On the monotone and primal-dual active set schemes for ℓp\ell^p-type problems, p∈(0,1]p \in (0,1]

Nonsmooth nonconvex optimization problems involving the $\ell^p$ quasi-norm, $p \in (0, 1]$, of a linear map are considered. A monotonically convergent scheme for a regularized version of the original problem is developed and necessary optimality conditions for the original problem in the form of a complementary system amenable for computation are given. Then an algorithm for solving the above mentioned necessary optimality conditions is proposed. It is based on a combination of the monotone scheme and a primal-dual active set strategy. The performance of the two algorithms is studied by means of a series of numerical tests in different cases, including optimal control problems, fracture mechanics and microscopy image reconstruction

Large deviations for some fast stochastic volatility models by viscosity methods

We consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. We study the asymptotics of a logarithmic functional of the process by methods of the theory of homogenisation and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity

Junction conditions for finite horizon optimal control problems on multi-domains with continuous and discontinuous solutions

This paper deals with junction conditions for Hamilton-Jacobi-Bellman (HJB) equations for finite horizon control problems on multi-domains. We consider two different cases where the final cost is continuous or lower semi-continuous. In the continuous case we extend the results of "Hamilton-Jacobi-Bellman equations on multi-domains" by the second and third authors in a more general framework with switching running costs and weaker controllability assumptions. The comparison principle has been established to guarantee the uniqueness and the stability results for the HJB system on such multi-domains. In the lower semi-continuous case, we characterize the value function as the unique lower semi-continuous viscosity solution of the HJB system, under a local controllability assumption

Existence and non-existence for time-dependent mean field games with strong aggregation

We investigate the existence of classical solutions to second-order quadratic Mean-Field Games systems with local and strongly decreasing couplings of the form $-\sigma m^\alpha$, $\alpha \ge 2/N$, where $m$ is the population density and $N$ is the dimension of the state space. We prove the existence of solutions under the assumption that $\sigma$ is small enough. For large $\sigma$, we show that existence may fail whenever the time horizon $T$ is large.Comment: 28 pages, 1 figur

On a monotone scheme for nonconvex nonsmooth optimization with applications to fracture mechanics

A general class of nonconvex optimization problems is considered, where the penalty is the composition of a linear operator with a nonsmooth nonconvex mapping, which is concave on the positive real line. The necessary optimality condition of a regularized version of the original problem is solved by means of a monotonically convergent scheme. Such problems arise in continuum mechanics, as for instance cohesive fractures, where singular behaviour is usually modelled by nonsmooth nonconvex energies. The proposed algorithm is successfully tested for fracture mechanics problems. Its performance is also compared to two alternative algorithms for nonsmooth nonconvex optimization arising in optimal control and mathematical imaging.Comment: arXiv admin note: text overlap with arXiv:1709.0650

Habits and demand changes after COVID-19

In this paper, we investigate how a transitory lockdown of a sector of the economy may have changed our habits and, therefore, altered the goods’ demand permanently. In a two-sector infinite horizon economy, we show that the demand of the goods produced by the sector closed during the lockdown could shrink or expand with respect to their pre-pandemic level depending on the lockdown’s duration and the habits’ strength. We also show that the end of a lockdown may be characterized by a price surge due to a combination of strong demand of both goods and rigidities in production

Some Results in Nonlinear PDEs: Large Deviations Problems, Nonlocal Operators, and Stability for Some Isoperimetric Problems

This thesis is concerned with various problems arising in the study of nonlinear elliptic PDE. It is divided into three parts. In the first part we consider the short time behaviour of stochastic systems affected by a stochastic volatility evolving at a faster time scale. Our mathematical framework is that of multiple time scale systems and singular perturbations. We are concerned with the asymptotic behaviour of a logarithmic functional of the process, which we study by methods of the theory of homogenization and singular perturbations for fully nonlinear PDEs. We point out three regimes depending on how fast the volatility oscillates relative to the horizon length. We provide some financial applications, namely we prove a large deviation principle for each regime and apply it to the asymptotics of option prices near maturity. In the second part we are concerned with the well-posedness of Neumann boundary value problems for nonlocal Hamilton-Jacobi equations related to jump processes in general (enough smooth) domains. We consider a nonlocal diffusive term of censored type of order less than 1 and Hamiltonian both in coercive form and in noncoercive Bellman form, whose growth in the gradient make them the leading term in the equation. We prove a comparison principle for bounded sub-and supersolutions in the context of viscosity solutions with generalized boundary conditions, and consequently by Perron's method we get the existence and uniqueness of continuous solutions. We give some applications in the evolutive setting, proving the large time behaviour of the associated evolutive problem under suitable assumptions on the data. In the last part we present some stability results for a class of integral inequalities, the Borell-Brascamp-Lieb inequality and we strengthen, in two different ways, these inequalities in the class of power concave functions. Then we present some applications to prove analogous quantitative results for certain type of isoperimetric inequalities satisfied by a wide class of variational functionals that can be written in terms of the solution of a suitable elliptic boundary value problem. As a toy model, we consider the torsional rigidity and prove quantitative results for its Brunn-Minkowski inequality and for its consequent (Urysohn type) isoperimetric inequality.Questa tesi si occupa di vari problemi che sorgono nello studio di equazioni alle derivate parziali ellittiche e paraboliche. La tesi è divisa in tre parti. Nella prima parte studiamo il comportamento per tempi brevi di sistemi dinamici a volatilità stocastica che evolve in una scala temporale più veloce.Ci occupiamo di perturbazioni singolari di sistemi a scala temporale multipla. Il nostro primo obiettivo è lo studio del comportamento asintotico di un funzionale logaritmico del processo stocastico, attraverso i metodi della teoria dell' omogeneizzazione e delle perturbazioni singolari per equazioni alle derivate parziali completamente non lineari. Individuiamo tre regimi a seconda della velocità con cui la volatilità oscilla rispetto alla lunghezza dell'orizzonte temporale. Inoltre forniamo alcune applicazioni finanziarie, in particolare proviamo un principio di grandi deviazioni in ogni regime e lo applichiamo per derivare una stima asintotica dei prezzi di opzioni vicino alla maturità e una formula asintotica per la volatilità di Black-Scholes implicita. Nella seconda parte studiamo la buona definizione di problemi al contorno di tipo Neumann, in domini generali (sufficientemente regolari), per equazioni tipo Hamilton-Jacobi con termini non locali che derivano da processi discontinui a salti. Consideriamo un termine diffusivo non locale di tipo censored, di ordine strettamente minore di 1, e un' Hamiltoniana, sia in forma coerciva sia di tipo Bellman non necessariamente coerciva, la cui crescita nel gradiente la rende il termine principale nell'equazione. Dimostriamo un principio di confronto per sotto e sopra soluzioni limitate (in senso di viscosità) con condizioni al contorno generalizzate, e di conseguenza tramite il metodo di Perron otteniamo l'esistenza e l'unicità di soluzioni continue. Diamo alcune applicazioni nel caso evolutivo, dimostrando la convergenza per tempi grandi della soluzione del problema evolutivo alla soluzione del problema stazionario associato, supponendo opportune ipotesi sui dati. Nell'ultima parte presentiamo alcuni risultati di stabilità per una classe di diseguaglianze integrali, le disuguaglianze Borrell-Brascamp-Lieb e rafforziamo, in due modi diversi, queste disuguaglianze nella classe di funzioni a potenza concava. Come applicazione di questo risultato, presentiamo analoghi risultati quantitativi per alcuni tipi di disuguaglianze isoperimetriche soddisfatte da un'ampia classe di funzionali variazionali che possono essere scritti in termini della soluzione di un opportuno problema al contorno ellittico. Come modello giocattolo, consideriamo la rigidità torsionale e dimostriamo risultati quantitativi per la sua disuguaglianza Brunn-Minkowski e per la sua conseguente disuguaglianza isoperimetrica di tipo Urysohn