Authors:Lijun Bo; Agostino Capponi Pages: 1 - 45 Abstract: We introduce a dynamic optimization framework in which collateral is used to mitigate losses arising at counterparty’s default. The investor faces two sources of risk: the default risk of the entity referencing the traded credit swap security, and counterparty risk generated from the default event of the trading counterparty. We show that the value function of the control problem coincides with the classical solution of a nonlinear dynamic programming equation. We provide an explicit characterization of the optimal investment strategy, and show that the investor does not trade if counterparty risk is sufficiently high. These findings suggest that moving credit swap trades into well-designed clearinghouses may stimulate economic activities. PubDate: 2018-02-01 DOI: 10.1007/s00245-016-9364-2 Issue No:Vol. 77, No. 1 (2018)

Authors:P. Jameson Graber; Alain Bensoussan Pages: 47 - 71 Abstract: We study a system of partial differential equations used to describe Bertrand and Cournot competition among a continuum of producers of an exhaustible resource. By deriving new a priori estimates, we prove the existence of classical solutions under general assumptions on the data. Moreover, under an additional hypothesis we prove uniqueness. PubDate: 2018-02-01 DOI: 10.1007/s00245-016-9366-0 Issue No:Vol. 77, No. 1 (2018)

Authors:Da Xu Pages: 73 - 97 Abstract: In this paper we consider space semi-discretization of some integro-differential equations using the harmonic analysis method. We study the problem of boundary observability, i. e., the problem of whether the initial data of solutions can be estimated uniformly in terms of the boundary observation as the net-spacing \(h\rightarrow 0\) . When \(h\rightarrow 0\) these finite-dimensional spaces increase and eventually cover the whole space. We thus recover the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero. We shall consider the piecewise Hermite cubic orthogonal spline collocation semi-discretization. PubDate: 2018-02-01 DOI: 10.1007/s00245-016-9367-z Issue No:Vol. 77, No. 1 (2018)

Authors:Tomasz Dlotko Pages: 99 - 128 Abstract: We consider the Navier–Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard \(L^2\) a priori estimates and we consider its regular approximations with the fractional power operator \((-P\Delta )^{1+\alpha }\) , \(\alpha >0\) small, where P is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard \(L^2\) a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator \((-P\Delta )^s\) with \(s>\frac{5}{4}\) . Using Dan Henry’s semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of such regular solutions of the approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type, the solutions corresponding to small initial data and small f are shown to be global in time and regular. PubDate: 2018-02-01 DOI: 10.1007/s00245-016-9368-y Issue No:Vol. 77, No. 1 (2018)

Authors:Seung Hak Han; William M. McEneaney Pages: 129 - 172 Abstract: We consider a two-point boundary value problem (TPBVP) in orbital mechanics involving a small body (e.g., a spacecraft or asteroid) and N larger bodies. The least action principle TPBVP formulation is converted into an initial value problem via the addition of an appropriate terminal cost to the action functional. The latter formulation is used to obtain a fundamental solution, which may be used to solve the TPBVP for a variety of boundary conditions within a certain class. In particular, the method of convex duality allows one to interpret the least action principle as a differential game, where an opposing player maximizes over an indexed set of quadratics to yield the gravitational potential. In the case where the time duration is less than a specific bound, there exists a unique critical point for the resulting differential game, which yields the fundamental solution given in terms of the solutions of associated Riccati equations. PubDate: 2018-02-01 DOI: 10.1007/s00245-016-9369-x Issue No:Vol. 77, No. 1 (2018)

Authors:Giles Auchmuty Pages: 173 - 195 Abstract: This paper describes different representations for solution operators of Laplacian boundary value problems on bounded regions in \({\mathbb R}^N, N \ge 2\) and in exterior regions when \(N = 3\) . Null Dirichlet, Neumann and Robin boundary conditions are allowed and the results hold for weak solutions in relevant subspaces of Hilbert–Sobolev space associated with the problem. The solutions of these problems are shown to be strong limits of finite rank perturbations of the fundamental solution of the problem. For exterior regions these expressions generalize multipole expansions. PubDate: 2018-02-01 DOI: 10.1007/s00245-016-9370-4 Issue No:Vol. 77, No. 1 (2018)

Authors:Beniamin Bogosel; Dorin Bucur; Ilaria Fragalà Abstract: This paper stems from the idea of adopting a new approach to solve some classical optimal packing problems for balls. In fact, we attack this kind of problems (which are of discrete nature) by means of shape optimization techniques, applied to suitable \(\Gamma \) -converging sequences of energies associated to Cheeger type problems. More precisely, in a first step we prove that different optimal packing problems are limits of sequences of optimal clusters associated to the minimization of energies involving suitable (generalized) Cheeger constants. In a second step, we propose an efficient phase field approach based on a multiphase \(\Gamma \) -convergence result of Modica–Mortola type, in order to compute those generalized Cheeger constants, their optimal clusters and, as a consequence of the asymptotic result, optimal packings. Numerical experiments are carried over in two and three space dimensions. Our continuous shape optimization approach to solve discrete packing problems circumvents the NP-hard character of these ones, and efficiently leads to configurations close to the global minima. PubDate: 2018-02-07 DOI: 10.1007/s00245-018-9476-y

Authors:Eduardo Hernández; Jianhong Wu; Denis Fernandes Abstract: We study the existence and uniqueness of mild and strict solutions for abstract neutral differential equations with state-dependent delay. Some examples related to partial neutral differential equations are presented. PubDate: 2018-01-31 DOI: 10.1007/s00245-018-9477-x

Authors:Sérgio S. Rodrigues Abstract: Given a nonstationary trajectory of the Navier–Stokes system, a finite-dimensional feedback boundary control stabilizing locally the system to the given trajectory is derived. Moreover the control is supported in a given open subset of the boundary of the domain containing the fluid. In a first step a controller (feedback operator) is derived which stabilizes the linear Oseen–Stokes system “around the given trajectory” to zero; for that a corollary of a suitable truncated boundary observability inequality, the regularizing property for the system, and some standard techniques of the optimal control theory are used. Then it is shown that the same controller also stabilizes, locally, the Navier–Stokes system to the given trajectory. PubDate: 2018-01-06 DOI: 10.1007/s00245-017-9474-5

Authors:Ana Cristina Barroso; Elvira Zappale Abstract: In this paper we investigate the possibility of obtaining a measure representation for functionals arising in the context of optimal design problems under non-standard growth conditions and perimeter penalization. Applications to modelling of strings are also provided. PubDate: 2018-01-05 DOI: 10.1007/s00245-017-9473-6

Authors:Ugur G. Abdulla; Bruno Poggi Abstract: We consider the inverse multiphase Stefan problem, where information on the heat flux on the fixed boundary is missing and must be found along with the temperature and free boundaries. Optimal control framework is pursued, where boundary heat flux is the control, and the optimality criteria consist of the minimization of the \(L_2\) -norm declination of the trace of the solution to the Stefan problem from the temperature measurement on the fixed right boundary. The state vector solves multiphase Stefan problem in a weak formulation, which is equivalent to Neumann problem for the quasilinear parabolic PDE with discontinuous coefficient. Full discretization through finite differences is implemented and discrete optimal control problem is introduced. We prove well-posedness in a Sobolev space framework and convergence of discrete optimal control problems to the original problem both with respect to the cost functional and control. Along the way, the convergence of the method of finite differences for the weak solution of the multiphase Stefan problem is proved. The proof is based on achieving a uniform \(L_{\infty }\) bound, and \(W_2^{1,1}\) -energy estimate for the discrete multiphase Stefan problem. PubDate: 2018-01-03 DOI: 10.1007/s00245-017-9472-7

Authors:D. T. V. An; J.-C. Yao; N. D. Yen Abstract: A parametric constrained convex optimal control problem, where the initial state is perturbed and the linear state equation contains a noise, is considered in this paper. Formulas for computing the subdifferential and the singular subdifferential of the optimal value function at a given parameter are obtained by means of some recent results on differential stability in mathematical programming. The computation procedures and illustrative examples are presented. PubDate: 2018-01-03 DOI: 10.1007/s00245-017-9475-4

Authors:M. Pellicer; B. Said-Houari Abstract: In this paper, we study the Moore–Gibson–Thompson equation in \(\mathbb {R}^N\) , which is a third order in time equation that arises in viscous thermally relaxing fluids and also in viscoelastic materials (then under the name of standard linear viscoelastic model). First, we use some Lyapunov functionals in the Fourier space to show that, under certain assumptions on some parameters in the equation, a norm related to the solution decays with a rate \((1+t)^{-N/4}\) . Since the decay of the previous norm does not give the decay rate of the solution itself then, in the second part of the paper, we show an explicit representation of the solution in the frequency domain by analyzing the eigenvalues of the Fourier image of the solution and writing the solution accordingly. We use this eigenvalues expansion method to give the decay rate of the solution (and also of its derivatives), which results in \((1+t)^{1-N/4}\) for \(N=1,2\) and \((1+t)^{1/2-N/4}\) when \(N\ge 3\) . PubDate: 2017-12-30 DOI: 10.1007/s00245-017-9471-8

Authors:To Fu Ma; Rodrigo Nunes Monteiro; Ana Claudia Pereira Abstract: This paper is concerned with the Timoshenko system, a recognized model for vibrations of thin prismatic beams. The corresponding autonomous system has been widely studied. However, there are only a few works dedicated to its non-autonomous counterpart. Here, we investigate the long-time dynamics of Timoshenko systems involving a nonlinear foundation and subjected to perturbations of time-dependent external forces. The main result establishes the existence of a pullback exponential attractor, which as a consequence, implies the existence of a minimal pullback attractor with finite fractal dimension. The upper-semicontinuity of attractors, as the non-autonomous forces tend to zero, is also studied. PubDate: 2017-12-28 DOI: 10.1007/s00245-017-9469-2

Authors:Sandra Cerrai; Arnaud Debussche Abstract: We are dealing with the validity of a large deviation principle for a class of reaction–diffusion equations with polynomial non-linearity, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale \(\epsilon \) and \(\delta (\epsilon )\) , respectively, with \(0<\epsilon ,\delta (\epsilon )\ll 1\) . We prove that, under the assumption that \(\epsilon \) and \(\delta (\epsilon )\) satisfy a suitable scaling limit, a large deviation principle holds in the space of continuous trajectories with values both in the space of square-integrable functions and in Sobolev spaces of negative exponent. Our result is valid, without any restriction on the degree of the polynomial nor on the space dimension. PubDate: 2017-12-26 DOI: 10.1007/s00245-017-9459-4

Authors:Xiaoshan Chen; Zhen-Qing Chen; Ky Tran; George Yin Abstract: This work develops asymptotic properties of a class of switching jump diffusion processes. The processes under consideration may be viewed as a number of jump diffusion processes modulated by a random switching mechanism. The underlying processes feature in the switching process depends on the jump diffusions. In this paper, conditions for recurrence and positive recurrence are derived. Ergodicity is examined in detail. Existence of invariant probability measures is proved. PubDate: 2017-12-26 DOI: 10.1007/s00245-017-9470-9

Authors:T. E. Duncan; B. Maslowski; B. Pasik-Duncan Abstract: A stochastic linear-quadratic control problem is formulated and solved for some stochastic equations in an infinite dimensional Hilbert space for both finite and infinite time horizons. The equations are bilinear in the state and the noise process where the noise is a scalar Gauss-Volterra process. The Gauss-Volterra noise processes are obtained from the integral of a Brownian motion with a suitable kernel function. These noise processes include fractional Brownian motions with the Hurst parameter \(H \in (\frac{1}{2},1)\) , Liouville fractional Brownian motions with \(H \in (\frac{1}{2},1)\) , and some multifractional Brownian motions. The family of admissible controls for the quadratic costs is a family of linear feedback controls. This restriction on the family of controls allows for a feasible implementation of the optimal controls. The bilinear equations have drift terms that are linear evolution operators. These equations can model stochastic partial differential equations of parabolic and hyperbolic types and two families of examples are given. PubDate: 2017-12-22 DOI: 10.1007/s00245-017-9468-3

Authors:Pablo Azcue; Nora Muler Abstract: This paper presents a continuous time cash management problem where the uncontrolled money stock follows a compound poisson process with two-sided jumps and negative drift. The main goal is to minimize the total expected discounted sum of the opportunity cost of holding cash and the adjustment cost coming from deposits and withdrawals. This adjustment cost can have a proportional and a fixed component. In this paper it is assumed that the controlled money stock cannot be negative. We show that the optimal value function is an a.e. strong solution of the corresponding Hamilton–Jacobi–Bellman equation and that it can be characterized as the largest a.e. strong supersolution. We prove that the optimal policy has an impulse multi-band structure with multiple trigger-target pairs. We present examples where the optimal value function is not differentiable and an example where the optimal impulse policy has more that one band. PubDate: 2017-12-21 DOI: 10.1007/s00245-017-9467-4

Authors:Nikolaos S. Papageorgiou; Vicenţiu D. Rădulescu; Dušan D. Repovš Abstract: We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values \(\lambda >0\) , the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter \(\lambda >0\) varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory. PubDate: 2017-12-13 DOI: 10.1007/s00245-017-9465-6

Authors:Exequiel Mallea-Zepeda; Elva Ortega-Torres; Élder J. Villamizar-Roa Abstract: We study an optimal boundary control problem for the two-dimensional stationary micropolar fluids system with variable density. We control the system by considering boundary controls, for the velocity vector and angular velocity of rotation of particles, on parts of the boundary of the flow domain. On the remaining part of the boundary, we consider mixed boundary conditions for the vector velocity (Dirichlet and Navier conditions) and Dirichlet boundary conditions for the angular velocity. We analyze the existence of a weak solution obtaining the fluid density as a scalar function in terms of the stream function. We prove the existence of an optimal solution and, by using the Lagrange multipliers theorem, we state first-order optimality conditions. We also derive, through a penalty method, some optimality conditions satisfied by the optimal controls. PubDate: 2017-12-13 DOI: 10.1007/s00245-017-9466-5