Approximations for Solutions of Lévy-Type Stochastic Differential Equations

Abstract

The problem of the construction of strong approximations with a given order of convergence for jump-diffusion equations is studied. General approximation schemes are constructed for Lévy-type stochastic differential equation. In particular, the article generalizes the results from [2 Gardoń , A. 2004 . The order of approximations for solutions of Ito-type stochastic differential equations with jumps . Stoch. Anal. Appl. 22 ( 3 ): 679 – 699 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 5 Kloeden , P.E. , and Platen , E. 1995 . Numerical Solutions of Stochastic Differential Equations . Springer-Verlag , Berlin . [Google Scholar]]. The Euler and the Milstein schemes are shown for finite and infinite Lévy measure.     

Convergence of Row Sequences of Simultaneous Padé–Faber Approximants

For meromorphic circumferentially mean p-valent functions, an analog of the classical distortion theorem is proved. It is shown that the existence of connected lemniscates of the function and a constraint on a cover of two given points lead to an inequality involving the Green energy of a discrete signedmeasure concentrated at the zeros of the given function and the absolute values of its derivatives at these zeros. This inequality is an equality for the superposition of a certain univalent function and an appropriate Zolotarev fraction.     

Padé Approximations of and preservation of quadratic Lyapunov functions

We investigate whether or not quadratic Lyapunov functions are preserved under Padé approximations. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spectral theory and Wiener-Itô decomposition for the image of a Jacobi field

Assume that K ⁺: H _? → T _? is a bounded operator, where H _? and T _? are Hilbert spaces and ρ is a measure on the space H _?. Denote by ρ_K the image of the measure ρ under K ⁺. We study the measure ρ_K under the assumption that ρ is the spectral measure of a Jacobi field and obtain a family of operators whose spectral measure is equal to ρ_K. We also obtain an analog of the Wiener-Itô decomposition for ρ_K. Finally, we illustrate the results obtained by explicit calculations carried out for the case, where ρ_K is a Lévy noise measure.     

Recursive computation of Padé–Legendre approximants and some acceleration properties

Summary. In this paper, after recalling the two definitions of the generalizations of the Padé approximants to orthogonal series, we will define the Padé–Legendre approximants of a Legendre series. We will propose two algorithms for the recursive computation of some sequences of these approximants. We will also estimate the speed of convergence of the columns of the Padé–Legendre table from the asymptotic behaviour of the coefficients of the Legendre series. Finally we will illustrate these results with some numerical examples. Received June 20, 1998 / Published online March 20, 2001     

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.     

On Spectral Inclusions and Approximations of α-times Resolvent Families

We present some basic properties of fractional resolvent families. Moreover, spectral inclusions and approximation for fractional resolvent families are considered here.     

Distributions of Poles to Painlevé Transcendents via Padé Approximations

A version of the Fair–Luke algorithm has been used to find the Padé approximate solutions to the Painlevé I, II, and IV equations. The distributions of poles in the complex plane are studied to check the dynamics of movable poles and the emergence of rational and truncated solutions, as well as various patterns formed by the poles. The high-order approximations allow us to check asymptotic expansions at infinity and estimate the range of asymptotic domains. The Coulomb gas interpretation of the pole ensembles is discussed in view of the patterns arising in Painlevé IV transcendents.     

Variational Solutions of Coupled Hamilton—Jacobi Equations

In this note we study variational solutions of weakly coupled Hamilton—Jacobi equations in the case where the Hamiltonians are convex. More precisely, we build the variational solution by an approximation scheme. Accepted 24 April 1998     

On the Existence of Solutions of Poisson Equation and Poincaré–Lelong Equation

One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of Kähler manifolds with nonnegative holomorphic bisectional curvature, $\mathrm{Ric}(x)\geq \left(a\ln\ln\left(10+r(x)\right)\right)\Big/\big.\left(\left(1+r^2(x)\right)\ln(10+r(x))\right)One of the main goals of this paper is to solve the Poincaré–Lelong equation on a class of K?hler manifolds with nonnegative holomorphic bisectional curvature, for some a > 67(n + 4)². We will also study the Poisson equation on complete noncompact manifolds which satisfy volume doubling and Poincaré inequality.     

Spectral Analysis for Stationary Solutions of the Cahn–Hilliard Equation in ℝ d

We consider the spectrum associated with three types of bounded stationary solutions for the Cahn–Hilliard equation on ? ^d , d ≥ 2: radial solutions, saddle solutions (only for d = 2), and planar periodic solutions. In particular, we establish spectral instability for each type of solution. The important case of multiply periodic solutions does not fit into the framework of our approach, and we do not consider it here.     

Existence of Solutions to Generalized Vector Quasi-Equilibrium Problems with Discontinuous Mappings

Let X, Y be two finite-dimensional topological vector spaces, Z a Hausdorff topological vector space, K C X and D C Z be two nonempty sets, C be a pointed, closed, and convex cone in Y with int C ≠θ Let S ： K → 2^K and T ： K → 2^D be two multivalued mappings, and φ ： K × D × K → Y be a trifunction. In this paper, we consider the generalized vector quasi-equilibrium problem, which is formulated by finding X∈ K and y∈ T（x） such that x∈ E S（x） and φ（x,y, u） （∈/） -int C for all u ∈ S（x）. We establish an existence result in which T is not supposed to have any continuity property. Our results extend and improve the corresponding results of Cubiotti, Yao and Guo.     

Discontinuous solutions of Hamilton–Jacobi–Bellman equation under state constraints

This article is devoted to the Hamilton–Jacobi partial differential equation $$\left\{\begin{array}{lll}\frac{\partial V}{\partial t} = H\left(t, x, - \frac{\partial V}{\partial x}\right) & \hbox{on} & [0, 1]\times {\overline{\Omega}}\\V(1, x) = g(x) & \hbox{on}& {\overline{\Omega}},\end{array}\right.$$ where the Hamiltonian ${{H:[0, 1] \times \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}}}$ is convex and positively homogeneous with respect to the last variable, ${{\Omega \subset \mathbb{R}^n}}$ is open and ${{g : \mathbb{R}^n \to \mathbb{R} \cup \{+ \infty\}}}$ is lower semicontinuous. Such Hamiltonians do arise in the optimal control theory. We apply the method of generalized characteristics to show uniqueness of lower semicontinuous solution of this first order PDE. The novelty of our setting lies in the fact that we do not ask regularity of the boundary of Ω and extend the Soner inward pointing condition in a nontraditional way to get uniqueness in the class of lower semicontinuous functions.     

An Algorithm for the Computation of Hermite–Padé Approximations to the Exponential Function: Divided Differences and Hermite–Padé Forms

We present the first of two different algorithms for the explicit computation of Hermite–Padé forms (HPF) associated with the exponential function. Some roots of the algebraic equation associated with a given HPF are good approximants to the exponential in some subsets of the complex plane: they are called Hermite–Padé approximants (HPA) to this function. Our algorithm is recursive and based upon the expression of HPF as divided differences of the function texp(xt) at multiple integer nodes. Using this algorithm, we find again the results obtained by Borwein and Driver for quadratic HPF. As an example, we give an interesting family of quadratic HPA to the exponential.     

Titchmarsh–Weyl Formula for the Spectral Density of a Class of Jacobi Matrices in the Critical Case

Functional Analysis and Its Applications - We consider a class of Jacobi matrices with unbounded entries in the so-called critical (double root, Jordan block) case. We prove a formula which relates...     

Asymptotic Solutions of Hamilton–Jacobi Equations with Semi-Periodic Hamiltonians

We study the long time behavior of viscosity solutions of the Cauchy problem for Hamilton–Jacobi equations in ? ⁿ . We prove that if the Hamiltonian H(x, p) is coercive and strictly convex in a mild sense in p and upper semi-periodic in x, then any solution of the Cauchy problem “converges” to an asymptotic solution for any lower semi-almost periodic initial function.     

Explicit Strong Solutions of SPDE’s with Applications to Non-Linear Filtering

A differential calculus for random fields is developed and combined with the S-transform to obtain an explicit strong solution of the Cauchy problem

Global Discontinuous Solutions to A Class of Generalizd Riemann Problems for Quasilinear Hyperbolic Systems（Ⅰ）

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