Abstract: We construct an abstract framework in which the dynamic programming principle (DPP) can be readily proven. It encompasses a broad range of common stochastic control problems in the weak formulation, and deals with problems in the “martingale formulation” with particular ease. We give two illustrations; first, we establish the DPP for general controlled diffusions and show that their value functions are viscosity solutions of the associated Hamilton–Jacobi–Bellman equations under minimal conditions. After that, we show how to treat singular control on the example of the classical monotone-follower problem. PubDate: 2019-06-20

Abstract: The value function associated with an optimal control problem subject to the Navier–Stokes equations in dimension two is analyzed. Its smoothness is established around a steady state, moreover, its derivatives are shown to satisfy a Riccati equation at the order two and generalized Lyapunov equations at the higher orders. An approximation of the optimal feedback law is then derived from the Taylor expansion of the value function. A convergence rate for the resulting controls and closed-loop systems is demonstrated. PubDate: 2019-06-18

Abstract: In a Hilbert space \({{\mathcal {H}}}\) , we study the convergence properties of a class of relaxed inertial forward–backward algorithms. They aim to solve structured monotone inclusions of the form \(Ax + Bx \ni 0\) where \(A:{{\mathcal {H}}}\rightarrow 2^{{\mathcal {H}}}\) is a maximally monotone operator and \(B:{{\mathcal {H}}}\rightarrow {{\mathcal {H}}}\) is a cocoercive operator. We extend to this class of problems the acceleration techniques initially introduced by Nesterov, then developed by Beck and Teboulle in the case of structured convex minimization (FISTA). As an important element of our approach, we develop an inertial and parametric version of the Krasnoselskii–Mann theorem, where joint adjustment of the inertia and relaxation parameters plays a central role. This study comes as a natural extension of the techniques introduced by the authors for the study of relaxed inertial proximal algorithms. An illustration is given to the inertial Nash equilibration of a game combining non-cooperative and cooperative aspects. PubDate: 2019-06-15

Abstract: In this paper we study the nonzero-sum constrained stochastic games for continuous-time jump processes with denumerable states and possibly unbounded transition rates. The optimality criterion under consideration is the expected average payoff criterion and the payoff functions of the players are allowed to be unbounded. Under the reasonable conditions, we introduce an approximating sequence of the auxiliary game models and show the existence of stationary constrained Nash equilibria for these approximating game models via employing the average occupation measures and constructing a suitable multifunction. Moreover, we obtain that any limit point of the stationary constrained Nash equilibria for the approximating sequence of the game models is a constrained Nash equilibrium for the original game model. Furthermore, we use a controlled birth and death system to illustrate our main results. PubDate: 2019-06-14

Abstract: We study the asymptotic behavior of Timoshenko systems with a fractional operator in the memory term depending on a parameter \(\theta \in [0,1]\) and acting only on one equation of the system. Considering exponentially decreasing kernels, we find exact decay rates. To be precise, we show that for \(\theta \in [0,1)\) , the system decay polynomially with rates that depend on the value of the difference of the wave propagation speeds. We also prove that these decay rates are optimal. Moreover, when \(\theta =1\) and the equations have the same propagation speeds we obtain the exponential decay of the solutions. PubDate: 2019-06-10

Abstract: We define a class of reflected backward stochastic differential equation (RBSDE) driven by a marked point process (MPP) and a Brownian motion, where the solution is constrained to stay above a given càdlàg process. The MPP is only required to be non-explosive and to have totally inaccessible jumps. Under suitable assumptions on the coefficients we obtain existence and uniqueness of the solution, using the Snell envelope theory. We use the equation to represent the value function of an optimal stopping problem, and we characterize the optimal strategy. PubDate: 2019-06-06

Abstract: In this paper, BDG-type inequality for G-stochastic calculus with respect to G-Lévy process is obtained, and solutions of the stochastic differential equations driven by the G-Lévy process under the non-Lipschitz condition are constructed. Furthermore, the mean square exponential stability and quasi-sure exponential stability of the solution using the G-Lyapunov function method is established. PubDate: 2019-06-05

Abstract: We study in this paper a compartmental SIR model for a population distributed in a bounded domain D of \(\mathbb {R}^d\) , \(\hbox {d}= 1\) , 2 or 3. We describe a spatial model for the spread of a disease on a grid of D. We prove two laws of large numbers. On the one hand, we prove that the stochastic model converges to the corresponding deterministic patch model as the size of the population tends to infinity. On the other hand, by letting both the size of the population tend to infinity and the mesh of the grid go to zero, we obtain a law of large numbers in the supremum norm, where the limit is a diffusion SIR model in D. PubDate: 2019-06-03

Abstract: In the present contribution we study the sliding mode control (SMC) problem for a diffuse interface tumor growth model coupling a viscous Cahn–Hilliard type equation for the phase variable with a reaction–diffusion equation for the nutrient. First, we prove the well-posedness and some regularity results for the state system modified by the state-feedback control law. Then, we show that the chosen SMC law forces the system to reach within finite time the sliding manifold (that we chose in order that the tumor phase remains constant in time). The feedback control law is added in the Cahn–Hilliard type equation and leads the phase onto a prescribed target \(\varphi ^*\) in finite time. PubDate: 2019-06-01

Abstract: We present a version of the stochastic maximum principle (SMP) for ergodic control problems. In particular we give necessary (and sufficient) conditions for optimality for controlled dissipative systems in finite dimensions. The strategy we employ is mainly built on duality techniques. We are able to construct a dual process for all positive times via the analysis of a suitable class of perturbed linearized forward equations. We show that such a process is the unique bounded solution to a backward SDE on infinite horizon from which we can write a version of the SMP. PubDate: 2019-06-01

Abstract: We show that the compliance functional in elasticity is differentiable with respect to horizontal variations of the load term, when the latter is given by a possibly concentrated measure; moreover, we provide an integral representation formula for the derivative as a linear functional of the deformation vector field. The result holds true as well for the p-compliance in the scalar case of conductivity. Then we study the limit problem as \(p \rightarrow + \infty \) , which corresponds to differentiate the Wasserstein distance in optimal mass transportation with respect to horizontal perturbations of the two marginals. Also in this case, we obtain an existence result for the derivative, and we show that it is found by solving a minimization problem over the family of all optimal transport plans. When the latter contains only one element, we prove that the derivative of the p-compliance converges to the derivative of the Wasserstein distance in the limit as \(p \rightarrow + \infty \) . PubDate: 2019-06-01

Abstract: In this paper, we consider both one-period and multi-period distributionally robust mean-CVaR portfolio selection problems. We adopt an uncertainty set which considers the uncertainties in terms of both the distribution and the first two order moments. We use the parametric method and the dynamic programming technique to come up with the closed-form optimal solutions for both the one-period and the multi-period robust portfolio selection problems. Finally, we show that our approaches are efficient when compared with both normal based portfolio selection models, and robust approaches based on known moments. PubDate: 2019-06-01

Abstract: We study an optimal investment problem with multiple entries and forced exits. A closed form solution of the optimisation problem is presented for general underlying diffusion dynamics and a general running payoff function in the case when forced exits occur on the jump times of a Poisson process. Furthermore, we allow the investment opportunity to be subject to the risk of a catastrophe that can occur at the jumps of the Poisson process. More precisely, we attach IID Bernoulli trials to the jump times and if the trial fails, no further re-entries are allowed. Interestingly, we find in the general case that the optimal investment threshold is independent of the success probability is the Bernoulli trials. The results are illustrated with explicit examples. PubDate: 2019-06-01

Abstract: The present paper represents a continuation of Migórski et al. (J Elast 127:151–178, 2017). There, the analysis of a new class of elliptic variational–hemivariational inequalities in reflexive Banach spaces, including existence and convergence results, was provided. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. In the current paper we complete this study with new results, including a convergence result with respect the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a one-dimensional mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a foundation, under the action of a body force. PubDate: 2019-06-01

Abstract: In this paper, we consider a non-overlapping domain decomposition method for solving optimal boundary control problems governed by parabolic equations. The whole domain is divided into non-overlapping subdomains, and the global optimal boundary control problem is decomposed into the local problems in these subdomains. The integral mean method is utilized to present an explicit flux calculation on the inter-domain boundary in order to communicate the local problems on the interface between subdomains. We establish the fully parallel and discrete schemes for solving these local problems. A priori error estimates in \(L^2\) -norm are derived for the state, co-state and control variables. Finally, we present numerical experiments to show the validity of the schemes and verify the derived theoretical results. PubDate: 2019-06-01

Abstract: In this paper, controlled linear stochastic evolution equations driven by square integrable Lévy processes are studied in the Hilbert space setting. The control operator may be unbounded which makes the results obtained in the abstract setting applicable to parabolic SPDEs with boundary or point control. The first part contains some preliminary technical results, notably a version of Itô formula which is applicable to weak/mild solutions of controlled equations. In the second part, the ergodic control problem is solved: The feedback form of the optimal control and the formula for the optimal cost are found. As examples, various parabolic type controlled SPDEs are studied. PubDate: 2019-06-01

Abstract: The optimal stopping problem arising in the pricing of American options can be tackled by the so called dual martingale approach. In this approach, a dual problem is formulated over the space of adapted martingales. A feasible solution of the dual problem yields an upper bound for the solution of the original primal problem. In practice, the optimization is performed over a finite-dimensional subspace of martingales. A sample of paths of the underlying stochastic process is produced by a Monte-Carlo simulation, and the expectation is replaced by the empirical mean. As a rule the resulting optimization problem, which can be written as a linear program, yields a martingale such that the variance of the obtained estimator can be large. In order to decrease this variance, a penalizing term can be added to the objective function of the pathwise optimization problem. In this paper, we provide a rigorous analysis of the optimization problems obtained by adding different penalty functions. In particular, a convergence analysis implies that it is better to minimize the empirical maximum instead of the empirical mean. Numerical simulations confirm the variance reduction effect of the new approach. PubDate: 2019-06-01

Abstract: We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of \(\mathbb {R}^n\) ( \(n\ge \) 3, \(p\in [2,n)\) ) periodically perforated by balls of radius \(O(\varepsilon ^\alpha )\) where \(\alpha >1\) and \(\varepsilon \) is the size of the period. The perforations are distributed along a \((n-1)\) -dimensional manifold \(\gamma \) , and we impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter \(\varepsilon ^{-\kappa }\) , \(\kappa \in \mathbb {R}\) and \(\varepsilon \) is a small parameter that we shall make to go to zero. We analyze different relations between the parameters \(p, \, n, \, \varepsilon , \, \alpha \) and \(\kappa \) , and obtain homogenized problems which are completely new in the literature even for the case \(p=2\) . PubDate: 2019-06-01

Abstract: The standard theory of stochastic approximation (SA) is extended to the case when the constraint set is a Riemannian manifold. Specifically, the standard ODE method for analyzing SA schemes is extended to iterations constrained to stay on a manifold using a retraction mapping. In addition, for submanifolds of a Euclidean space, a framework is developed for a projected SA scheme with approximate retractions. The framework is also extended to non-differentiable constraint sets. PubDate: 2019-05-16

Abstract: We propose a variational regularisation approach for the problem of template-based image reconstruction from indirect, noisy measurements as given, for instance, in X-ray computed tomography. An image is reconstructed from such measurements by deforming a given template image. The image registration is directly incorporated into the variational regularisation approach in the form of a partial differential equation that models the registration as either mass- or intensity-preserving transport from the template to the unknown reconstruction. We provide theoretical results for the proposed variational regularisation for both cases. In particular, we prove existence of a minimiser, stability with respect to the data, and convergence for vanishing noise when either of the abovementioned equations is imposed and more general distance functions are used. Numerically, we solve the problem by extending existing Lagrangian methods and propose a multilevel approach that is applicable whenever a suitable downsampling procedure for the operator and the measured data can be provided. Finally, we demonstrate the performance of our method for template-based image reconstruction from highly undersampled and noisy Radon transform data. We compare results for mass- and intensity-preserving image registration, various regularisation functionals, and different distance functions. Our results show that very reasonable reconstructions can be obtained when only few measurements are available and demonstrate that the use of a normalised cross correlation-based distance is advantageous when the image intensities between the template and the unknown image differ substantially. PubDate: 2019-05-14