A Regularized Hybrid Steepest Descent Method for Variational Inclusions

This article is concerned with a generalization of the hybrid steepest descent method from variational inequalities to the multivalued case. This will be reached by replacing the multivalued operator by its Yosida approximate, which is always Lipschitz continuous. It is worth mentioning that the hybrid steepest descent method is an algorithmic solution to variational inequality problems over the fixed point set of certain nonexpansive mappings and has remarkable applicability to the constrained nonlinear inverse problems like image recovery and MIMO communication systems (see, e.g., [9 I. Yamada , M. Yukawa , and M. Yamagishi ( 2011 ). Minimizing the moreau envelope of nonsmooth convex functions over the fixed point set of certain quasi-nonexpansive mappings . In Fixed Point Algorithms for Inverse Problems in Science and Engineering ( H.H. Bauschke , R. Burachik , P.L. Combettes , V. Elser , D.R. Luke , and H. Wolkowicz , eds.), Springer-Verlag , New York , Chapter 17 , pp. 343 – 388 . [Google Scholar], 10 I. Yamada , Ogura , and N. Shirakawa ( 2002 ). A numerically robust hybrid steepest descent method for the convexly constrained generalized inverse problems . In Inverse Problems, Image Analysis and Medical Imaging. Contemporary Mathematics ( Z. Nashed and O. Scherzer , eds.), American Mathematical Society , Providence , RI , Vol. 313 , pp. 269 – 305 . [Google Scholar]]).     

A New Approach to Pricing Double-Barrier Options with Arbitrary Payoffs and Exponential Boundaries

Abstract

We consider in this article the arbitrage free pricing of double knock-out barrier options with payoffs that are arbitrary functions of the underlying asset, where we allow exponentially time-varying barrier levels in an otherwise standard Black–Scholes model. Our approach, reminiscent of the method of images of electromagnetics, considerably simplifies the derivation of analytical formulae for this class of exotics by reducing the pricing of any double-barrier problem to that of pricing a related European option. We illustrate the method by reproducing the well-known formulae of Kunitomo and Ikeda (1992 Kunitomo, N. and Ikeda, M. 1992. Pricing options with curved boundaries. Mathematical Finance, 2: 276–298.  [Google Scholar]) for the standard knock-out double-barrier call and put options. We give an explanation for the rapid rate of convergence of the doubly infinite sums for affine payoffs in the stock price, as encountered in the pricing of double-barrier call and put options first observed by Kunitomo and Ikeda (1992 Kunitomo, N. and Ikeda, M. 1992. Pricing options with curved boundaries. Mathematical Finance, 2: 276–298.  [Google Scholar]).     

Stochastic Expansion for the Pricing of Call Options with Discrete Dividends

Abstract

In the context of an asset paying affine-type discrete dividends, we present closed analytical approximations for the pricing of European vanilla options in the Black–Scholes model with time-dependent parameters. They are obtained using a stochastic Taylor expansion around a shifted lognormal proxy model. The final formulae are, respectively, first-, second- and third- order approximations w.r.t. the fixed part of the dividends. Using Cameron–Martin transformations, we provide explicit representations of the correction terms as Greeks in the Black–Scholes model. The use of Malliavin calculus enables us to provide tight error estimates for our approximations. Numerical experiments show that this approach yields very accurate results, in particular compared with known approximations of Bos, Gairat and Shepeleva (2003 Bos, R., Gairat, A. and Shepeleva, D. 2003. Dealing with discrete dividends. Risk Magazine, 16: 109–112.  [Google Scholar], Dealing with discrete dividends, Risk Magazine, 16, pp. 109–112) and Veiga and Wystup (2009 Veiga, C. and Wystup, U. 2009. Closed formula for options with discrete dividends and its derivatives. Applied Mathematical Finance, 16(6): 517–531. [Taylor & Francis Online] , [Google Scholar], Closed formula for option with discrete dividends and its derivatives, Applied Mathematical Finance, 16(6), pp. 517–531), and quicker than the iterated integration procedure of Haug, Haug and Lewis (2003 Haug, E. G., Haug, J. and Lewis, A. 2003. Back to basics: a new approach to the discrete dividend problem. Wilmott Magazine, : 37–47.  [Google Scholar], Back to basics: a new approach to the discrete dividend problem, Wilmott Magazine, pp. 37–47) or than the binomial tree method of Vellekoop and Nieuwenhuis (2006 Vellekoop, M. and Nieuwenhuis, J. 2006. Effcient pricing of derivatives on assets with discrete dividends. Applied Mathematical Finance, 13(3): 265–284. [Taylor & Francis Online] , [Google Scholar], Efficient pricing of derivatives on assets with discrete dividends, Applied Mathematical Finance, 13(3), pp. 265–284).     

Non–existence of theta–shaped self–similarly shrinking networks moving by curvature

We prove that there are no networks homeomorphic to the Greek “Theta” letter (a double cell) embedded in the plane with two triple junctions with angles of 120 degrees, such that under the motion by curvature they are self–similarly shrinking.

This fact completes the classification of the self–similarly shrinking networks in the plane with at most two triple junctions, see [5 Chen, X., Guo, J.-S. (2007). Self-similar solutions of a 2-D multiple-phase curvature flow. Phys. D. 229(1):22–34.[Crossref], [Web of Science ®] , [Google Scholar], 10 Hättenschweiler, J. (2007). Mean curvature flow of networks with triple junctions in the plane. Master’s thesis. ETH Zürich. [Google Scholar], 25 Schnürer, O. C., Azouani, A., Georgi, M., Hell, J., Nihar, J., Koeller, A., Marxen, T., Ritthaler, S., Sáez, M., Schulze, F., Smith, B. (2011). Evolution of convex lens–shaped networks under the curve shortening flow. Trans. Am. Math. Soc. 363(5):2265–2294.[Crossref], [Web of Science ®] , [Google Scholar], 2 Baldi, P., Haus, E., Mantegazza, C. (2016). Networks self-similarly moving by curvature with two triple junctions. Networks self-similarly moving by curvature with two triple junctions. 28(2017):323–338. [Google Scholar]].     

Unifying strongly clean power series rings

It is unknown whether a power series ring over a strongly clean ring is, itself, always strongly clean. Although a number of authors have shown that the above statement is true in certain special cases, the problem remains open, in general. In this article, we look at a class of strongly clean rings, which we call the optimally clean rings, over which power series are strongly clean. This condition is motivated by work in [10 Diesl, A. J., Dorsey, T. J., Garg, S., Khurana, D. (2012). A note on completeness and strongly clean rings, preprint. [Google Scholar]] and [11 Diesl, A. J., Dorsey, T. J., Iberkleid, W., LaFuente-Rodriguez, R., McGovern, W (2013). Strongly clean triangular matrices over abelian rings, preprint. [Google Scholar]]. We explore the properties of optimally clean rings and provide many examples, highlighting the role that this new class of rings plays in investigating the question of strongly clean power series.     

Larson–Sweedler Theorem and Some Properties of Discrete Type in (G-Cograded) Multiplier Hopf Algebras

We extend the Larson–Sweedler theorem to group-cograded multiplier Hopf algebras introduced in Abd El-hafez et al. (2004 Abd El-hafez , A. T. , Delvaux , L. , Van Daele , A. ( 2004 ). Group-cograded multiplier Hopf (?-)algebra. Math. QA/0404026 . To appear in Algebras and Representation Theory . [CSA]  [Google Scholar]), by showing that a group-cograded multiplier bialgebra with finite-dimensional unital components is a group-cograded multiplier Hopf algebra if and only if it possesses a nondegenerate left cointegral. We also generalize the theory of multiplier Hopf algebras of discrete type in Van Daele and Zhang (1999 Van Daele , A. , Zhang , Y. ( 1999 ). Multiplier Hopf algebras of discrete type . J. Algebra 214 : 400 – 417 . [CSA] [CROSSREF]  [Google Scholar]) to group-cograded multiplier Hopf algebras. Our results are applicable to Hopf group-coalgebras in the sense of Turaev (2000 Turaev , V. G. ( 2000 ). Homotopy field theory in dimension 3 and crossed group-categories . Preprint GT/0005291. [CSA]  [Google Scholar]). Finally, we study regular multiplier Hopf algebras of η -discrete type.     

Characterizing the Development of Specialized Mathematical Content Knowledge for Teaching in Algebraic Reasoning and Number Theory

This article characterizes the development of a deep and connected body of mathematical knowledge categorized by Ball and Bass' (2003b) Ball, D. L. and Bass, H. Toward a practice-based theory of mathematical knowledge for teaching. Proceedings of the 2002 Annual Meeting of the Canadian Mathematics Education Study Group. Edmonton, AB, Canada. Edited by: Davis, B. and Simmt, E. pp.3–14. CMESG/GCEDM.  [Google Scholar] model of Mathematical Knowledge for Teaching (MKT), as Specialized Content Knowledge for Teaching (SCK) in algebraic reasoning and number sense. The research employed multiple cases across three years from two content courses designed for elementary and middle-level mathematics specialists. Qualitative data were collected and a grounded theory approach to data analysis was employed. The resulting framework characterizes developmental levels of deep and connected mathematical content knowledge for teaching algebraic reasoning and number theory content. The framework consists of four intertwined components related to a teacher's ability to (1) solve problems and justify his/her reasoning, (2) use multiple representations, (3) recognize, use, and generalize conceptually similar tasks, and (4) pose problems. Implications for mathematics teacher education programs are discussed as well as directions for further research.     

The Cauchy Problem for the Klein–Gordon–Schrödinger System in Low Dimensions Below the Energy Space

ABSTRACT

We show that the Klein–Gordon–Schrödinger system in one, two, and three dimensions has a global solution below the energy space. The proof uses the I-method recently introduced by Colliander et al. (2001 Colliander , J. , Keel , M. , Staffilani , G. , Takaoka , H. , Tao , T. ( 2001 ). Global well-posedness for Schrödinger equations with derivative . SIAM J. Math. Anal. 33 ( 3 ): 649 – 669 . [CROSSREF]  [Google Scholar]) and mixed type Strichartz estimates for the solutions of Schrödinger and Klein–Gordon equations, respectively.     

Mass-Selection in Alignment Models with Non-Deterministic Effects

In this paper we consider a kinetic model for alignment of cells or filaments with probabilistic turning. For this equation existence of solutions is known, see [6 Kang , K. , Perthame , B. , Stevens , A. , Velazquez , J. J. L. ( 2009 ). An integro- differential equation model for alignment and orientational aggregation . J. Diff Equations 246 : 1387 – 1421 . [Google Scholar]]. To understand its qualitative behavior, especially with respect to the selection of orientations and mass distributions for long times, the model is approximated by a diffusion equation in the limit of small deviations of the interactions between the cell bundles. For this new equation existence of steady states is shown. In contrast to the kinetic equation discussed in [6 Kang , K. , Perthame , B. , Stevens , A. , Velazquez , J. J. L. ( 2009 ). An integro- differential equation model for alignment and orientational aggregation . J. Diff Equations 246 : 1387 – 1421 . [Google Scholar]] with deterministic turning, where local stability of two opposite orientations was shown but no selection of mass could be observed, for the new approximating problem with probabilistic turning additionally mass selection takes place. In the limit of small diffusion, steady states can only be constructed, if the aligning masses are either equal or the total mass is concentrated in one direction. By numerical simulations we tested stability of these steady states and for situations with 4 symmetrically placed smooth distributions of alignment. Convergence of the numerical code was proved. The simulations suggest, that only the 2- and the 1-peak steady states can be stable, whereas the 4 peak steady state is always unstable. We conjecture that the noise in the system is responsible for this final selection of masses. There exist other steady states with an arbitrary number of aligned bundles of cells or filaments, but we suspect that, as numerically shown for the 4 peak case, these multi-peak states are all unstable.     

Global Existence for a Kinetic Model of Chemotaxis via Dispersion and Strichartz Estimates

We investigate further the existence of solutions to kinetic models of chemotaxis. These are nonlinear transport-scattering equations with a quadratic nonlinearity which have been used to describe the motion of bacteria since the 80's when experimental observations have shown they move by a series of ‘run and tumble’. The existence of solutions has been obtained in several papers Chalub et al. (2004 Chalub , F. A. C. C. , Markowich , P. A. , Perthame , B. , Schmeiser , C. ( 2004 ). Kinetic models for chemotaxis and their drift-diffusion limits . Monatsh. Math. 142 : 123 – 141 .[Crossref], [Web of Science ®] , [Google Scholar]), Hwang et al. (2005a Hwang , H. J. , Kang , K. , Stevens , A. ( 2005a ). Global solutions of nonlinear transport equations for chemosensitive movement . SIAM J. Math. Anal. 36 ( 4 ): 1177 – 1199 . [Google Scholar] b Hwang , H. J. , Kang , K. , Stevens , A. ( 2005b ). Drift-diffusion limits of kinetic models for chemotaxis: a generalization . Discrete Contin. Dyn. Syst. Ser. B 5 ( 2 ): 319 – 334 . [Google Scholar]) using direct and strong dispersive effects.

Here, we use the weak dispersion estimates of Castella and Perthame (1996 Castella , F. , Perthame , B. ( 1996 ). Estimations de Strichartz pour les équations de transport cinétique. [Strichartz’ estimates for kinetic transport equations.] C. R. Acad. Sci. Paris Sér. I 322 ( 6 ): 535 – 540 . [Google Scholar]) to prove global existence in various situations depending on the turning kernel. In the most difficult cases, where both the velocities before and after tumbling appear, with the known methods, only Strichartz estimates can give a result, with a smallness assumption.     

A Note on the Rate of Complete Convergence for Weighted Sums of Arrays of Banach Space Valued Random Elements

A rate of complete convergence for weighted sums of arrays of rowwise independent Banach space valued random elements was obtained by Ahmed et al. [1 Ahmed , S.E. , Giuliano Antonini , R. , and Volodin , A. 2002 . On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes . Statist. Probab. Lett. 58 : 185 – 194 . [Google Scholar]]. Recently, Sung and Volodin [2 Sung , S.H. , and Volodin , A.I. 2006. On the rate of complete convergence for weighted sums of arrays of random elements. J. Korean Math. Soc. 43:815–828.[Crossref], [Web of Science ®] , [Google Scholar]], Chen et al. [3 Chen , P. , Sung , S.H. , and Volodin , A.I. 2006 . Rate of complete convergence for arrays of Banach space valued random elements . Siberian Adv. Math. 16 : 1 – 14 . [Google Scholar]], and Kim and Ko [4 Kim , T.S. , and Ko , M.H. 2008 . On the complete convergence of moving average process with Banach space valued random elements . J. Theor. Probab. 21 : 431 – 436 . [Google Scholar]] solved an open question posed by Ahmed et al. In this article, we improve and complement the result of Ahmed et al. The method used in this article is simpler than those in Ahmed et al., Sung and Volodin, Chen et al., and Kim and Ko.     

On the Characterization of f-Ideals

In this paper, we give the characterization of unmixed f-ideals of degree d ≥ 2 generalizing the results given in [1 Abbasi , G. Q. , Ahmad , S. , Anwar , I. , Baig , W. A. ( 2012 ). f-Ideals of degree 2. Algebra Colloquium 19 (Spec 1):921–926 . [Google Scholar]].     

Algebraic Entropy of Generalized Shifts on Direct Products

For a set Γ, a function λ: Γ → Γ and a nontrivial abelian group K, the \emphgeneralized shift σ_λ: K ^Γ → K ^Γ is defined by (x _i ) _i∈Γ ? (x _λ(i)) _i∈Γ [3 Ayatollah Zadeh Shirazi , F. , Heidari Ardi , F. , Karami Kabir , N. ( 2008 ). A note on shift theory . Math. Pannon. 19 : 187 – 195 . [Google Scholar]]. In this article we compute the algebraic entropy of σ_λ; it is either zero or infinite, depending exclusively on the properties of λ. This solves two problems posed in [2 Akhavin , M. , Ayatollah Zadeh Shirazi , F. , Dikranjan , D. , Giordano Bruno , A. , Hosseini , A. ( 2009 ). Algebraic entropy of shift endomorphisms on abelian groups . Quaest. Math. 32 : 529 – 550 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Conjugacy Classes of P-Cycles of Type D in Alternating Groups

We classify the conjugacy classes of p-cycles of type D in alternating groups. This finishes the open cases in [3 Andruskiewitsch , N. , Fantino , F. , Graña , M. , Vendramin , L. ( 2011 ). Finite-dimensional pointed Hopf algebras with alternating groups are trivial . Ann. Mat. Pura Appl 190 : 225 – 245 .[Web of Science ®] , [Google Scholar]]. Also we determine all the subracks of those conjugacy classes which are not of type D.     

Semilocal Convergence of Steffensen-Type Algorithms for Solving Nonlinear Equations

In this article, we provide a semilocal analysis for the Steffensen-type method (STTM) for solving nonlinear equations in a Banach space setting using recurrence relations. Numerical examples to validate our main results are also provided in this study to show that STTM is faster than other methods ([7 I. K. Argyros , J. Ezquerro , J. M. Gutiérrez , M. Hernández , and S. Hilout ( 2011 ). On the semilocal convergence of efficient Chebyshev-Secant-type methods . J. Comput. Appl. Math. 235 : 3195 – 3206 .[Crossref], [Web of Science ®] , [Google Scholar], 13 J. A. Ezquerro and M. A. Hernández ( 2009 ). An optimization of Chebyshev's method . J. Complexity 25 : 343 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]]) using similar convergence conditions.     

Self-Interacting Markov Chains: Some Asymptotics

Abstract

Guided by the self-interaction mechanisms introduced in Benaim et al. [2 Benaim , M. , Ledoux , M. , and Raimond , O. 2002 . Self-interacting diffusions . Probab. Theory Related Fields 122 : 1 – 41 . [Google Scholar]] and in [5 Del Moral , P. , and Miclo , L. 2006 . Self-interacting Markov chains . Stochastic Anal. Appl. 24 : 615 – 660 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], we present a more general definition of self-interacting Markov chains (SIMCs) (than in Del Moral and Miclo [5 Del Moral , P. , and Miclo , L. 2006 . Self-interacting Markov chains . Stochastic Anal. Appl. 24 : 615 – 660 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] and Benaim et al. [2 Benaim , M. , Ledoux , M. , and Raimond , O. 2002 . Self-interacting diffusions . Probab. Theory Related Fields 122 : 1 – 41 . [Google Scholar]]). We then establish, for particular self-interaction mechanisms, a stability theorem with error estimation, two central limit theorems, two functional central limit theorems, and the large deviation principle.     

Splitting of operations for alternative and Malcev structures

In this paper, we define pre-Malcev algebras and alternative quadri-algebras and prove that they generalize pre-Lie algebras and quadri-algebras, respectively, to the alternative setting. We use the results and techniques from [4 Bai, C., Bellier, O., Guo, L., Ni, X. (2013). Splitting of operations, Manin products, and Rota-Baxter operators. Int. Math. Res. Not. 2013(3):485–524. [Google Scholar], 14 Gubarev, V. Y., Kolesnikov, P. S. (2013). Embedding of dendriform algebras into Rota-Baxter algebras. Cent. Eur. J. Math. 11(2):226–245.[Crossref], [Web of Science ®] , [Google Scholar]] to discuss and give explicit computations of different constructions in terms of bimodules, splitting of operations, and Rota–Baxter operators.     

A Note on the Dimension Theory of ℤ n -Graded Rings

In this note we want to generalize some of the results in [1 Brewer , J. , Montgomery , P. , Rutter E. , Heinzer , W. ( 1973 ). Krull dimension of polynomial rings in “Conference on Commutative Algebra, Lawrence 1972.” . Springer Lecture Notes in Mathematics 311 : 26 – 45 .[Crossref] , [Google Scholar]] from polynomial rings in several indeterminates to arbitrary ? ⁿ -graded commutative rings. We will prove an analogue of Jaffard's Special Chain Theorem and a similar result for the height of a prime ideal 𝔭 over its graded core 𝔭*.     

Reflected BSDE and Locally Periodic Homogenization of Semilinear PDEs with Nonlinear Neumann Boundary Condition

We study the homogenization of semilinear partial differential equations (PDEs) with nonlinear Neumann boundary condition, locally periodic coefficients, and highly oscillating drift and nonlinear term. Our method is entirely probabilistic, as in a periodic case by Ouknine and Pardoux [14 Ouknine , Y. , and Pardoux , É. 2002 . Homogenization of PDEs with non linear boundary condition, Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999). Progresses of Probability, 52, Birkhäuser, Basel , pp. 229 – 242 . [Google Scholar]] and builds on our earlier work [5 Diakhaby , A. , and Ouknine , Y. 2006 . Locally periodic homogenization of reflected diffusion . Journal of Applied Mathematics and Stochastic Analysis . [Google Scholar]], which gives us the locally periodic counterpart of Theorem 2.2 in Tanaka [21 Tanaka , H. 1984 . Homogenization of diffusion processes with boundary conditions . Stochastic Analysis and Applications 7 : 411 – 437 . Advanced Probability and Related Topics 7, Dekker, New York . [Google Scholar]].