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:G. P. Panasenko; R. Stavre Abstract: A two-dimensional time dependent model of an interaction between a thin elastic plate and a Newtonian viscous fluid described by the non-steady Stokes equations is considered. It depends on a small parameter \(\varepsilon \) that is the ratio of the thicknesses of the plate and the fluid layer. The Young’s modulus of the plate and its density may be great or small parameters equal to some powers (positive or negative) of \(\varepsilon \) while the density and the viscosity of the fluid are supposed to be of order one. An asymptotic expansion is constructed and justified for various magnitudes of the rigidity and density of the plate. The limit problems are studied in all these cases. They are Stokes equations with some special coupled or uncoupled boundary conditions modeling the interaction with the plate. The estimates of the difference between the exact solution and a truncated asymptotic expansion are established. These estimates justify the asymptotic approximations. PubDate: 2018-02-24 DOI: 10.1007/s00245-018-9480-2

Authors:Charafeddine Mouzouni Abstract: We explore a mechanism of decision-making in mean field games with myopic players. At each instant, agents set a strategy which optimizes their expected future cost by assuming their environment as immutable. As the system evolves, the players observe the evolution of the system and adapt to their new environment without anticipating. With a specific cost structures, these models give rise to coupled systems of partial differential equations of quasi-stationary nature. We provide sufficient conditions for the existence and uniqueness of classical solutions for these systems, and give a rigorous derivation of these systems from N-players stochastic differential games models. Finally, we show that the population can self-organize and converge exponentially fast to the ergodic mean field games equilibrium, if the initial distribution is sufficiently close to it and the Hamiltonian is quadratic. PubDate: 2018-02-22 DOI: 10.1007/s00245-018-9484-y

Authors:Nacira Agram; Bernt Øksendal Abstract: The original version of this article unfortunately contained a few mistakes in Theorems and notation. The corrected information is given below. PubDate: 2018-02-21 DOI: 10.1007/s00245-018-9483-z

Authors:Xin Guo; Yi Zhang Abstract: We consider a piecewise deterministic Markov decision process, where the expected exponential utility of total (nonnegative) cost is to be minimized. The cost rate, transition rate and post-jump distributions are under control. The state space is Borel, and the transition and cost rates are locally integrable along the drift. Under natural conditions, we establish the optimality equation, justify the value iteration algorithm, and show the existence of a deterministic stationary optimal policy. Applied to special cases, the obtained results already significantly improve some existing results in the literature on finite horizon and infinite horizon discounted risk-sensitive continuous-time Markov decision processes. PubDate: 2018-02-17 DOI: 10.1007/s00245-018-9485-x

Authors:Anup Biswas; Subhamay Saha Abstract: Zero-sum games with risk-sensitive cost criterion are considered with underlying dynamics being given by controlled stochastic differential equations. Under the assumption of geometric stability on the dynamics, we completely characterize all possible saddle point strategies in the class of stationary Markov controls. In addition, we also establish existence-uniqueness result for the value function of the Hamilton–Jacobi–Isaacs equation. PubDate: 2018-02-15 DOI: 10.1007/s00245-018-9479-8

Authors:Ido Polak; Nicolas Privault Abstract: We construct Cournot games with limited demand, resulting into capped sales volumes according to the respective production shares of the players. We show that such games admit three distinct equilibrium regimes, including an intermediate regime that allows for a range of possible equilibria. When information on demand is modeled by a delayed diffusion process, we also show that this intermediate regime collapses to a single equilibrium while the other regimes approximate the deterministic setting as the delay tends to zero. Moreover, as the delay approaches zero, the unique equilibrium achieved in the stochastic case provides a way to select a natural equilibrium within the range observed in the no lag setting. Numerical illustrations are presented when demand is modeled by an Ornstein–Uhlenbeck process and price is an affine function of output. PubDate: 2018-02-15 DOI: 10.1007/s00245-018-9481-1

Authors:Nikolaos S. Papageorgiou; Calogero Vetro; Francesca Vetro Abstract: We consider a parametric Dirichlet problem driven by the sum of a p-Laplacian ( \(p>2\) ) and a Laplacian (a (p, 2)-equation). The reaction consists of an asymmetric \((p-1)\) -linear term which is resonant as \(x \rightarrow - \infty \) , plus a concave term. However, in this case the concave term enters with a negative sign. Using variational tools together with suitable truncation techniques and Morse theory (critical groups), we show that when the parameter is small the problem has at least three nontrivial smooth solutions. PubDate: 2018-02-14 DOI: 10.1007/s00245-018-9482-0

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