ALBERT

Description:    An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option (both call and put) in discrete time using linear programming duality. Under the assumption that the underlying stock price follows a discrete time and discrete state Markov process, namely a geometric random walk, we formulate the pricing problem as an infinite dimensional linear programming (LP) problem using the excessive-majorant property of the value function. This formulation allows us to solve complementary slackness conditions in closed-form, revealing an optimal stopping strategy which highlights the set of stock-prices where the option should be exercised. The analysis for the call option reveals that such a critical value exists only in some cases, depending on a combination of state-transition probabilities and the economic discount factor (i.e., the prevailing interest rate) whereas it ceases to be an issue for the put. Content Type Journal Article Pages 1-26 DOI 10.1007/s00245-012-9182-0 Authors Robert J. Vanderbei, Department of Operations Research and Financial Engineering, Princeton University, Princeton, USA Mustafa Ç. Pınar, Department of Industrial Engineering, Bilkent University, Ankara, Turkey Efe B. Bozkaya, Faculty of Administrative Sciences, Sabancı University, Istambul, Turkey Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    This paper deals with existence and uniqueness of a solution in viscosity sense, for a system of m variational partial differential inequalities with inter-connected obstacles. A particular case is the Hamilton-Jacobi-Bellmann system of the Markovian stochastic optimal m -states switching problem. The switching cost functions depend on ( t , x ). The main tool is the notion of systems of reflected backward stochastic differential equations with oblique reflection. Content Type Journal Article Pages 1-34 DOI 10.1007/s00245-012-9184-y Authors S. Hamadène, LMM, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France M. A. Morlais, LMM, Université du Maine, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    We construct a stochastic maximum principle (SMP) which provides necessary conditions for the existence of Nash equilibria in a certain form of N -agent stochastic differential game (SDG) of a mean-field type. The information structure considered for the SDG is of a possible asymmetric and partial type. To prove our SMP we take an approach based on spike-variations and adjoint representation techniques, analogous to that of S. Peng (SIAM J. Control Optim. 28(4):966–979, 1990 ) in the optimal stochastic control context. In our proof we apply adjoint representation procedures at three points. The first-order adjoint processes are defined as solutions to certain mean-field backward stochastic differential equations, and second-order adjoint processes of a first type are defined as solutions to certain backward stochastic differential equations. Second-order adjoint processes of a second type are defined as solutions of certain backward stochastic equations of a type that we introduce in this paper, and which we term conditional mean-field backward stochastic differential equations. From the resulting representations, we show that the terms relating to these second-order adjoint processes of the second type are of an order such that they do not appear in our final SMP equations. A comparable situation exists in an article by R. Buckdahn, B. Djehiche, and J. Li (Appl. Math. Optim. 64(2):197–216, 2011 ) that constructs a SMP for a mean-field type optimal stochastic control problem; however, the approach we take of using these second-order adjoint processes of a second type to deal with the type of terms that we refer to as the second form of quadratic-type terms represents an alternative to a development, to our setting, of the approach used in their article for their analogous type of term. Content Type Journal Article Pages 1-40 DOI 10.1007/s00245-012-9177-x Authors John Joseph Absalom Hosking, Project-Team MATHFI, Inria Paris-Rocquencourt, Domaine de Voluceau, Rocquencourt, B.P. 105, 78153 Le Chesnay Cedex, France Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    In this paper the problem of accomplishing multiple objectives by a number of agents represented as dynamic systems is considered. Each agent is assumed to have a goal which is to accomplish one or more objectives where each objective is mathematically formulated using an appropriate objective function. Sufficient conditions for accomplishing objectives are derived using particular convergent approximations of minimum and maximum functions depending on the formulation of the goals and objectives. These approximations are differentiable functions and they monotonically converge to the corresponding minimum or maximum function. Finally, an illustrative pursuit-evasion game example with two evaders and two pursuers is provided. Content Type Journal Article Pages 1-19 DOI 10.1007/s00245-012-9179-8 Authors Dušan M. Stipanović, Coordinated Science Laboratory, Department of Industrial and Enterprise Systems Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA Claire J. Tomlin, Department of Electrical Engineering and Computer Science, University of California at Berkeley, Berkeley, USA George Leitmann, College of Engineering, University of California at Berkeley, Berkeley, USA Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    We present a new algorithm to compute the Snell envelope in the specific case where the criteria to optimize is associated with a small probability or a rare event. This new approach combines the Stochastic Mesh approach of Broadie and Glasserman with a particle approximation scheme based on a specific change of measure designed to concentrate the computational effort in regions pointed out by the criteria. The theoretical analysis of this new algorithm provides non asymptotic convergence estimates. Finally, the numerical tests confirm the practical interest of this approach. Content Type Journal Article Pages 1-22 DOI 10.1007/s00245-012-9173-1 Authors Pierre Del Moral, Centre INRIA Bordeaux et Sud-Ouest & Institut de Mathématiques de Bordeaux, Université de Bordeaux I, 351 cours de la Libération, 33405 Talence cedex, France Peng Hu, Centre INRIA Bordeaux et Sud-Ouest & Institut de Mathématiques de Bordeaux, Université de Bordeaux I, 351 cours de la Libération, 33405 Talence cedex, France Nadia Oudjane, EDF R&D Clamart, Clamart, France Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    We discuss several variants of the TV-regularization model used in image recovery. The proposed alternatives are either of nearly linear growth or even of linear growth, but with some weak ellipticity properties. The main feature of the paper is the investigation of the analytic properties of the corresponding solutions. Content Type Journal Article Pages 1-31 DOI 10.1007/s00245-012-9174-0 Authors Michael Bildhauer, Department of Mathematics, Saarland University, P.O. Box 15 11 50, 66041 Saarbrücken, Germany Martin Fuchs, Department of Mathematics, Saarland University, P.O. Box 15 11 50, 66041 Saarbrücken, Germany Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    We study optimal control of Markov processes with age-dependent transition rates. The control policy is chosen continuously over time based on the state of the process and its age. We study infinite horizon discounted cost and infinite horizon average cost problems. Our approach is via the construction of an equivalent semi-Markov decision process. We characterise the value function and optimal controls for both discounted and average cost cases. Content Type Journal Article Pages 1-15 DOI 10.1007/s00245-012-9171-3 Authors Mrinal K. Ghosh, Department of Mathematics, Indian Institute of Science, Bangalore-12, India Subhamay Saha, Department of Mathematics, Indian Institute of Science, Bangalore-12, India Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    In the present paper we establish results concerning the decay of the energy related to the damped Korteweg–de Vries equation posed on infinite domains. We prove the exponential decay rates of the energy when a initial value problem and a localized dissipative mechanism are in place. If this mechanism is effective in the whole line, we get a similar result in H k -level, k ∈ℕ. In addition, the decay of the energy regarding a initial boundary value problem posed on the right half-line, is obtained considering convenient a smallness condition on the initial data but a more general dissipative effect. Content Type Journal Article Pages 1-31 DOI 10.1007/s00245-011-9156-7 Authors Marcelo M. Cavalcanti, Department of Mathematics, State University of Maringá, Avenida Colombo, 5790, CEP 87020-900 Maringá, PR, Brazil Valéria N. Domingos Cavalcanti, Department of Mathematics, State University of Maringá, Avenida Colombo, 5790, CEP 87020-900 Maringá, PR, Brazil Andrei Faminskii, Faculty of Science, Peoples’ Friendship University of Russia, Miklukho-Maklai str. 6, Moscow, 117198 Russia Fábio Natali, Department of Mathematics, State University of Maringá, Avenida Colombo, 5790, CEP 87020-900 Maringá, PR, Brazil Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    This paper is devoted to the study of numerical approximation schemes for a class of parabolic equations on (0,1) perturbed by a non-linear rough signal. It is the continuation of Deya (Electron. J. Probab. 16:1489–1518, 2011 ) and Deya et al. (Probab. Theory Relat. Fields, to appear), where the existence and uniqueness of a solution has been established. The approach combines rough paths methods with standard considerations on discretizing stochastic PDEs. The results apply to a geometric 2-rough path, which covers the case of the multidimensional fractional Brownian motion with Hurst index H 〉1/3. Content Type Journal Article Pages 1-40 DOI 10.1007/s00245-011-9157-6 Authors Aurélien Deya, Institut Élie Cartan Nancy, Université de Nancy 1, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616

Description:    We study the local risk-minimization approach for defaultable claims with random recovery at default time , seen as payment streams on the random interval 〚0, τ ∧ T 〛, where T denotes the fixed time-horizon. We find the pseudo-locally risk-minimizing strategy in the case when the agent information takes into account the possibility of a default event (local risk-minimization with G -strategies) and we provide an application in the case of a corporate bond. We also discuss the problem of finding a pseudo-locally risk-minimizing strategy if we suppose the agent obtains her information only by observing the non-defaultable assets. Content Type Journal Article Pages 1-22 DOI 10.1007/s00245-011-9155-8 Authors Francesca Biagini, Department of Mathematics, LMU, Theresienstraße 39, 80333 Munich, Germany Alessandra Cretarola, Dipartimento di Matematica e Informatica, Università degli Studi di Perugia, via Vanvitelli 1, 06123 Perugia, Italy Journal Applied Mathematics & Optimization Online ISSN 1432-0606 Print ISSN 0095-4616