analyze aggregation monotonicity

This study addresses the monotonicity of normalized implied volatility coordinates within a finite quoted option chain under no-arbitrage conditions and derives model-independent variance identities. Relying solely on static no-arbitrage assumptions—including discrete strike comparisons, convexity, monotonicity, and call-put parity—the authors provide the first purely discrete proof of monotonicity for normalized coordinates in both the Black–Scholes and Bachelier frameworks, without requiring continuous quotes or differentiability. The main contributions are twofold: (1) a rigorous discrete verification of monotonicity in these two canonical implied volatility models, and (2) the introduction of a normal variance identity that serves as the natural counterpart to Fukasawa’s log-normal result, thereby establishing a model-independent theoretical foundation for volatility derivatives.

This work addresses the detrimental effects of non-affine gradient aggregation in convex optimization, which disrupts the monotonicity of update operators and consequently undermines algorithmic convergence and generalization. We establish, for the first time, that only positive affine aggregations preserve monotonicity, and we derive sufficient conditions to restore it. Leveraging monotone operator theory, convex optimization, and algorithmic stability analysis, we rigorously quantify how non-affine aggregation degrades both convergence and stability. Our unified theoretical framework elucidates the common failure modes of modern learning systems under constraints related to adaptivity, privacy, robustness, and fairness, thereby providing principled guidance for designing algorithms that simultaneously satisfy such constraints and guarantee convergence.

This paper identifies an error in a corollary of Theorem 2.8 in Bayraktar & Yu (2018), undermining their market viability conclusion under proportional transaction costs. To address feasibility, it proposes strict consistent local martingale systems (SCLMS) — replacing the conventional strict consistent pricing systems — as the dual criterion, and constructs a unified verification framework based on two weak no-arbitrage conditions: NUPBR (no unbounded profit with bounded risk) and the newly introduced NLABP (no local acceptable profit). It establishes, for the first time, the robust equivalence between SCLMS and both NUPBR and NLABP. The introduction of NLABP extends the scope of arbitrage-free theory to broader settings with transaction costs. Finally, the paper derives necessary and sufficient conditions for market viability under proportional transaction costs, providing a novel theoretical foundation for utility maximization in frictional markets.

This paper addresses the axiomatization and representation of max/min-stable functionals under first-order stochastic dominance. We introduce a novel integration of stability concepts with first-order stochastic dominance, establishing an exact representation theorem for nondegenerate, lower-semicontinuous max-stable, min-stable, and jointly stable functionals—expressed as suprema of binary functions. From this, we uniformly derive the complete axiomatic system for Lambda-quantiles. Our contributions comprise three fully characterized axiomatizations, filling a fundamental gap in the theory of stability within stochastic dominance frameworks. Moreover, the results provide a rigorous mathematical foundation for financial risk measures—particularly Lambda-VaR—and for quantile-based voting rules in social choice theory. The analysis bridges abstract functional representation with concrete applications in risk management and collective decision-making.

This study addresses the pricing of take-or-pay swing options in energy markets, where the underlying asset follows a non-Markovian ARCH-type process with convex (or semiconvex) coefficients. To tackle this discrete-time stochastic optimal control problem, we establish— for the first time under non-Markovian dynamics—a propagation-of-convexity theory for the value function with respect to the asset price, relaxing the classical convex-coefficient assumption to semiconvexity and thereby substantially broadening applicability. Leveraging tools from convex analysis, stochastic control, Stein’s identity, and regularization techniques, we rigorously prove the convexity and monotonicity of the value function in key parameters, and derive verifiable convex-order dominance criteria. Numerical experiments confirm both the validity and robustness of the theoretical results.

This study investigates the stability of combinatorial markets when side payments are introduced, focusing on equilibrium properties under constrained utility transfers among agents. By constructing an explicit side-payment mechanism, the work proposes a partition-based notion of stability under restricted transferable utility—termed the T-core—and systematically classifies distinct market settings. Integrating combinatorial market modeling, cooperative game-theoretic core theory, and analyses of both transferable (TU) and non-transferable utility (NTU) stability, the paper uncovers key structural differences: when side payments are permitted among buyers, higher-order market formulations become equivalent; among sellers, equivalence holds only from the third order onward. Furthermore, under personalized pricing, all stability concepts collapse to NTU stability.

This study addresses the computational complexity of von Neumann–Morgenstern (vNM) stable sets in one-to-one matching markets, which arises from the definition of domination. By generalizing the classical Decomposition Lemma to arbitrary internally stable matching pairs, the work uncovers an intrinsic connection between internal stability and the cyclic structure of the market, and constructs a reduced environment in which all undominated outcomes are concentrated. Building on this reduction, it establishes an equivalence between vNM stable sets and the core of the simplified market, thereby proving their unique existence and providing an efficient constructive algorithm for their computation. This paper presents the first structural characterization and practical algorithmic solution for vNM stable sets in such settings.

This study addresses collective pricing and hedging when risk exchanges form a finitely generated convex cone, extending the classical Fundamental Theorem of Asset Pricing. By introducing the notion of collective strong replicability and integrating tools from functional analysis, convex analysis, and equivalent martingale measure theory, the work establishes a novel framework accommodating infinite-dimensional trading opportunities and finite-dimensional convex-cone exchanges. Key contributions include proving that collective no-arbitrage implies closedness of the aggregate feasibility cone, characterizing the set of collective prices as a relatively open convex set, providing equivalent conditions for collective and strong completeness, and demonstrating that when the exchange structure is merely a convex cone rather than a vector space, the pricing–hedging duality undergoes a fundamental transformation—thereby yielding an enhanced version of the Collective Second Fundamental Theorem of Asset Pricing.

This study addresses the nonexistence of traditional maximal elements when preference relations exhibit cycles and the set of alternatives is infinite. Within a unified order-theoretic and topological framework, the paper establishes the first topological characterization of von Neumann–Morgenstern (vNM) stable maximality by integrating Upper MacNeille information monotonicity, compact topology, Nachbin closedness, and upper semicontinuity. Leveraging tools from order theory and topology, the authors prove that under consistency and information monotonicity conditions, there exists a specific compact topology ensuring that the set of maximal elements is nonempty and vNM stable. This result yields necessary and sufficient conditions for the existence of such stable solutions.