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We consider the problem of model selection type aggregation in the context of density estimation. We first show that empirical risk minimization is sub-optimal for this problem and it shares this property with the exponential weights…

We consider the problem of finding a stopping time that minimises the $L^1$-distance to $\theta$, the time at which a L\'evy process attains its ultimate supremum. This problem was studied in [12] for a Brownian motion with drift and a…

We use the technique developed in [32]-[33] to construct the upper and the lower bounds for classes of problems containing non-local terms, including problems in micromagnetics and problems arising in the variational study of the Method of…

In this paper we discuss the question how to bound supremum of a stochastic process with the index set of a product type. There is a tempting idea to approach the question by the analysis of the process on each of the marginal index spaces…

Totally asymmetric simple exclusion processes on lattices with junctions, where particles interact with hard-core exclusion and move on parallel lattice branches that at the junction combine into a single lattice segment, are investigated.…

Robustness and generalization ability of machine learning models are of utmost importance in various application domains. There is a wide interest in efficient ways to analyze those properties. One important direction is to analyze…

We characterize the value of swing contracts in continuous time as the unique viscosity solution of a Hamilton-Jacobi-Bellman equation with suitable boundary conditions. The case of contracts with penalties is straightforward, and in that…

We study the finite element approximation of the solid isotropic material with penalization method (SIMP) for the topology optimization problem of minimizing the compliance of a linearly elastic structure. To ensure the existence of a local…

We consider vector spin glass models with self-overlap correction. Since the limit of free energy is an infimum, we use arguments analogous to those for generic models to show the following: 1) the averaged self-overlap converges; 2) the…

It has recently been shown in [arXiv:2310.06745] that, upon constraining the system to stay in a balanced state, the Parisi formula for the mean-field Potts model can be written as an optimization problem over permutation-invariant…

This paper considers the problem of minimizing the time average of a stochastic process subject to time average constraints on other processes. A canonical example is minimizing average power in a data network subject to multi-user…

In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of $u$ or of its gradient and we find the best values of the constants…

The recent approach based on Hamiltonian systems and the implicit parametri\-za\-tion theorem, provides a general fixed domain approximation method in shape optimization problems, using optimal control theory. In previous works, we have…

In this paper we characterize sharp time-data tradeoffs for optimization problems used for solving linear inverse problems. We focus on the minimization of a least-squares objective subject to a constraint defined as the sub-level set of a…

We consider the optimal prediction problem of stopping a spectrally negative L\'evy process as close as possible to a given distance $b \geq 0$ from its ultimate supremum, under a squared error penalty function. Under some mild conditions,…

In mathematical modelling, the data and solutions are represented as measurable functions and their quality is oftentimes captured by the membership to a certain function space. One of the core questions for an analysis of a model is the…

We consider an optimal switching problem with random lag and possibility of component failure. The random lag is modeled by letting the operation mode follow a regime switching Markov-model with transition intensities that depend on the…

In this paper we present a sufficient condition for the existence of a solution for an equilibrium problem on an Hadamard manifold and under suitable assumptions on the sectional curvature, we propose a framework for the convergence…

We examine the supervised learning problem in its continuous setting and give a general optimality condition through techniques of functional analysis and the calculus of variations. This enables us to solve the optimality condition for the…

This paper considers an optimal control problem for a linear mean-field stochastic differential equation having regime switching with quadratic functional in the large time horizons. Our main contribution lies in establishing the strong…