of capacity solves a family of parabolic double-obstacle problems. We prove existence, uniqueness and local Sobolev regularity of the strong solution, characterize investment, waiting and disinvestment regions by continuous, strictly separated free boundaries, and verify optimality of the reflected capacity process. Numerically, joint demand improvements shift both boundaries super-additively, 1.5–2.7 times as strongly at the disinvestment boundary, depending on factor correlation.

We study the short-maturity implied volatility and the skew stickiness ratio for baskets of assets with continuous, possibly rough, stochastic volatility. The fluctuation of the instantaneous basket variance has two sources: fluctuations of the constituent variances and fluctuations of the basket weights. We derive a near-the-money implied volatility expansion that separates these contributions. We then specialize the result to volatility models given by general functions of Gaussian Volterra factors and obtain an explicit basket skew coefficient in terms of the short-time kernel asymptotics, the factor sensitivities, and the return-factor correlations. A density expansion justifies differentiation of the near-the-money expansion at the money. Finally, using a Malliavin representation of the dynamics of total implied variance, we prove that the short-maturity skew stickiness ratio converges to the universal limit $H + 3/2$ for Gaussian factor basket models with $H \in (0, 1/2]$.

Hedging a derivative by trading the underlying asset changes the payoff that the hedging intended to replicate. We study this phenomenon when trading generates price impact and execution costs. In a binomial model, we characterize replication through a fixed-point equation. In continuous time, we derive a nonlinear pricing PDE whose implicit terminal condition captures the nature of the moving target problem of the hedger. For monotone convex Lipschitz payoffs (such as calls and puts) we establish exact replication under midpoint execution costs. Numerical experiments illustrate: (i) how price impact shifts the effective strike, (ii) the non-linear dependence of the option price on the number of contracts, (iii) how execution costs smooth terminal holdings, (iv) the extent to which the hedger's own trading can bring an otherwise worthless option into the money, and (v) we explain the spread and the shape of the limit order book in the options market based on the price impact and the shape of the limit order book of the underlying.

In quantitative finance, standard regression losses are misaligned with the economics of return prediction. As the conditional mean of financial log-returns is close to zero, symmetric losses such as the mean squared and mean absolute errors make the constant zero forecast a near-optimal solution, penalizing models with genuine but noisy directional skill. This applies both during training, where predictions shrink toward zero, and during evaluation, where trivial forecasters can lead loss-based rankings. Under a Gaussian linear prediction model, we show that all symmetric monotonic losses share a universal breakeven directional accuracy against the zero predictor, which rises sharply and becomes unobtainable as the prediction noise approaches the standard deviation of the returns. We introduce the CZAR (Composite Zero-Agnostic Return) loss function, a piecewise quadratic loss built around five key requirements: convexity in the prediction, asymmetry with true return direction that vanishes at zero, near-linear penalization of undershoots and wrong-direction predictions, divergence for large errors, and an adaptive loss floor for evaluation. CZAR is provably convex in the prediction at fixed true value, has closed-form gradient and Hessian suitable for custom objectives in gradient-boosted libraries, and its four hyperparameters reduce to a single choice through correlated defaults. In idealized tests, the minimum directional accuracy required for a CZAR-evaluated forecaster to outperform the zero predictor under mean log loss remains near the 50% chance level, whereas the corresponding threshold for symmetric losses rises sharply with prediction noise. This advantage persists under heavy-tailed return distributions. In a LightGBM experiment on intraday BTC log-returns, CZAR-trained models reduce the `zero-returns bias' and improve directional accuracy on large-magnitude returns.

We ask whether transposition changes the stationary heavy tail of the co-moving recursion $\mathbf x_{t+1}=Q_tBQ_t^\top\mathbf x_t+Q_t\boldsymbol\eta_t$ in $d=3$. The model has independent Haar rotations, Gaussian body-frame noise with covariance $\Sigma$, and an explicit positive metric $\mathsf g$. The tail index $\alpha_\star$ depends only on singular values, so $\alpha_\star(B)=\alpha_\star(B^\top)$. The structure theorem and polar-twist mechanism are unconditional. In the $\mathsf g=I$ chart, write $B=S+\widehat{\boldsymbol\omega}$: $\Delta_6(B)=-16\det[\boldsymbol\omega,S\boldsymbol\omega,S^2\boldsymbol\omega]$. Then $B\not\sim_{O(3)}B^\top$, $\Delta_6(B)\ne0$, $\operatorname{rank}[\boldsymbol\omega,S\boldsymbol\omega,S^2\boldsymbol\omega]=3$, and controllability of $(S,\boldsymbol\omega)$ are equivalent; intrinsic operator chirality is a Kalman rank condition. Unlike the exponent, the tail level need not be transpose-invariant. The exact polar-twist identity rotates the anisotropic noise frame under mirroring, showing how level asymmetry can enter. A nonzero asymmetry is established only conditionally and numerically, not by a closed-form example: under Assumption R on the SPD chart, $\Delta\log C=\langle A,G\rangle+o(\|A\|)$ as $A\to0$, with $G=2H\circ[\Sigma,\partial_\Sigma\log C]$; finite-threshold simulations are consistent with this first-order law. Strict-Haar averaging makes every lagged second-order cross-moment transpose-blind; a fourth radial moment retains an odd channel. No finite scalar-weighted radial-moment combination uniformly cancels the even quadratic form while retaining the odd covector.

We solve the perpetual liquidation problem for a geometric multi-skew Brownian motion carrying local-time pushes upward at a support level and downward at a resistance level, a model of technical analysis that is Markov in the price alone. Three geometries arise, separated by a closed-form criterion: the continuation band lies below resistance, straddles it, or pins its upper edge there. Selling into resistance is a conclusion rather than an assumption. Identification is asymmetric: exercise behaviour determines the support parameters exactly, while the strength of resistance is recoverable only on an interval with a closed-form endpoint.

Building on the forest expansion of Alòs, Gatheral and Radoičić and on the explicit Bergomi-Guyon smile expansion derived by Bourgey and Gatheral (2026), we derive fixed-point approximations, in terms of the implied total variance at a small number of magic strikes, for the fair values of power payoff contracts; the variance and gamma contracts arise as the endpoints of this one-parameter family, priced by a single set of formulae. We prove that such approximations exist at every order: at order $n$, $2\lceil n/2\rceil+1$ magic strikes suffice. Numerical tests under the Heston and rough Bergomi models demonstrate good accuracy even for strongly skewed smiles, and the method applies directly to interpolated market smiles without strike inversion. For the volatility contract, we show that the maturity-T smile does not determine the leading correction to the Rolloos-Arslan approximation.

This paper introduces oracle-parametrized automated market makers (OP-AMMs), i.e., automated market makers whose quoted price depends jointly on the pool reserves and an external oracle price. In doing so, we extend the information-agnostic AMM framework to settings, such as tokenized securities, for which price discovery occurs off-chain. Under a strict oracle-contraction condition, we show that the quoted price of any OP-AMM interpolates between the oracle price and an implicit autarkic price determined by the pool reserves. We then derive a general loss-versus-rebalancing (LVR) decomposition that separates the residual exposure to market lags from the losses induced by oracle errors. This analysis is further extended to stale, discrete-update oracles and to sandwich attacks around oracle updates. Using this framework, we find conditions under which OP-AMMs simultaneously increase local capital efficiency and reduce normalized LVR relative to information-agnostic AMMs. However, sufficiently noisy or stale oracles can reverse these gains. A counterfactual backtest using one-second SPY NBBO data is provided to demonstrate these trade-offs. In particular, we map the Pareto-efficient frontier of oracle-parametrized constant function market maker (OP-CFMM) designs across stylized oracle regimes.

We consider a problem of optimal proportional reinsurance-dividend distribution under a Brownian risk model, where both the drift and volatility coefficients are subject to endogenous regime-switching. Dividend payments are subject to fixed transaction costs. The problem is formulated as a two-dimensional stochastic control problem, and we prove that the value function is the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation with nonlocal operator. For almost all parameter configurations, we explicitly characterize the optimal strategy that maximizes the expected total discounted dividends net of transaction costs until ruin. The optimal dividend policy is a two-barrier impulsive strategy, while the optimal reinsurance proportion is given in feedback form. Numerical examples are provided to illustrate the optimality results.

We study simple predictable processes whose coefficients are represented by neural networks. On finite measure spaces, we establish density results for neural networks in Orlicz spaces. For filtrations generated by a stochastic process, measurable random variables, including at stopping times, can be approximated by neural networks depending on finitely many observations. Every stochastic integral with respect to a semimartingale can then be approximated, in the semimartingale topology, by integrals of such simple predictable processes. We show that restricting trading strategies to this class leaves the minimal mean-variance hedging error under partial information unchanged and obtain a no-free-lunch characterization in terms of equivalent martingale measures. Finally, the Bichteler-Dellacherie characterization of semimartingales remains valid even upon restricting the predictable integrands to those whose coefficients are represented by neural networks.

This article is concerned with the asymptotic shape of quantile surfaces, defined as the set of quantiles at a given level $\alpha$ generated by a controlled one-dimensional distribution. Specifically, when the distribution arises as a linear combination of log-normal random variables and the control is a vector of positive coefficients, we prove that quantile surfaces are globally concave in the left tail ($\alpha\to0$) and globally convex in the right tail ($\alpha\to1$). Moreover, these surfaces exhibit asymptotic separation of scale and shape.

Reweighting a probability law by a positive numéraire and pushing forward the payoff-to-numéraire ratio yields the change-of-numéraire reweighting: the swap-rate law under the annuity measure, the numéraire-inversion involution of martingale optimal transport, and the population form of self-normalized importance sampling. We characterize exactly when it is Wasserstein-stable. Along weakly convergent inputs with uniformly integrable numéraires and a strictly positive limiting numéraire mean, convergence of the reweighted laws is equivalent to a uniform integrability condition, computed under the inputs, on one explicit family in the payoff and numéraire. At first order the numéraire cancels, the sole obstruction being payoff mass where the numéraire vanishes; absent such mass, prices and first moments pass to the limit under a uniformly integrable payoff alone, with no assumption on the reciprocal numéraire. Two-atom examples with bounded inputs attain the threshold exactly. Applications characterize a standing moment assumption in the transport literature and delimit which annuity-measure statistics survive.

The analytic tractability of affine pricing models is usually expressed through two complementary formulations: a coordinate-space pricing operator and an exponential-affine transform representation governed by generalized Riccati equations. We develop Affine Holonomy Group Quantization (AHGQ) as a geometric framework in which these two formulations arise from the same underlying structure. The construction separates the affine pricing symbol into a homogeneous quadratic sector and a complementary affine sector. The first generates a finite-dimensional symplectic transport and a centrally extended Lie group, while the second is represented by a multiplicative holonomy carried by a thin-path groupoid. Their combination determines an affine Poincaré–Cartan form. Its characteristic dynamics reduce in momentum variables to the generalized Riccati system and its scalar amplitude, whereas the coordinate representation recovers the standard affine pricing operator. Representative Gaussian and square-root models illustrate the construction. The contribution is structural: AHGQ gives a common geometric origin to the coordinate and transform representations of continuous-path, time-homogeneous affine pricing models.

In a finite discrete-time market, trading decisions may be predictable with respect to a filtration that does not adapt asset prices. The first fundamental theorem then characterizes absence of arbitrage by measures under which the optional projection of discounted prices is a martingale. We examine the corresponding completeness question. For a fixed projection, every claim measurable with respect to its terminal price history is attainable precisely when all equivalent martingale measures of that fixed process agree on the claim sigma-field. We give the finite-dimensional proof, retaining the distinction between the trading filtration and the claim sigma-field. If the projection is common to all optional martingale measures of the original prices, projected completeness implies uniqueness of their restrictions, but the converse fails even in a three-state model with a unique optional martingale measure. For the binomial model under its product martingale measure, a delay of $k$ periods yields a complete projected market with effective horizon $(T-k)^+$. An explicit replication construction and finite examples distinguish this completeness from replication at the original prices.

Commodity option surfaces contain information beyond the at-the-money volatility level. We develop a surface-driven stochastic-volatility framework for soybean futures options using daily Chicago Mercantile Exchange Group Volatility Index (CME CVOL) indicators from October 2013 to August 2025. The ATM level and convexity are positive, right-skewed, and well described by log Ornstein Uhlenbeck dynamics, whereas the additive skew changes sign and is modeled on the real line by an Ornstein Uhlenbeck process. These empirical features motivate a joint mean-reverting surface-factor system. Under a log OU futures volatility model, leading-order smile relations map the ATM level to the latent volatility state, convexity to an effective surface-implied volatility- of-volatility state, and skew to an effective leverage state. We derive the associated risk-neutral pricing equation and Feynman Kac representation, benchmark prices by Monte Carlo simulation, validate the pricing PDE by finite differences, and train a neural-network surrogate for fast repeated valuation. The empirical results show substantial variation in surface states across market regimes, with the 2021 La Nina drought window exhibiting elevated volatility and skewness. The numerical results show close agreement between finite-difference and Monte Carlo prices over the validation grid, while the neural-network surrogate reproduces the pricing map at substantially lower inference cost. Overall, the results support the use of level, skew, and convexity jointly as dynamic inputs for commodity-option pricing and model diagnostics.

The equal-weighted portfolio is a passive, rule-based strategy that has historically been difficult to outperform, delivering higher returns than the capitalization-weighted "market" benchmark across many markets and periods. Stochastic portfolio theory (SPT) reveals that this relative performance is regime dependent, with the equal-weighted portfolio underperforming during periods of increasing market concentration and high correlations, particularly market bubbles. These observations have motivated us to formulate and solve a stochastic control problem in which an investor actively allocates between the equal-weighted and market portfolios. The investor bases their allocation decisions on forecasts made under a flexible stochastic diversity–dispersion (SDD) model. Using a quadratic surrogate for implementation frictions, we characterize the optimal allocation through a linear forward–backward SDE and obtain an explicit "aiming in front of a moving target" representation of the optimal trading rate, in the spirit of Gârleanu and Pedersen. The penalty parameters are calibrated in sample to match the cumulative wealth effect of proportional transaction costs, while out-of-sample performance is evaluated with those costs deducted directly from portfolio wealth. Using historical S&P 500 data, we show that a mean-reverting SDD specification reproduces several empirical features of market diversity and dispersion. In out-of-sample backtests from 1995 to 2024, the resulting strategies deliver higher cumulative net returns than both the equal-weighted and market portfolios, and higher information ratios than the equal-weighted portfolio after 15-basis-point proportional transaction costs.

We study equilibrium and risk evaluation in large sparse economic networks with heterogeneous responses, shocks, and bilateral exposures. Our approach approximates the distribution of equilibrium outcomes through local computations on a limiting rooted network. Under marked local weak convergence, we prove convergence in probability of the empirical equilibrium distribution in two settings: uniformly contractive responses and bounded monotone nonexpansive responses whose lower and upper root iterations coalesce. In the latter setting, the limit is independent of the measurable finite-network equilibrium selection, even when equilibria are not unique. Under bounded domain and continuity assumptions, this convergence extends to law-invariant risk measures, including conditional value-at-risk, with recursion-depth error bounds under additional regularity. Numerical experiments on production networks and payment clearing systems illustrate the accuracy and computational benefits of the local approximation.

We provide a model for the nested optimisation problem of market making and rebate design problems in option markets and find optimal strategies. A single market maker trades multiple European call options in a local-stochastic volatility option market with both make and take strategies, modeled, respectively, as continuous and impulse controls. Her objective is to maximize, over all admissible make-take strategies, net profit of option portfolio value and cumulative rebate revenue, subject to a penalty on residual portfolio delta and vega. In addition, we demonstrate how an exchange can incentivize a market maker to improve market liquidity by setting suitable fee rebates, thereby resolving its own liquidity attraction problem. To this end, we propose a three-step rebate design scheme with flexibility to accommodate specific liquidity targets imposed by an exchange. Numerical results are provided to validate the effectiveness of the proposed scheme.

We study the infinite-horizon optimal investment and consumption problem in a general class of continuous financial markets, where uncertainty is driven by a continuous non-decreasing stochastic clock representing accumulated variance. This framework encompasses classical Markovian and non-Markovian stochastic volatility models as well as singular realized-variance models in which no spot volatility process exists. We characterize the value process and optimal investment and consumption strategies in terms of a non-linear infinite-horizon backward stochastic differential equation driven jointly by calendar time and the stochastic clock. We develop a general well-posedness theory for this new class of IVC-BSDEs based on the method of sub- and supersolutions, establishing existence, uniqueness, and stability under natural conditions that might be of independent interest beyond the financial application at hand. We are moreover able to identify the sign of the $Z$-component of the solution using Malliavin calculus. We then apply our results to Volterra Heston models with locally integrable kernels, covering both rough and hyper-rough regimes. Exploiting the affine structure of the model, we verify the optimality of the candidate strategies in incomplete markets and obtain an explicit representation of the solution in the complete market case. Owing to the generality of the framework and the weak assumptions imposed on the stochastic clock, our results unify and extend several existing results for optimal investment and consumption, including classical Markovian stochastic volatility models.

We examine the conditional volatility dynamics of the USD 1Yx10Y forward swap rate using GARCH(1,1), GJR-GARCH(1,1), and a two-regime Markov-switching GARCH (MSGARCH) model. The analysis uses daily data from 2007 to 2023 and incorporates market-implied measures (ATM swaption volatility and the SRVIX in- dex) together with a broad set of diagnostic tests. Standard GARCH and GJR- GARCH models show stable short-run parameters, but the intercept {\omega} varies markedly across rolling windows, causing instability in the implied long-run vari- ance. This pattern, confirmed by the Nyblom test, motivates adopting a regime- switching specification. MSGARCH mitigates this issue by keeping regime-specific parameters stable and capturing time variation through filtered regime probabili- ties. It delivers the highest log-likelihood and lowest AIC, whereas BIC favours the more parsimonious GJR-GARCH. One-step-ahead backtesting indicates comparable short-horizon accuracy across models, but MSGARCH offers a clearer structural in- terpretation by isolating high- and low-volatility regimes aligned with major market events.

We study affine stochastic Volterra equations on the cone of symmetric positive semidefinite matrices. For scalar kernels acting entrywise on the matrix dynamics, we establish weak existence by exploiting stochastic invariance results for Volterra equations on convex domains and derive a conditional Fourier–Laplace transform formula characterized by matrix-valued Riccati–Volterra equations. As an application, we extend the Gibson–Schwartz commodity model by replacing its variance-covariance structure with a Volterra–Wishart process. The resulting model allows for memory in the variances and for stochastic instantaneous correlation, while retaining affine tractability. Its joint Fourier–Laplace transform admits an exponential-affine representation governed by a matrix Riccati–Volterra equation. While the existence theory considered here excludes kernels that are singular at the origin, shifted fractional kernels remain admissible and provide a tractable specification with power-law memory.

We characterize the worst- and best-case values of a mean-variance functional over a 2-Wasserstein ball. Using quantile representations and the geometry of attainable means and standard deviations, we reduce both infinite-dimensional problems to scalar equations and construct the extremal laws as location-scale transformations of the reference distribution. Their values depend on the reference law only through its first two moments. We then derive dual representations and formulate proportional risk sharing under heterogeneous beliefs as a finite-dimensional optimization problem. Under homogeneous beliefs, we show that the classical proportional allocation remains optimal for every ambiguity radius and $\alpha$-maxmin weight. Finally, we study a coupled distortion-variance functional and characterize its worst-case quantile through a convex-envelope construction, allowing the extremal law to change in shape as well as location and scale.

Mitigating drawdown, the decline in wealth from its running peak, presents a canonical problem in path-dependent risk control. In this paper, we develop a finite-horizon control framework that enforces a prescribed maximum percentage drawdown limit in multi-asset stochastic systems. Our first result is an exact robust-invariance theorem characterizing every control action that preserves a prescribed drawdown limit against all supported returns. We show that every robustly safe control admits a drawdown-modulated form: the product of the current drawdown cushion and a feasible normalized direction. This yields a complete parameterization of robustly drawdown-safe policies. Additionally, under stagewise-independent returns, we show that optimizing over all robustly safe causal policies reduces to a one-dimensional Bellman recursion and yields an optimal robustly safe state-feedback policy. Finally, we characterize the linear time-invariant (LTI) gains satisfying a prescribed drawdown limit and prove that optimal drawdown modulation achieves no lower expected return under the same limit. Strict expected-return improvement holds for horizons of at least two stages whenever the LTI policy has positive expected one-stage net return.

We study random signed Takagi–Landsberg bridges with independent Rademacher Faber–Schauder coefficients. At index $H=1/2$, we prove that every fixed deterministic refining sequence of partitions with vanishing mesh yields quadratic variation $t$, almost surely and uniformly in $t\in[0,1]$. For partitions that need not be refining, the mesh condition $o(1/\log n)$ is sufficient, and its order is sharp. Our proofs use an operator representation of the quadratic sums and moment bounds for Rademacher chaos. We also show that asymmetric signs can destroy this invariance. At every index $H\ne1/2$, we construct a deterministic refining sequence along which the critical $p$-power sums, with $p=1/H$, have two distinct finite accumulation points almost surely. At $H=1/4$, uniform thirds grids provide an explicit alternative to the dyadic limit. Thus $H=1/2$ is the unique index in the symmetric family at which critical variation is invariant along every fixed deterministic refining sequence. Nevertheless the variation index equals $1/H$ almost surely across a large sequence of partitions.

We study a representative-agent Epstein-Zin economy with geometric dividends and a hidden finite-state Markov drift. We allow the price-dividend ratio to contain an additional positive, absolutely continuous valuation factor and, within the class $\mathfrak C$ defined below and under the regularity, admissibility, and positivity conditions of our main theorem, equilibrium forces this factor to be constant, yielding belief-Markovian prices. In the two-state case, under the stated positivity condition and strictly positive transition intensities, we prove existence, uniqueness, endpoint smoothness, interior analyticity, and uniform bounds for the positive solution of the pricing equation. For $0<\theta\leq1$, this solution supports an equilibrium under any continuous short rate satisfying the model's one-sided portfolio condition. Finally, in the two-state subregion $\eta>0$, we derive belief-dependent stock volatility, a European option-pricing PDE, and a leading short-maturity conditional risk-neutral log-return skewness expansion.

Nested clustered optimization allocates within each cluster from the cluster's own covariance block and then across the resulting cluster portfolios. Block inversion says the unconstrained minimum-variance portfolio has the same two-tier shape, with each block replaced by its Schur complement against every other asset. Conditioning instead on one knot from each other cluster truncates the conditioning set in the manner of a Vecchia approximation, and we give the rank-one model of cross-cluster dependence under which it is exact. Damping the complement by $\gamma\in[0,1]$ then gives a bridge with nested clustered optimization at $\gamma=0$ and the global optimum at $\gamma=1$, with no linear solve larger than a cluster or the number of clusters. Under estimation error the optimal $\gamma$ can be strictly interior and full coupling can remain optimal, and we give the local theorem at the minimum-variance end with exact examples of both, including a symmetric family in which the optimum is a closed form.

A substantial body of empirical evidence suggests that stochastic mortality models ignoring long range dependence tend to underestimate life expectancy, which may lead to profound implications for pension schemes and funding arrangements. This paper addresses the modelling of stochastic mortality via a mixture of a fractional Lévy process and standard Brownian motion. Our stochastic mortality model exhibits nice analytical tractability in actuarial valuations and flexibility in the choice of underlying Lévy specifications. The long range dependence feature embedded in our stochastic mortality model is well reflected in our empirical studies on the mortality shocks during World War II and COVID-19. For efficient numerical pricing of longevity derivatives, we construct an effective singular value decomposition approximation scheme to overcome the computational challenges arising from the non-Markovian nature of the fractional Lévy process. Truncation errors in singular value decomposition approximation can be reduced by an effective residual correction scheme.