Authors:Francesca Biagini; Alessandro Gnoatto; Maximilian Härtel Pages: 405 - 441 Abstract: We develop the HJM framework for forward rates driven by affine processes on the state space of symmetric positive semidefinite matrices. In this setting we find an explicit representation for the long-term yield in terms of the model parameters. This generalises the results of El Karoui et al. (Rev Deriv Res 1(4):351–369, 1997) and Biagini and Härtel (Int J Theor Appl Financ 17(3):1–24, 2012), where the long-term yield is investigated under no-arbitrage assumptions in a HJM setting using Brownian motions and Lévy processes respectively. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9379-8 Issue No:Vol. 77, No. 3 (2018)

Authors:P. Tamilalagan; P. Balasubramaniam Pages: 443 - 462 Abstract: In this manuscript, we investigate the solvability and optimal controls for fractional stochastic differential equations driven by Poisson jumps in Hilbert space by using analytic resolvent operators. Sufficient conditions are derived to prove that the system has a unique mild solution by using the classical Banach contraction mapping principle. Then, the existence of optimal control for the corresponding Lagrange optimal control problem is investigated. Finally, the derived theoretical result is validated by an illustrative example. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9380-2 Issue No:Vol. 77, No. 3 (2018)

Authors:Gerd Wachsmuth Pages: 463 - 497 Abstract: We consider a set \(\mathcal {C}\) with pointwise constraints in a vector-valued Sobolev space. We characterize its tangent and normal cone. Under the additional assumption that the pointwise constraints are affine and satisfy the linear independence constraint qualification, we show that the set \(\mathcal {C}\) is polyhedric. The results are applied to the optimal control of a string in a polyhedral tube. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9381-1 Issue No:Vol. 77, No. 3 (2018)

Authors:Hong-Ming Yin; Wei Wei Pages: 499 - 513 Abstract: In this paper we study an optimization problem arising from a sterilization process for packaged foods by using a microwave heating method. The goal of the optimal control is to find the optimal frequency function such that the temperature profile at the final stage has a relative uniform distribution in the food product. The underlying state variables are electric, magnetic fields and temperature which satisfy the coupled nonlinear Maxwell’s system and a nonlinear heat equation. The control variable for the system is chosen to be the electric frequency function. We show that there exists an optimal frequency which minimizes the cost functional. Moreover, an optimality condition is also derived. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9382-0 Issue No:Vol. 77, No. 3 (2018)

Authors:Nguyen Dinh Tuan Pages: 515 - 539 Abstract: In this note, we develop first- and second-order necessary optimality conditions for local weak solutions in nonsmooth vector optimization problems subject to mixed constraints in infinite-dimensional settings. To this aim, we use some set-valued directional derivatives of the Hadamard type and tangent sets, and impose (first-order) Hadamard differentiability assumptions of the data at the point of consideration. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9383-z Issue No:Vol. 77, No. 3 (2018)

Authors:Tomasz Klimsiak; Andrzej Rozkosz; Leszek Słomiński Pages: 541 - 566 Abstract: We consider a system of seminlinear parabolic variational inequalities with time-dependent convex obstacles. We prove the existence and uniqueness of its solution. We also provide a stochastic representation of the solution and show that it can be approximated by the penalization method. Our proofs are based upon probabilistic methods from the theory of Markov processes and the theory of backward stochastic differential equations. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9387-8 Issue No:Vol. 77, No. 3 (2018)

Authors:Natalie Attard Pages: 567 - 597 Abstract: Two players are observing a right-continuous and quasi-left-continuous strong Markov process X. We study the optimal stopping problem \(V^{1}_{\sigma }(x)=\sup _{\tau } \mathsf {M}_{x}^{1}(\tau ,\sigma )\) for a given stopping time \(\sigma \) (resp. \(V^{2}_{\tau }(x)=\sup _{\sigma } \mathsf {M}_{x}^{2}(\tau ,\sigma )\) for given \(\tau \) ) where \(\mathsf {M}_{x}^{1}(\tau ,\sigma ) = \mathsf {E}_{x} [G_{1}(X_{\tau })I(\tau \le \sigma ) + H_{1}(X_{\sigma })I(\sigma < \tau )]\) with \(G_1,H_1\) being continuous functions satisfying some mild integrability conditions (resp. \(\mathsf {M}_{x}^{2}(\tau ,\sigma ) = \mathsf {E}_{x} [G_{2}(X_{\sigma })I(\sigma < \tau ) + H_{2}(X_{\tau })I(\tau \le \sigma )]\) with \(G_2,H_2\) being continuous functions satisfying some mild integrability conditions). We show that if \(\sigma = \sigma _{D_{2}} = \inf \{t \ge 0: X_t \in D_2\}\) (resp. \(\tau = \tau _{D_{1}} = \inf \{t \ge 0: X_t \in D_1\}\) ) where \(D_{2}\) (resp. \(D_1\) ) has a regular boundary, then \(V^{1}_{\sigma _{D_{2}}}\) (resp. \(V^{2}_{\tau _{D_{1}}}\) ) is finely continuous. If \(D_{2}\) (resp. \(D_1\) ) is also (finely) closed then \(\tau _*^{\sigma _{D_2}} = \inf \{t \ge 0: X_{t} \in D_{1}^{\sigma _{D_{2}}}\}\) (resp. \(\sigma _{*}^{\tau _{D_1}} = \inf \{t \ge 0: X_{t} \in D_{2}^{\tau _{D_{1}}}\}\) ) where \(D_{1}^{\sigma _{D_{2}}} = \{V^{1}_{\sigma _{D_{2}}} = G_{1}\}\) (resp. \(D_{2}^{\tau _{D_{1}}} = \{V^{2}_{\tau _{D_{1}}} = G_{2}\}\) ) is optimal for player one (resp. player two). We then derive a partial superharmonic characterisation for \(V^{1}_{\sigma _{D_2}}\) (resp. \(V^{2}_{\tau _{D_1}}\) ) which can be exploited in examples to construct a pair of first entry times that is a Nash equilibrium. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9388-7 Issue No:Vol. 77, No. 3 (2018)

Authors:Arkadiusz Misztela Pages: 599 - 611 Abstract: An example of a nonunique solution of the Cauchy problem of Hamilton–Jacobi–Bellman (HJB) equation with surprisingly regular Hamiltonian is presented. The Hamiltonian H(t, x, p) is locally Lipschitz continuous with respect to all variables, convex in p and with linear growth with respect to p and x. The HJB equation possesses two distinct lower semicontinuous solutions with the same final conditions; moreover, one of them is the value function of the corresponding Bolza problem. The definition of lower semicontinuous solution was proposed by Frankowska (SIAM J. Control Optim. 31:257–272, 1993) and Barron and Jensen (Commun. Partial Differ. Equ. 15(12):1713–1742, 1990). Using the example an analysis and comparison of assumptions in some uniqueness results in HJB equations is provided. PubDate: 2018-06-01 DOI: 10.1007/s00245-016-9393-x Issue No:Vol. 77, No. 3 (2018)

Authors:Gyoocheol Shim; Jung Lim Koo; Yong Hyun Shin Pages: 197 - 228 Abstract: We study an optimal job-switching and consumption/investment problem of an infinitely-lived economic agent who exhibits constant relative risk aversion. We consider two kinds of jobs, one of which allows the agent to receive higher income but makes him suffer higher level of utility loss than the other. The job-switching opportunities are reversible in the sense that one can move from the current job to the other at any time. We provide the closed form solution for the optimal job-switching and consumption/investment policies by using the dynamic programming approach, and show various properties of the solution. We compare the optimal consumption/investment policies to those without job-switching opportunities. As a special case of our problem, we also compare the solution in the case where the agent has a reversible retirement option with that in the case where he has an irreversible retirement option. PubDate: 2018-04-01 DOI: 10.1007/s00245-016-9371-3 Issue No:Vol. 77, No. 2 (2018)

Authors:Wenzhao Zhang Pages: 275 - 296 Abstract: In this paper, we consider the continuous-time nonzero-sum constrained stochastic games with the discounted cost criteria. The state space is denumerable and the action space of each player is a general Polish space, while the transition rates and cost functions are allowed to be unbounded from below and from above. The strategies for each player may be history-dependent and randomized. Models with these features seemingly have not been handled in the previous literature. By constructing a sequence of continuous-time finite-state game models to approximate the original denumerable-state game model, we prove the existence of constrained Nash equilibria for the constrained games with denumerable states. PubDate: 2018-04-01 DOI: 10.1007/s00245-016-9374-0 Issue No:Vol. 77, No. 2 (2018)

Authors:Ahmed A. Keddi; Tijani A. Apalara; Salim A. Messaoudi Pages: 315 - 341 Abstract: In this paper we study the well-posedness and the asymptotic stability of a one-dimensional thermoelastic Bresse system, where the heat conduction is given by Cattaneo’s law effective in the shear angle displacements. We establish the well-posedness of the system and prove that the system is exponentially stable depending on the parameters of the system. Furthermore, we show that in general, the system is not exponential stable. In this regards, we prove that the solution decays polynomially. PubDate: 2018-04-01 DOI: 10.1007/s00245-016-9376-y Issue No:Vol. 77, No. 2 (2018)

Abstract: A discrete-time hybrid control model with Borel state and action spaces is introduced. In this type of models, the dynamic of the system is composed by two sub-dynamics affecting the evolution of the state; one is of a standard-type that runs almost every time and another is of a special-type that is active under special circumstances. The controller is able to use two different type of actions, each of them is applied to each of the two sub-dynamics, and the activations of these sub-dynamics are possible according to an activation rule that can be handled by the controller. The aim for the controller is to find a control policy, containing a mix of actions (of either standard- or special-type), with the purpose of minimizing an infinite-horizon discounted cost criterion whose discount factor is dependent on the state-action history and may be equal to one at some stages. Two different sets of conditions are proposed to guarantee (i) the finiteness of the cost criterion, (ii) the characterization of the optimal value function and (iii) the existence of optimal control policies; to do so, we employ the dynamic programming approach. A useful characterization that signalizes the accurate times between changes of sub-dynamics in terms of the so-named contact set is also provided. Finally, we introduce two examples that illustrate our results and also show that control models such as discrete-time impulse control models and discrete-time switching control models become special cases of our present hybrid model. PubDate: 2018-05-30 DOI: 10.1007/s00245-018-9503-z

Authors:Dang H. Nguyen; George Yin Abstract: This paper develops near-optimal sustainable harvesting strategies for the predator in a predator-prey system. The objective function is of long-run average per unit time type in the path-wise sense. To date, ecological systems under environmental noise are usually modeled as stochastic differential equations driven by a Brownian motion. Recognizing that the formulation using a Brownian motion is only an idealization, in this paper, it is assumed that the environment is subject to disturbances characterized by a jump process with rapid jump rates. Under broad conditions, it is shown that the systems under consideration can be approximated by a controlled diffusion system. Based on the limit diffusion system, control policies of the original systems are constructed. Such an approach enables us to develop sustainable harvesting policies leading to near optimality. To treat the underlying problems, one of the main difficulties is due to the long-run average objective function. This in turn, requires the handling of a number of issues related to ergodicity. New approaches are developed to obtain the tightness of the underlying processes based on the population dynamic systems. PubDate: 2018-05-28 DOI: 10.1007/s00245-018-9504-y

Authors:Patricio Guzmán Abstract: In this paper we prove the local exact controllability to the trajectories of the Cahn–Hilliard equation, which is a nonlinear fourth-order parabolic equation, by means of a control supported on an interior open interval. To prove this result we derive a Carleman estimate that allows us to conclude, thanks to a duality argument, the null controllability of the linearized equation around a given solution. Then, we apply a local inversion theorem to extend the control result to the nonlinear equation. PubDate: 2018-05-24 DOI: 10.1007/s00245-018-9500-2

Authors:Fabián Flores-Bazán; Gabriel Cárcamo; Stephanie Caro Abstract: Many formulations of quadratic allocation problems, portfolio optimization problems, the maximum weight clique problem, among others, take the form as the well-known standard quadratic optimization problem, which consists in minimizing a homogeneous quadratic function on the usual simplex in the non negative orthant. We propose to analyze the same problem when the simplex is substituted by a convex and compact base of any pointed, closed, convex cone (so, the cone of positive semidefinite matrices or the cone of copositive matrices are particular instances). Three main duals (for which a semi-infinite formulation of the primal problem is required) are associated, and we establish some characterizations of strong duality with respect to each of the three duals in terms of copositivity of the Hessian of the quadratic objective function on suitable cones. Such a problem reveals a hidden convexity and the validity of S-lemma. In case of bidimensional quadratic optimization problems, copositivity of the Hessian of the objective function is characterized, and the case when every local solution is global. PubDate: 2018-05-23 DOI: 10.1007/s00245-018-9502-0

Authors:Bruno Després; Emmanuel Trélat Abstract: The polynomial approximation of the hypograph of a function can be recast as a two-sided space–time \(L^1\) minimization problem, related with the Lasso problem. In this paper, we solve this problem within an optimal control framework, which in turn provides insights to develop efficient computation algorithms. We prove existence and uniqueness of the optimal solution and we characterize it by means of the Pontryagin maximum principle. We establish convergence properties as the polynomial degree tends to \(+\infty \) . We provide numerical simulations to illustrate our results. In passing, we study the geometry and, in particular, the extremal points of the convex set of polynomials of one variable having two-sided constraints on an interval. PubDate: 2018-05-19 DOI: 10.1007/s00245-018-9501-1

Authors:Rajesh Mahadevan; A. K. Nandakumaran; Ravi Prakash Abstract: While considering boundary value problems with oscillating coefficients or in oscillating domains, it is important to associate an asymptotic model which accounts for the average behaviour. This model permits to obtain the average behaviour without costly numerical computations implied by the fine scale of oscillations in the original model. The asymptotic analysis of boundary value problems in oscillating domains has been extensively studied and involves some key issues such as: finding uniformly bounded extension operators for function spaces on oscillating domains, the choice of suitable sequences of test functions for passing to the limit in the variational formulation of the model equations etc. In this article, we study a boundary value problem for the Laplacian in a domain, a part of whose boundary is highly oscillating (periodically), involving non-homogeneous non-linear Neumann or Robin boundary condition on the periodically oscillating boundary. The non-homogeneous Neumann condition or the Robin boundary condition on the oscillating boundary adds a further difficulty to the limit analysis since it involves taking the limits of surface integrals where the surface changes with respect to the parameter. Previously, some model problems have been studied successfully in Gaudiello (Ricerche Mat 43(2):239–292, 1994) and in Mel’nyk (Math Methods Appl Sci 31(9):1005–1027, 2008) by converting the surface term into a volume term using auxiliary boundary value problems. Some problems of this nature have also been studied using an extension of the notion of two-scale convergence (Allaire et al. in Proceedings of the international conference on mathematical modelling of flow through porous media, Singapore, 15–25, 1996, Neuss-Radu in C R Acad Sci Paris Sr I Math 322:899–904, 1996). In this article, we use a different approach to handle of such terms based on the unfolding operator. PubDate: 2018-05-10 DOI: 10.1007/s00245-018-9499-4

Authors:Abderrahim Hantoute; Bao Tran Nguyen Abstract: We characterize in Hilbert spaces the boundary of the values of maximal monotone operators, by means only of the values at nearby points, which are close enough to the reference point but distinct of it. This allows to write the values of such operators using finite convex combinations of the values at at most two nearby points. We also provide similar characterizations for the normal cone to prox-regular sets. PubDate: 2018-05-04 DOI: 10.1007/s00245-018-9498-5

Authors:Archil Gulisashvili; Frederi Viens; Xin Zhang Abstract: We consider the class of Gaussian self-similar stochastic volatility models, and characterize the small-time (near-maturity) asymptotic behavior of the corresponding asset price density, the call and put pricing functions, and the implied volatility. Away from the money, we express the asymptotics explicitly using the volatility process’ self-similarity parameter H, and its Karhunen–Loève characteristics. Several model-free estimators for H result. At the money, a separate study is required: the asymptotics for small time depend instead on the integrated variance’s moments of orders \(\frac{1}{2}\) and \( \frac{3}{2}\) , and the estimator for H sees an affine adjustment, while remaining model-free. PubDate: 2018-04-30 DOI: 10.1007/s00245-018-9497-6

Authors:João Pedro Vidal Nunes; José Carlos Dias; João Pedro Ruas Abstract: This paper proves the existence, uniqueness, monotonicity and continuity of the early exercise boundary attached to American-style standard options under the jump to default extended constant elasticity of variance model of Carr and Linetsky (Financ Stoch 10(3):303–330, 2006). PubDate: 2018-04-23 DOI: 10.1007/s00245-018-9496-7