We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated with fully nonlinear elliptic equations of order $2\,s$, with $s\in (1/2,\,1)$, and a coercive gradient term with subcritical power $0< p<2\,s$. Due to the nonlocal nature of the diffusion, new blow-up phenomena arise within the range $0< p<2\,s$, involving a continuum family of solutions and/or solutions blowing-up to $-\infty$ on the boundary. This is in striking difference with the local case studied by Lasry–Lions for the subquadratic case $1< p<2$.

In this paper we state some sharp maximum principle, i.e. we characterize the geometry of the sets of minima for supersolutions of equations involving the $k$-th fractional truncated Laplacian or the $k$-th fractional eigenvalue which are fully nonlinear integral operators whose nonlocality is somehow $k$-dimensional.

Semi-supervised and unsupervised machine learning methods often rely on graphs to model data, prompting research on how theoretical properties of operators on graphs are leveraged in learning problems. While most of the existing literature focuses on undirected graphs, directed graphs are very important in practice, giving models for physical, biological or transportation networks, among many other applications. In this paper, we propose a new framework for rigorously studying continuum limits of learning algorithms on directed graphs. We use the new framework to study the PageRank algorithm and show how it can be interpreted as a numerical scheme on a directed graph involving a type of normalised graph Laplacian. We show that the corresponding continuum limit problem, which is taken as the number of webpages grows to infinity, is a second-order, possibly degenerate, elliptic equation that contains reaction, diffusion and advection terms. We prove that the numerical scheme is consistent and stable and compute explicit rates of convergence of the discrete solution to the solution of the continuum limit partial differential equation. We give applications to proving stability and asymptotic regularity of the PageRank vector. Finally, we illustrate our results with numerical experiments and explore an application to data depth.

In this article, we establish a Lyapunov-type inequality for the following extremal Pucci’s equation:

$$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}{\mathcal{M}}_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6EC}}^{+}(D^{2}u)+b(x)|Du|+a(x)u=0 & \text{in}~\unicode[STIX]{x1D6FA},\\ u=0 & \text{on}~\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA},\end{array}\right.\end{eqnarray}$$

$\unicode[STIX]{x1D6FA}$

$\mathbb{R}^{N}$

$N\geq 2$

We consider a two-player zero-sum stochastic differential game with a random planning horizon and diffusive state variable dynamics. The random planning horizon is a function of a non-negative continuous random variable, which is assumed to be independent of the Brownian motion driving the state variable dynamics. We study this game using a combination of dynamic programming and viscosity solution techniques. Under some mild assumptions, we prove that the value of the game exists and is the unique viscosity solution of a certain nonlinear partial differential equation of Hamilton–Jacobi–Bellman–Isaacs type.

This paper deals with the periodic homogenization of nonlocal parabolic Hamilton–Jacobi equations with superlinear growth in the gradient terms. We show that the problem presents different features depending on the order of the nonlocal operator, giving rise to three different cell problems and effective operators. To prove the locally uniform convergence to the unique solution of the Cauchy problem for the effective equation we need a new comparison principle among viscosity semi-solutions of integrodifferential equations that can be of independent interest.

We prove Hölder continuous regularity of bounded, uniformly continuous, viscosity solutions of degenerate fully nonlinear equations defined in all of ℝn space. In particular, the result applies also to some operators in Carnot groups.

In this paper, we study the fractional Dirichlet problem with the homogeneous exterior data posed on a bounded domain with Lipschitz continuous boundary. Under an extra assumption on the domain, slightly weaker than the exterior ball condition, we are able to prove existence and uniqueness of solutions which are Hölder continuous on the boundary. In proving this result, we use appropriate barrier functions obtained by an approximation procedure based on a suitable family of zero-th order problems. This procedure, in turn, allows us to obtain an approximation scheme for the Dirichlet problem through an equicontinuous family of solutions of the approximating zero-th order problems on ${\bar \Omega}$. Both results are extended to an ample class of fully non-linear operators.

In this paper we show that conforming Galerkin approximations for p-harmonic functions tend to ∞-harmonic functions in the limit p → ∞ and h → 0, where h denotes the Galerkin discretization parameter.

We consider a two-dimensional optimal dividend problem in the context of two insurance companies with compound Poisson surplus processes, who collaborate by paying each other's deficit when possible. We study the stochastic control problem of maximizing the weighted sum of expected discounted dividend payments (among all admissible dividend strategies) until ruin of both companies, by extending results of univariate optimal control theory. In the case that the dividends paid by the two companies are equally weighted, the value function of this problem compares favorably with the one of merging the two companies completely. We identify the optimal value function as the smallest viscosity supersolution of the respective Hamilton–Jacobi–Bellman equation and provide an iterative approach to approximate it numerically. Curve strategies are identified as the natural analogue of barrier strategies in this two-dimensional context. A numerical example is given for which such a curve strategy is indeed optimal among all admissible dividend strategies, and for which this collaboration mechanism also outperforms the suitably weighted optimal dividend strategies of the two stand-alone companies.

A new approach to jump diffusion is introduced, where the jump is treated as a vertical shift of the price (or volatility) function. This method is simpler than the previous methods and it is applied to the portfolio model with a stochastic volatility. Moreover, closed-form solutions for the optimal portfolio are obtained. The optimal closed-form solutions are derived when the value function is not smooth, without relying on the method of viscosity solutions.

The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem.

We provide a deterministic-control-based interpretation for a broad class of fullynonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in asmooth domain. We construct families of two-person games depending on a small parameterε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary conditionby introducing some specific rules near the boundary. We show that the value functionconverges, in the viscosity sense, to the solution of the PDE as ε tendsto zero. Moreover, our construction allows us to treat both the oblique and the mixed typeDirichlet–Neumann boundary conditions.

This article is the starting point of a series of works whose aim is the study of deterministic control problems where the dynamic and the running cost can be completely different in two (or more) complementary domains of the space ℝN. As a consequence, the dynamic and running cost present discontinuities at the boundary of these domains and this is the main difficulty of this type of problems. We address these questions by using a Bellman approach: our aim is to investigate how to define properly the value function(s), to deduce what is (are) the right Bellman Equation(s) associated to this problem (in particular what are the conditions on the set where the dynamic and running cost are discontinuous) and to study the uniqueness properties for this Bellman equation. In this work, we provide rather complete answers to these questions in the case of a simple geometry, namely when we only consider two different domains which are half spaces: we properly define the control problem, identify the different conditions on the hyperplane where the dynamic and the running cost are discontinuous and discuss the uniqueness properties of the Bellman problem by either providing explicitly the minimal and maximal solution or by giving explicit conditions to have uniqueness.

In this paper we study systems of weakly coupled Hamilton-Jacobi equations with implicit obstacles that arise in optimal switching problems. Inspired by methods from the theory of viscosity solutions and weak $\text{KAM}$ theory, we extend the notion of Aubry set for these systems. This enables us to prove a new result on existence and uniqueness of solutions for the Dirichlet problem, answering a question of F. Camilli, P. Loreti, and N. Yamada.

The paper deals with deterministic optimal control problems with state constraints and non-linear dynamics. It is known for such problems that the value function is in general discontinuous and its characterization by means of a Hamilton-Jacobi equation requires some controllability assumptions involving the dynamics and the set of state constraints. Here, we first adopt the viability point of view and look at the value function as its epigraph. Then, we prove that this epigraph can always be described by an auxiliary optimal control problem free of state constraints, and for which the value function is Lipschitz continuous and can be characterized, without any additional assumptions, as the unique viscosity solution of a Hamilton-Jacobi equation. The idea introduced in this paper bypasses the regularity issues on the value function of the constrained control problem and leads to a constructive way to compute its epigraph by a large panel of numerical schemes. Our approach can be extended to more general control problems. We study in this paper the extension to the infinite horizon problem as well as for the two-player game setting. Finally, an illustrative numerical example is given to show the relevance of the approach.

We study the partial differential equation

        max{Lu − f, H(Du)} = 0

where u is the unknownfunction, L is a second-order elliptic operator, f is agiven smooth function and H is a convex function. This is a modelequation for Hamilton-Jacobi-Bellman equations arising in stochastic singular control. Weestablish the existence of a unique viscosity solution of the Dirichlet problem that has aHölder continuous gradient. We also show that if H is uniformlyconvex, the gradient of this solution is Lipschitz continuous.

This paper is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a “junction”, that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present article provide new powerful tools for the analysis of such problems.

This paper concerns continuous dependence estimates for Hamilton-Jacobi-Bellman-Isaacsoperators. We establish such an estimate for the parabolic Cauchy problem in the wholespace  [0, +∞) × ℝn and, under some periodicity and eitherellipticity or controllability assumptions, we deduce a similar estimate for the ergodicconstant associated to the operator. An interesting byproduct of the latter result will bethe local uniform convergence for some classes of singular perturbation problems.

In this paper we study the limit as p → ∞ of minimizers of thefractional Ws,p-norms. In particular, weprove that the limit satisfies a non-local and non-linear equation. We also prove theexistence and uniqueness of solutions of the equation. Furthermore, we prove the existenceof solutions in general for the corresponding inhomogeneous equation. By making strong useof the barriers in this construction, we obtain some regularity results.