Explicit Solution of a Non-Linear Filtering Problem for Lévy Processes with Application to Finance

In this paper we explicitly solve a non-linear filtering problem with mixed observations, modelled by a Brownian motion and a generalized Cox process, whose jump intensity is given in terms of a Lévy measure. Motivated by empirical observations of R. Cont and P. Tankov we propose a model for financial assets, which captures the phenomenon of time inhomogeneity of the jump size density. We apply the explicit formula to obtain the optimal filter for the corresponding filtering problem.     

An Implementation for the Algorithm of Janet bases of Linear Differential Ideals in the Maple System

In this paper, an algorithm for computing the Janet bases of linear differential equations is described, which is the differential analogue of the algorithm JanetBasis improved by Gerdt. An implementation of the algorithm in Maple is given. The implemented algorithm includes some subalgorithms: Janet division,Pommaret division, the judgement of involutive divisor and reducible, the judgement of conventional divisor and reducible, involutive normal form and conventional normal form, involutive autoreduction and conventional autoreduction, PJ-autoreduction and so on. As an application, the Janet Bases of the determining system of classical Lie symmetries of some partial differential equations are obtained using our package.     

Acceleration waves analyzed by a new continuum model of solids incorporating microscopic thermal vibrations

Acceleration waves propagating in isotropic solids at finite temperatures are studied by applying the method of singular surfaces to a new continuum model derived statistical-mechanically from a three-dimensional lattice model. The continuum model explicitly takes into account the microscopic thermal vibrations of the constituent atoms as one of the field variables. The propagation speeds and the ratios of mechanical and thermal amplitudes for both longitudinal and transverse waves are consistently determined. The differential equations that govern the time variation of the amplitudes of the waves are also derived. The analytical results, which are valid over a wide temperature range that includes the melting point, are evaluated numerically for several materials, and their physical implications are discussed. One of the findings to be emphasized is that of the singularities of the characteristic quantities at the melting point.Received: 13 March 2003, Accepted: 20 June 2003PACS: 62.30. + d, 65.40.-bM. Sugiyama: Correspondence to Dedicated to Prof. Ingo Müller on the occasion of his 65th birthday.     

A Note on the Oscillation of Second Order Differential Equations

We give a sufficient condition for the oscillation of linear homogeneous second order differential equation y + p(x)y + q(x)y = 0, where and is positive real number.     

On the periodic nature of the solutions of the reciprocal difference equation with maximum

We prove that every positive solution of the third order difference equation

     

Linearized stability analysis of discrete Volterra equations

Various different types of stability are defined, in a unified framework, for discrete Volterra equations of the type x(n)=f(n)+∑ⁿ_j=0K(n,j,x(n)) (n?0). Under appropriate assumptions, stability results are obtainable from those valid in the linear case (K(n,j,x(n))=B(n,j)x(j)), and a linearized stability theory is studied here by using the fundamental and resolvent matrices. Several necessary and sufficient conditions for stability are obtained for solutions of the linear equation by considering the equations in various choices of Banach space , the elements of which are sequences of vectors (, , n,j?0, etc.). We show that the theory, including a number of new results as well as results already known, can be presented in a systematic framework, in which results parallel corresponding results for classical Volterra integral equations of the second kind.     

Trichotomy of a system of two difference equations

In this paper we study the boundedness and the asymptotic behavior of the positive solutions of the system of difference equations

     

regularizing effect in a nonlinear, degenerate parabolic equation in one space dimension

In this paper we provide and analyze a nonlinear degenerate parabolic equation in one space dimension with the following smoothing property: If the initial data is only uniformly continuous, at positive times, the solution has bounded second derivatives (it belongs to ). We call this surprising phenomenon a regularizing effect. So far, such phenomena have only been observed in uniformly parabolic equations.

     

Analysis of finite element approximation for time-dependent Maxwell problems

We provide an error analysis of finite element methods for solving time-dependent Maxwell problem using Nedelec and Thomas-Raviart elements. We study the regularity of the solution and develop some new error estimates of Nedelec finite elements. As a result, the optimal -error bound for the semidiscrete scheme is obtained.

     

A new approximation technique for div-curl systems

In this paper, we describe an approximation technique for div-curl systems based in where is a domain in . We formulate this problem as a general variational problem with different test and trial spaces. The analysis requires the verification of an appropriate inf-sup condition. This results in a very weak formulation where the solution space is and the data reside in various negative norm spaces. Subsequently, we consider finite element approximations based on this weak formulation. The main approach of this paper involves the development of ``stable pairs" of discrete test and trial spaces. With this approach, we enlarge the test space so that the discrete inf-sup condition holds and we use a negative-norm least-squares formulation to reduce to a uniquely solvable linear system. This leads to optimal order estimates for problems with minimal regularity which is important since it is possible to construct magnetostatic field problems whose solutions have low Sobolev regularity (e.g., with ). The resulting algebraic equations are symmetric, positive definite and well conditioned. A second approach using a smaller test space which adds terms to the form for stabilization will also be mentioned. Some numerical results are also presented.