The Mean-Variance Hedging in a Bond Market with Jumps

We construct a market of bonds with jumps driven by a general marked point process as well as by a ? ⁿ -valued Wiener process based on Björk et al. [6 Björk , T. , Kabanov , Y. , and Runggaldier , W. 1997 . Bond market structure in the presence of marked point processes . Math. Finance 7 : 211 – 223 .[Crossref], [Web of Science ®] , [Google Scholar]], in which there exists at least one equivalent martingale measure Q ⁰. Then we consider the mean-variance hedging of a contingent claim H ∈ L ²(? _{T 0} ) based on the self-financing portfolio based on the given maturities T ₁,…, T _n with T ₀ < T ₁ < … <T _n  ≤ T*. We introduce the concept of variance-optimal martingale (VOM) and describe the VOM by a backward semimartingale equation (BSE). By making use of the concept of ?*-martingales introduced by Choulli et al. [8 Choulli , T. , Krawczyk , L. , and Stricker , C. 1998 . ?-martingales and their applications in mathematical finance . The Annals of Probability 26 ( 2 ): 853 – 876 . [Google Scholar]], we obtain another BSE which has a unique solution. We derive an explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by the solutions of these two BSEs.     

Local and Global Existence for an Aggregation Equation

The purpose of this work is to develop a satisfactory existence theory for a general class of aggregation equations. An aggregation equation is a non-linear, non-local partial differential equation that is a regularization of a backward diffusion process. The non-locality arises via convolution with a potential. Depending on how regular the potential is, we prove either local or global existence for the solutions. Aggregation equations have been used recently to model the dynamics of populations in which the individuals attract each other (Bodnar and Velazquez, 2005 Bodnar , M. , Velazquez , J. J. L. ( 2005 ). Derivation of macroscopic equations for individual cell-based models: a formal approach . Math. Methods Appl. Sci. 28 ( 15 ): 1757 – 1779 .[Crossref], [Web of Science ®] , [Google Scholar]; Holm and Putkaradze, 2005 Holm , D. D. , Putkaradze , V. ( 2005 ). Aggregation of finite size particles with variable mobility . Phys. Rev. Lett. 95 : 226106 . [Google Scholar]; Mogilner and Edelstein-Keshet, 1999 Mogilner , A. , Edelstein-Keshet , L. ( 1999 ). A non-local model for a swarm . J. Math. Biol. 38 ( 6 ): 534 – 570 .[Crossref], [Web of Science ®] , [Google Scholar]; Morale et al., 2005 Morale , D. , Capasso , V. , Oelschläger , K. ( 2005 ). An interacting particle system modelling aggregation behavior: from individuals to populations . J. Math. Biol. 50 ( 1 ): 49 – 66 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]; Topaz and Bertozzi, 2004 Topaz , C. M. , Bertozzi , A. L. ( 2004 ). Swarming patterns in a two-dimensional kinematic model for biological groups . SIAM J. Appl. Math. 65 ( 1 ): 152 – 174 (electronic) .[Crossref], [Web of Science ®] , [Google Scholar]; Topaz et al., 2006 Topaz , C. M. , Bertozzi , A. L. , Lewis , M. A. ( 2006 ). A nonlocal continuum model for biological aggregation . Bull. Math. Biol. 68 ( 7 ): 1601 – 1623 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]).     

Stochastic motion under G-framework: II. Kinematics

Continuing the study of stochastic motion that we started [11 Zhang, H., and Kannan, D. 2014. Stochastic motion under G-framework: I. Nelson stochastic derivatives. Stoch Anal Appl 32:1067–1096.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], we present in this article the kinematics of such a motion. We begin by defining the quadratic derivative of an S2-process, and show that this derivative of the Brownian motion captures the variance uncertainty. We show, under certain vanishing derivatives and independence conditions, the martingale properties of an S1-process. Starting with an S1-process, we derive the equation of motion, an Itô equation corresponding to a G-diffusion process.     

Determining an Unbounded Potential from Cauchy Data in Admissible Geometries

In [4 Dos Santos Ferreira , D. , Kenig , C.E. , Salo , M. , Uhlmann , G. ( 2009 ). Limiting Carleman weights and anisotropic inverse problems . Invent. Math. 178 : 119 – 171 .[Crossref], [Web of Science ®] , [Google Scholar]] anisotropic inverse problems were considered in certain admissible geometries, that is, on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. In particular, it was proved that a bounded smooth potential in a Schrödinger equation was uniquely determined by the Dirichlet-to-Neumann map in dimensions n ≥ 3. In this article we extend this result to the case of unbounded potentials, namely those in L ^n/2. In the process, we derive L ^p Carleman estimates with limiting Carleman weights similar to the Euclidean estimates of Jerison and Kenig [8 Jerison , D. , Kenig , C.E. ( 1985 ). Unique continuation and absence of positive eigenvalues for Schrödinger operators . Ann. Math. 121 : 463 – 494 .[Crossref], [Web of Science ®] , [Google Scholar]] and Kenig et al. [9 Kenig , C.E. , Ruiz , A. , Sogge , C.D. ( 1987 ). Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators . Duke Math. J. 55 : 329 – 347 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Gorenstein coresolving categories

Let 𝒜 be an abelian category. A subcategory 𝒳 of 𝒜 is called coresolving if 𝒳 is closed under extensions and cokernels of monomorphisms and contains all injective objects of 𝒜. In this paper, we introduce and study Gorenstein coresolving categories, which unify the following notions: Gorenstein injective modules [8 Enochs, E. E., Jenda, O. M. G. (1995). Gorenstein injective and projective modules. Math. Z. 220:611–633.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein FP-injective modules [20 Mao, L. X., Ding, N. Q. (2008). Gorenstein FP-injective and Gorenstein flat modules. J. Algebra Appl. 7:491–506.[Crossref], [Web of Science ®] , [Google Scholar]], Gorenstein AC-injective modules [3 Bravo, D., Gillespie, J. (2016). Absolutely clean, level, and Gorenstein AC-injective complexes. Commun. Algebra 44:2213–2233.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], and so on. Then we define a resolution dimension relative to the Gorenstein coresolving category 𝒢?_𝒳(𝒜). We investigate the properties of the homological dimension and unify some important properties possessed by some known homological dimensions. In addition, we study stability of the Gorenstein coresolving category 𝒢?_𝒳(𝒜) and apply the obtained properties to special subcategories and in particular to module categories.     

GRADINGS OF SIMPLE JORDAN ALGEBRAS AND THEIR RELATION TO THE GRADINGS OF SIMPLE ASSOCIATIVE ALGEBRAS

In this paper we describe all group gradings of the simple Jordan algebra of a non-degenerate symmetric form on a vector space over a field of characteristic different from 2. If we use the notion of the Clifford algebra, then we are able to recover some of the gradings on matrix algebras obtained in an entirely different way in [BSZ] Bahturin, Y., Seghal, S. and Zaicev, M. in press. Group Gradings of Associative Algebras. J. Algebra, [Web of Science ®] , [Google Scholar].

     

Calderón problem for connections

In this paper, we consider the problem of identifying a connection ? on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian ?*? over conformally transversally anisotropic (CTA) manifolds. This was proved in [9 Dos Santos Ferreira, D., Kenig, C., Salo, M., Uhlmann, G. (2009). Limiting Carleman weights and anisotropic inverse problems. Invent. Math. 178:119–171.[Crossref], [Web of Science ®] , [Google Scholar]] for line bundles in the case of the transversal manifold being simple—we generalize this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of [11 Dos Santos Ferreira, D., Kurylev, Y., Lassas, M., Salo, M. (2016). The Calderón problem in transversally anisotropic geometries. J. Eur. Math. Soc., 18:2579–2626.[Crossref], [Web of Science ®] , [Google Scholar]]. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.     

Convergence of the Structure Function of a Multifractal Random Walk in a Mixed Asymptotic Setting

Some asymptotic properties of a Brownian motion in multifractal time, also called multifractal random walk, are established. We show the almost sure and L ¹ convergence of its structure function. This is an issue directly connected to the scale invariance and multifractal property of the sample paths. We place ourselves in a mixed asymptotic setting where both the observation length and the sampling frequency may go together to infinity at different rates. The results we obtain are similar to the ones that were given by Ossiander and Waymire [19 Ossiander , M. , and Waymire , E.C. 2000 . Statistical estimation for multiplicative cascades . Ann. Stat. 28 : 1533 – 1560 .[Crossref], [Web of Science ®] , [Google Scholar]] and Bacry et al. [1 Bacry , E. , Gloter , A. , Hoffmann , M. , and Muzy , J.F. Multifractal analysis in a mixed asymptotic framework . Ann. Appl. Prob. (to appear) . [Google Scholar]] in the simpler framework of Mandelbrot cascades.     

Eigenfunctions for Partially Rectangular Billiards

In this note, we further develop the methods of Burq and Zworski (2005 Burq , N. , Zworski , M. ( 2005 ). Bouncing ball modes and quantum chaos . SIAM Review 47 ( 5 ): 43 – 49 [CROSSREF] [CSA] [Crossref] , [Google Scholar]) to study eigenfunctions for billiards which have rectangular components: these include the Bunimovich billiard, the Sinai billiard, and the recently popular pseudointegrable billiards (Bogomolny et al., 1999 Bogomolny , E. , Gerland , U. , Schmit , C. ( 1999 ). Models of intermediate spectral statistics . Phys. Rev. E 59 : 1315 – 1318 [CROSSREF] [CSA] [Crossref], [Web of Science ®] , [Google Scholar]). The results are an application of a “black-box” point of view as presented in Burq and Zworski (2004 Burq , N. , Zworski , M. ( 2004 ). Geometric control in the presence of a black box . JAMS 17 : 443 – 471 [CROSSREF] [CSA] [Web of Science ®] , [Google Scholar]).