Equations In Infinite Dimensional Spaces Research Articles

On Existence and Uniqueness of Solutions for Ordinary Differential Equations in Locally Convex Topological Linear Spaces

ABSTRACT This article discusses differential equations in infinite-dimensional spaces. We introduce a general version of the Lipschitz-type continuity criterion without the notion of a distance and prove an existence and uniqueness result for ordinary differential equations in locally convex topological linear spaces. It is shown that remarkable techniques of classical analysis, such as the method of successive approximations, are still adaptable tools in the study of the abstract models.

Relationships between the maximum principle and dynamic programming for infinite dimensional stochastic control systems

The Pontryagin type maximum principle and Bellman's dynamic programming principle serve as two of the most important tools in solving optimal control problems. There is a huge literature on the study of relationship between them. The main purpose of this paper is to investigate the relationship between the Pontryagin type maximum principle and the dynamic programming principle for control systems governed by stochastic evolution equations in infinite dimensional space, with the control variables entering into both the drift and the diffusion terms. To do so, we first prove the dynamic programming principle for those systems without employing the martingale solutions. Then we establish the desired relationships in both cases that value function associated is smooth or nonsmooth. For the nonsmooth case, in particular, by employing the relaxed transposition solution, we discover the connection between the superdifferentials and subdifferentials of value function and the first-order and second-order adjoint equations.

On Weak and Strong Solutions of Paired Stochastic Functional Differential Equations in Infinite-Dimensional Spaces

In this paper, we study the questions of the existence of global weak solutions and local strong solutions of paired stochastic functional differential equations in a Hilbert space, one of which is an equation with an unbounded operator, and the other is an ordinary differential equation. We proved the existence and uniqueness theorems in the case of coefficients with polynomial growth.

Positive Solutions of a Class of Operator Equations

Positive solutions of a class of matrix equations were studied by Bhatia, et al., Bull. London Math. Soc., 32, 214 (2000), SIAM J. Matrix Anal. Appl., 14, 132 (1993) and 27, 103–114 (2005), by Kwong, Linear Algebra Appl., 108, 177–197 (1988), and by Cvetkovi? and Milovanovi?, [Linear Algebra Appl., 429, 2401–2414 (2008)]. Following the idea used in the last paper, we study a class of operator equations in infinite-dimensional spaces and prove that the positivity of solutions can be established for this class of equations under the condition that a certain rational function is positive semidefinite.

A note on the initial value problem of fractional evolution equations

In this paper, we discuss the continuous dependence of mild solutions on initial values and orders for the initial value problem of fractional evolution equations in infinite dimensional spaces. The results obtained in this paper improve and extend some related conclusions on this topic. This paper can be considered as a contribution to this emerging field.

Taylor's series for non-differentiable functions

In the deterministic context a series of well established results allow to reformulate delay differential equations (DDEs) as evolution equations in infinite dimensional spaces. Several models in the theoretical economic literature have been studied using this reformulation. On the other hand, in the stochastic case only few results of this kind are available and only for specific problems. The contribution of the present letter is to present a way to reformulate in infinite dimension a prototype controlled stochastic DDE, where the control variable appears delayed in the diffusion term. As application, we present a model for quadratic risk minimization hedging of European options with execution delay and a time-to-build model with shock. Some comments concerning the possible employment of the dynamic programming after the reformulation in infinite dimension conclude the letter.

Monte Carlo Euler approximations of HJM term structure financial models

We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.

Nonlinear functions and the norm of the propagators of operational equations

We explore the relation between nonlinear functions and the norm functions of the propagators of operational equations in infinite-dimensional spaces. A general criterion is obtained to judge whether a given nonlinear function is the norm function of some C0-semigroup, the propagator of a wellposed Cauchy problem for an operational equation in an infinite-dimensional space, which gives a solution to an open problem posed in Lin et al. (2010) [6]. Moreover, applying our abstract results, we show the existence of two valuable C0-semigroups, which one has been seeking.

Input–output admissibility and exponential trichotomy of difference equations

We propose a new method for the study of the asymptotic behavior of difference equations in infinite-dimensional spaces, providing characterizations for the property of uniform exponential trichotomy. We deduce the structure of the stable, unstable and bounded subspace and prove the uniqueness of the projection families. We introduce a new admissibility concept with respect to a discrete input–output system and prove that this is a necessary and sufficient condition for the existence of uniform exponential trichotomy. Throughout the paper, there is no assumption on the coefficients and the obtained results are applicable to any class of difference equations.

Robust stability of a class of positive quasi-polynomials in Banach spaces

In this paper, we study the stability radii of positive quasipolynomials associated with linear functional difference equations in infinite-dimensional spaces. It is shown that the positive, real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by a simple example.

Average and deviation for the stochastic FitzHugh--Nagumo system

An averaged system for the slow-fast stochastic FitzHugh--Nagumo system is derived in this paper. The rate of convergence in probability is obtained as a byproduct. Moreover the deviation between the original system and the averaged system is studied. A martingale approach proves that the deviation is described by a Gaussian process. The deviation gives a more accurate asymptotic approximation than previous work. References S. Cerrai and M. Freidlin, Averaging principle for a class of stochastic reaction--diffusion equations, to appear in Probab. Th. and Rel. Fields . doi:10.1007/s00440-008-0144-z R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane. Biophys J. , 1 (1961). H. Kesten and G. C. Papanicolaou, A limit theorem for turbulent diffusion, Commun. Math. Phys. , 65 (1979), 79--128. doi:10.1007/3-540-08853-9 R. Z. Khasminskii, On the principle of averaging the Ito stochastic differential equations (Russian), Kibernetika , 4 (1968), 260--279. P. E. Kloeden and E. Platen. Numerical solution of stochastic differential equations , volume 23 of Applications of Mathematics . Springer--Verlag, 1992. M. Metivier, Stochastic Partial Differential Equations in Infinite Dimensional Spaces , Scuola Normale Superiore, Pisa, 1988. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions , Cambridge University Press, 1992. J. Simon, Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl. , 146 (1987), 65--96. doi:10.1007/BF01762360 V. M. Volosov, Averaging in systems of ordinary differential equations. Russ. Math. Surv. , 17 (1962), 1--126. doi:10.1070/RM1962v017n06ABEH001130

Stationary solutions of retarded Ornstein–Uhlenbeck processes in Hilbert spaces

In this note, we shall study a class of retarded Langevin equations in infinite-dimensional spaces. An informative description of stationary solutions is given by means of Green operators, Lyapunov equations, the second moment bounds and retarded point spectrum. The covariance operator of the stationary solution is described as a solution of some retarded operator equation.

Necessary Conditions of Optimality for Second-Order Nonlinear Impulsive Differential Equations

We discuss the existence of optimal controls for a Lagrange problem of systems governed by the second-order nonlinear impulsive differential equations in infinite dimensional spaces. We apply a direct approach to derive the maximum principle for the problem at hand. An example is also presented to demonstrate the theory.

Inexact-Newton method for solving operator equations in infinite-dimensional spaces

In this paper, we develop an inexact-Newton method for solving nonsmooth operator equations in infinite-dimensional spaces. The linear convergence and superlinear convergence of inexact-Newton method under some conditions are shown. Then, we characterize the order of convergence in terms of the rate of convergence of the relative residuals. The present inexact-Newton method could be viewed as the extensions of previous ones with same convergent results in finite-dimensional spaces.

Delay difference equations in infinite-dimensional spaces

We present explicit stability conditions for the solutions of nonlinear delay difference equations defined in infinite-dimensional spaces , where X is a Banach space; are linear bounded operators and the nonlinearities satisfy a local comparison condition. By virtue of the new estimates for the norm of functions of quasi-Hermitian operators as well as the Freezing Method for difference systems, explicit stability and boundedness conditions are given. Applications to infinite dimensional delay difference systems are discussed.

Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces

Solutions of semilinear elliptic differential equations in infinite-dimensional spaces are obtained by means of forward and backward infinite-dimensional stochastic evolution equations. The backward equation is considered on an infinite time horizon and a suitable growth condition replaces the final condition. Elliptic equations are intended in a mild sense, suitable also for applications to optimal control. We finally notice that, due to the lack of smoothing properties, the elliptic partial differential equation considered here could not be treated by analytic methods.

Semi-numerical Approximation Structures for Nonlinear Noncompact Operators in Banach Spaces

The A-proper operators, introduced by F. E. Browder and W. V. Petryshyn (Browder, F. E., Petryshyn, W. V. (1968). The topological degree and Galerkin approximations for noncompact operators in Banach spaces. Bull. Amer. Math. Soc. 74:641–646; Browder, F. E., Petryshyn, W. V. (1969). Approximation methods and the generalized topological degree for nonlinear mappings in Banach spaces. J. Funct. Anal. 3:217–245.), arise naturally in the solution to an equation in infinite dimensional space via the finite dimensional approximation. In the last years, much attention has been paid to the study of linear semi-numerical approximation schemes of Galerkin type. In this article the uniform limits, sums, and compositions of A-proper maps are investigated, as well as their generalized topological degree with respect to various nonlinear approximation schemes.

Nonlinear Kolmogorov equations in infinite dimensional spaces: the backward stochastic differential equations approach and applications to optimal control

Solutions of semilinear parabolic differential equations in infinite dimensional spaces are obtained by means of forward and backward infinite dimensional stochastic evolution equations. Parabolic equations are intended in a mild sense that reveals to be suitable also towards applications to optimal control.

Asymptotic behavior of solutions of a class of differential equations in infinite-dimensional spaces

Asymptotic behavior of solutions of a class of differential equations in infinite-dimensional spaces

P-th moment growth bounds of infinite-dimensional stochastic evolution equations

The aim of this paper is to investigate the p-th moment growth bounds wilh a general rate function λ(t) of the strong solution for a class of stochastic differential equations in infinite dimensional space under various sufficient hypotheses. The results derived here extend the usual situations to some extent, containing for example the polynomial or iterated logarithmic growth cases studied by many authors. In particular, more generalised sufficient conditions, ensuring the p-th moment upper-bound of sample paths given by solutions of a class of nonlinear stochastic evolution equations, are captured. Applications to parabolic itô equations are also considered.