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Vasily Melnikov

Publications and source records attributed to Vasily Melnikov.

8 recordsLinked to original sources
arXiv2026

Complemented Copies of $c_{0}$ in Positive Tensor Products of Banach Lattices

A result of Cembranos states that a non-trivial injective tensor product of a $C$-space contains a complemented copy of $c_{0}$, and in particular fails the Grothendieck property. We establish a positive analogue of the Cembranos theorem for tensor products of Banach lattices. If $E$ and $F$ are infinite dimensional Banach lattices, with $E$ containing $c_{0}$ and $E^{\ast}$ or $F^{\ast}$ having the bounded positive approximation property, then the Wittstock tensor product $E\widetilde{\otimes}_{\vert{\varepsilon}\vert}F$ contains a complemented copy of $c_{0}$. If $F$ is in addition reflexive, then $E\widetilde{\otimes}_{\vert{\varepsilon}\vert}F$ fails the positive Grothendieck property.

math.FA↗
arXiv2026

A General Theory of Risk Sharing

We introduce a new paradigm for risk sharing that generalizes earlier models based on discrete agents and extends them to allow for sharing risk within a continuum of agents. Agents are represented by points of a measure space and have potentially heterogeneous risk preferences modeled by risk measures on a separable probability space. We derive the dual representation of the value function using a Strassen-type theorem for the weak-star topology and provide a characterization of the acceptance set using Aumann integration. These results are illustrated by explicit formulas when risk preferences are within the family of entropic and expected shortfall risk measures, and applications to Pareto efficiency in large markets.

q-fin.RM↗
arXiv2025

Optimal Risk Sharing Without Preference Convexity: An Aggregate Convexity Approach

We consider the optimal risk sharing problem with a continuum of agents, modeled via a non-atomic measure space. Individual preferences are not assumed to be convex. We show the multiplicity of agents induces the value function to be convex, allowing for the application of convex duality techniques to risk sharing without preference convexity. A computationally tractable formula for the conjugate of the value function is derived, yielding an explicit dual representation of the value function. Applications of our results include a version of the two fundamental theorems of welfare economics for a large class of non-convex preferences, and non-existence results for Pareto optima when preferences are distortion risk measures whose corresponding distortion function fails to majorize the identity.

econ.TH↗
arXiv2025

Fatou limits of stochastic integrals

The convergence of stochastic integrals is essential to stochastic analysis, especially in applications to mathematical finance, where they model the gains associated with a self-financing strategy. However, Fatou convergence of $(X^{n})_{n=1}^{\infty}$ $\unicode{x2014}$a notion introduced for its amenability to compactness principles$\unicode{x2014}$implies little about the sequence of Itô integrals $\left(\int_{0}^{\cdot}YdX^{n}\right)_{n=1}^{\infty}$ for a fixed integrand $Y$. Under a boundedness condition, we find convex combinations $(\widetilde{X}^{n})_{n=1}^{\infty}$ of $(X^{n})_{n=1}^{\infty}$ with Fatou limit $\widetilde{X}$, such that $\left(\int_{0}^{\cdot}Yd\widetilde{X}^{n}\right)_{n=1}^{\infty}$ converges in a Fatou-like sense to $\int_{0}^{\cdot}Yd\widetilde{X}$ for all continuous semimartingales $Y$. The result is sharp, in the sense that continuity of $Y$ cannot be relaxed to being the left limits process of a semimartingale.

math.PR↗
arXiv2024

Limit theorems for $σ$-localized Émery convergence

Given a bounded sequence $\{X^{n}\}_{n}$ of semimartingales on a time interval $[0,T]$, we find a sequence of convex combinations $\{Y^{n}\}_{n}$ and a limiting semimartingale $Y$ such that $\{Y^{n}\}_{n}$ converges to $Y$ in a $σ$-localized modification of the Émery topology. More precisely, $\{Y^{n}\}_{n}$ converges to $Y$ in the Émery topology on an increasing sequence $\{D_{n}\}_{n}$ of predictable sets covering $Ω\times[0,T]$. We also prove some technical variants of this theorem, including a version where the complement of $\{D_{n}\}_{n}$ forms a disjoint sequence. Applications include a complete characterization of sequences admitting convex combinations converging in the Émery topology, and a supermartingale counterpart of Helly's selection theorem.

math.PR↗
arXiv2024

Relative weak compactness in infinite-dimensional Fefferman-Meyer duality

Let $E$ be a Banach space such that $E'$ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(μ,E)$ to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in $L^{1}(μ,E)$. In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for $H^{1}(μ,E)$-bounded sequences, which decomposes a subsequence into a relatively weakly compact part, a pointwise weakly convexly convergent part, and a null part converging to zero uniformly on compacts in measure. As a corollary, we investigate a parameterized version of the vector-valued Komlós theorem without the assumption of $H^{1}(μ,E)$-boundedness.

math.FA↗
arXiv2024

Risk Measure Duality Without Structure

We study risk measures on vector spaces of random variables which a priori have little structure, such as spaces lacking law invariance or a lattice structure. Ensuring the existence of a tractable dual representation (one which does not contain non-sigma-additive measures) is one of the main problems in risk measure theory, and we address it under minimal conditions. The existence of a tractable dual representation is shown to be equivalent to a Fatou-like property when the domain of the risk measure satisfies a topological regularity condition. Without the topological regularity condition, the Fatou property implies the existence of a tractable dual representation whenever the risk measure is viewed with constraints. We also present counterexamples demonstrating the sharpness of the assumptions made.

q-fin.RM↗