arXiv subjects

Camilo Hernández

Publications and source records attributed to Camilo Hernández.

10 recordsLinked to original sources

arXiv2026

Optimal transportation under McKean-Vlasov dynamics

We study a broad class of stochastic transport problems under controlled McKean-Vlasov dynamics, allowing control of both drift and volatility, with weak terminal constraints. In contrast to the classical Schrödinger problem, the reference dynamics depend on the evolving law of the process, leading to nonlinear optimization problems on path space, while the terminal constraint is formulated through weak optimal transport, permitting a flexible treatment of distributional targets. This framework encompasses several problems of current interest, including the mean field Schrödinger problem, martingale optimal transport, and Schrödinger-Bass bridges. We establish three main results. First, we prove the existence of optimal controls under general coercivity, continuity, and convexity assumptions. Second, we derive a convex dual representation that extends weak Kantorovich duality and stochastic optimal transport duality to controlled McKean-Vlasov dynamics. Finally, for the mean field Schrödinger problem, we prove convergence, at the level of value functions, of the associated finite-particle Schrödinger bridge problems to the mean field limit, thereby identifying the nonlinear problem as the exact infinite-particle limit of finite-particle Schrödinger bridges. As a byproduct, we establish the existence of an optimal mean field Schrödinger potential, providing a variational foundation for duality in nonlinear stochastic transport.

math.OC↗

arXiv2026

Closed-loop equilibria for Stackelberg games: a story about stochastic targets

We provide a general approach to reformulating any continuous-time stochastic Stackelberg differential game under closed-loop strategies as a single-level optimisation problem with target constraints. More precisely, we consider a Stackelberg game in which the leader and the follower can both control the drift and the volatility of a stochastic output process, in order to maximise their respective expected utility. The aim is to characterise the Stackelberg equilibrium when the players adopt 'closed-loop strategies', i.e. their decisions are based solely on the historical information of the output process, excluding especially any direct dependence on the underlying driving noise, often unobservable in real-world applications. We first show that, by considering the second-order backward stochastic differential equation associated with the continuation utility of the follower as a controlled state variable for the leader, the latter's unconventional optimisation problem can be reformulated as a more standard stochastic control problem with target constraints. Thereafter, adapting the methodology developed by Soner and Touzi (2002a) or Bouchard, Elie and Imbert (2010), the optimal strategies, as well as the corresponding value of the Stackelberg equilibrium, can be characterised through the solution of a well-specified system of Hamilton- Jacobi-Bellman equations. For a more comprehensive insight, we illustrate our approach through a simple example, facilitating both theoretical and numerical detailed comparisons with the solutions under different information structures studied in the literature.

math.OC↗

arXiv2026

Marginal flows of non-entropic weak Schrödinger bridges

This paper introduces a dynamic formulation of divergence-regularized optimal transport with weak targets on the path space. In our formulation, the classical relative entropy penalty is replaced by a general convex divergence, and terminal constraints are imposed in a weak sense. We establish well-posedness and a convex dual formulation, together with a dual existence result and explicit structural characterizations of primal and dual optimizers. Specifically, the optimal path measure admits an explicit density relative to a reference diffusion, generalizing the classical Schr{ö}dinger system. In the case of zero transport cost, which corresponds to a non-entropic dynamic Schr{ö}dinger problem, we further characterize the flow of time marginals of the optimal bridge, recovering known results in the entropic setting and providing new descriptions for non-entropic divergences, including the $χ^2$-divergence

math.PR↗

arXiv2024

Propagation of chaos for mean field Schrödinger problems

In this work, we study the mean field Schrödinger problem from a purely probabilistic point of view by exploiting its connection to stochastic control theory for McKean-Vlasov diffusions. Our main result shows that the mean field Schrödinger problem arises as the limit of ``standard'' Schrödinger problems over interacting particles. Due to the stochastic maximum principle and a suitable penalization procedure, the result follows as a consequence of novel (quantitative) propagation of chaos results for forward-backwards particle systems. The approach described in the paper seems flexible enough to address other questions in the theory. For instance, our stochastic control technique further allows us to solve the mean field Schrödinger problem and characterize its solution, the mean field Schrödinger bridge, by a forward-backward planning equation.

math.PR↗

arXiv2023

Time-inconsistent contract theory

This paper investigates the moral hazard problem in finite horizon with both continuous and lump-sum payments, involving a time-inconsistent sophisticated agent and a standard utility maximiser principal. Building upon the so-called dynamic programming approach in Cvitanić, Possamaï, and Touzi [18] and the recently available results in Hernández and Possamaï [43], we present a methodology that covers the previous contracting problem. Our main contribution consists in a characterisation of the moral hazard problem faced by the principal. In particular, it shows that under relatively mild technical conditions on the data of the problem, the supremum of the principal's expected utility over a smaller restricted family of contracts is equal to the supremum over all feasible contracts. Nevertheless, this characterisation yields, as far as we know, a novel class of control problems that involve the control of a forward Volterra equation via Volterra-type controls, and infinite-dimensional stochastic target constraints. Despite the inherent challenges associated to such a problem, we study the solution under three different specifications of utility functions for both the agent and the principal, and draw qualitative implications from the form of the optimal contract. The general case remains the subject of future research.

econ.TH↗

arXiv2021

On quadratic multidimensional type-I BSVIEs, infinite families of BSDEs and their applications

This paper investigates multidimensional extended type-I BSVIEs and infinite families of BSDEs in the case of quadratic generators. We establish existence and uniqueness results in the case of fully quadratic as well as Lipschitz-quadratic quadratic generators. We also present and discuss a type of flow property satisfied by this family of BSVIEs. As a preliminary step, we establish the well-posedness of a class of infinite families of BSDEs, as introduced in Hernåndez and Possamaï [20], which are of interest in their own right. Our approach relies on the strategy developed by Tevsadze [49] for quadratic BSDEs and the treatment of Lipschitz extended type-I BSVIEs in [20]. We motivate their analysis of both of these objects by a series of practical applications.

math.PR↗

arXiv2021

Me, myself and I: a general theory of non-Markovian time-inconsistent stochastic control for sophisticated agents

We develop a theory for continuous-time non-Markovian stochastic control problems which are inherently time-inconsistent. Their distinguishing feature is that the classical Bellman optimality principle no longer holds. Our formulation is cast within the framework of a controlled non-Markovian forward stochastic differential equation, and a general objective functional setting. We adopt a game-theoretic approach to study such problems, meaning that we seek for sub-game perfect Nash equilibrium points. As a first novelty of this work, we introduce and motivate a refinement of the definition of equilibrium that allows us to establish a direct and rigorous proof of an extended dynamic programming principle, in the same spirit as in the classical theory. This in turn allows us to introduce a system consisting of an infinite family of backward stochastic differential equations analogous to the classical HJB equation. We prove that this system is fundamental, in the sense that its well-posedness is both necessary and sufficient to characterise the value function and equilibria. As a final step we provide an existence and uniqueness result. Some examples and extensions of our results are also presented.

math.OC↗

arXiv2020

A unified approach to well-posedness of type-I backward stochastic Volterra integral equations

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite dimensional system of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semilinear partial differential equation of Hamilton-Jacobi-Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.

math.PR↗

arXiv2020

Approximate super-resolution of positive measures in all dimensions

We study the problem of reconstructing a positive discrete measure on a compact set $K \subseteq \mathbb{R}^n$ from a finite set of moments (possibly known only approximately) via convex optimization. We give new uniqueness results, new quantitative estimates for approximate recovery and a new sum-of-squares based hierarchy for approximate super-resolution on compact semi-algebraic sets.

math.OC↗

arXiv2017

Compressive sensing and truncated moment problems on spheres

We propose convex optimization algorithms to recover a good approximation of a point measure $μ$ on the unit sphere $S\subseteq \mathbb{R}^n$ from its moments with respect to a set of real-valued functions $f_1,\dots, f_m$. Given a finite subset $C\subseteq S$ the algorithm produces a measure $μ^*$ supported on $C$ and we prove that $μ^*$ is a good approximation to $μ$ whenever the functions $f_1,\dots, f_m$ are a sufficiently large random sample of independent Kostlan-Shub-Smale polynomials. More specifically, we give sufficient conditions for the validity of the equality $μ=μ^*$ when $μ$ is supported on $C$ and prove that $μ^*$ is close to the best approximation to $μ$ supported on $C$ provided that all points in the support of $μ$ are close to $C$.

math.OC↗