Exit times for multivariate autoregressive processes

We study exit times from a set for a family of multivariate autoregressive processes with normally distributed noise. By using the large deviation principle, and other methods, we show that the asymptotic behavior of the exit time depends only on the set itself and on the covariance matrix of the stationary distribution of the process. The results are extended to exit times from intervals for the univariate autoregressive process of order n$n$, where the exit time is of the same order of magnitude as the exponential of the inverse of the variance of the stationary distribution.     

Jensen's inequality for filtration consistent nonlinear expectation without domination condition

In this paper, the general filtration consistent nonlinear expectation defined on the integrable variable space is considered, based on the results in [F. Coquet, Y. Hu, J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory Related Fields 123 (2002) 1-27]. Under a natural continuous assumption for the nonlinear expectation, which weakens the domination assumption in [F. Coquet, Y. Hu, J. Memin, S. Peng, Filtration consistent nonlinear expectations and related g-expectation, Probab. Theory Related Fields 123 (2002) 1-27], the author obtains the necessary and sufficient conditions under which Jensen's inequality for filtration consistent nonlinear expectation holds in general, respectively on scalar function and bivariate function. These two results generalize the known results on Jensen's inequality for g-expectation in [Z. Chen, R. Kulperger, L. Jiang, Jensen's inequality for g-expectation: Part 1, C. R. Acad. Sci. Paris Ser. I 337 (11) (2003) 725-730; Z. Chen, R. Kulperger, L. Jiang, Jensen's inequality for g-expectation: Part 2, C. R. Acad. Sci. Paris Ser. I 337 (12) (2003) 797-800; L. Jiang, On Jensen's inequality of bivariate function for g-expectation, J. Shandong Univ. 38 (5) (2003) 13-22 (in Chinese); L. Jiang, Z. Chen, On Jensen's inequality for g-expectation, Chinese Ann. Math. Ser. B 25 (3) (2004) 401-412; L. Jiang, Jensen's inequality for backward stochastic differential equation, Chinese Ann. Math. Ser. B 27 (5) (2006) 553-564; S. Fan, Jensen's inequality for g-expectation on convex (concave) function, Chinese Ann. Math. Ser. A 27 (5) (2006) 635-644 (in Chinese)].     

Applications of integration by parts formula for infinite-dimensional semimartingales

This work is devoted to the study of the behavior of stochastic evolution equations, obtained from stochastic flows, under the multiplicative transformations corresponding to the flows of Cameron-Martin transformations.     

Exit problems for nonlinear stochastic evolution equations on Hilbert spaces

This paper extends exit theorems of Da Prato and Zabczyk to nonconstant diffusion coefficients. It uses extensively general, exponential estimates due to Peszat.     

Control of an extending nonlinear elastic cable with an active vibration control strategy

An active control strategy based on the fuzzy sliding mode control (FSMC) is developed in this research for controlling the large-amplitude vibrations of an extending nonlinear elastic cable. The geometric nonlinearity of the cable and the fixed–fixed boundary of the cable are considered. For effectively and accurately control the motion of the cable with the active control strategy developed, the governing equation of the elastic cable is established and transformed into a multi-dimensional dynamic system with the 3rd order Galerkin method. The active control strategy is developed on the basis of the dynamic system, and the control strategy is applicable to multi-dimensional dynamic systems. In the numerical simulation, large-amplitude vibrations of the cable are effectively controlled with the control strategy. The results of the research demonstrate significances for controlling the cable vibrations of an elevator in practice.     

Exit asymptotics for small diffusion about an unstable equilibrium

A dynamical system perturbed by white noise in a neighborhood of an unstable fixed point is considered. We obtain the exit asymptotics in the limit of vanishing noise intensity. This is a refinement of a result by Kifer (1981).     

Nonlinear filtering equations for two-parameter semimartingales

Nonlinear filtering equations for two-parameter semimartingales of a Brownian sheet are obtained by an extended reference probability method, also applicable to the known Gaussian linear models.     

Exponential stability for nonlinear filtering

We study the a.s. exponential stability of the optimal filter w.r.t. its initial conditions. A bound is provided on the exponential rate (equivalently, on the memory length of the filter) for a general setting both in discrete and in continuous time, in terms of Birkhoff's contraction coefficient. Criteria for exponential stability and explicit bounds on the rate are given in the specific cases of a diffusion process on a compact manifold, and discrete time Markov chains on both continuous and discrete-countable state spaces. A similar question regarding the optimal smoother is investigated and a stability criterion is provided.     

The laws of large numbers for Pareto-type random variables under sub-linear expectation

In this paper, some laws of large numbers are established for random variables that satisfy the Pareto distribution, so that the relevant conclusions in the traditional probability space are extended to the sub-linear expectation space. Based on the Pareto distribution, we obtain the weak law of large numbers and strong law of large numbers of the weighted sum of some independent random variable sequences.     

Large deviation principle for stochastic integrals and stochastic differential equations driven by infinite-dimensional semimartingales

The paper concerns itself with establishing large deviation principles for a sequence of stochastic integrals and stochastic differential equations driven by general semimartingales in infinite-dimensional settings. The class of semimartingales considered is broad enough to cover Banach space-valued semimartingales and the martingale random measures. Simple usable expressions for the associated rate functions are given in this abstract setup. As illustrated through several concrete examples, the results presented here provide a new systematic approach to the study of large deviation principles for a sequence of Markov processes.     

Validity of nonlinear geometric optics with times growing logarithmically

The profiles (a.k.a. amplitudes) which enter in the approximate solutions of nonlinear geometric optics satisfy equations, sometimes called the slowly varying amplitude equations, which are simpler than the original hyperbolic systems. When the underlying problem is conservative one often finds that the amplitudes are defined for all time and are uniformly bounded. The approximations of nonlinear geometric optics typically have percentage error which tends to zero uniformly on bounded time intervals as the wavelength tends to zero. Under suitable hypotheses when the amplitude is uniformly bounded in space and time we show that the percentage error tends to zero uniformly on time intervals which grow logarithmically. The proof relies in an essential way on the fact that one has a corrector to the leading term of geometric optics.

     

Some conditional expectation identities for the multivariate normal

We give formulas for the conditional expectations of a product of multivariate Hermite polynomials with multivariate normal arguments. These results are extended to include conditional expectations of a product of linear combination of multivariate normals. A unified approach is given that covers both Hermite and modified Hermite polynomials, as well as polynomials associated with a matrix whose eigenvalues may be both positive and negative.     

EXISTENCE OF SOLUTIONS OF NONLINEAR TWO AND THREE-POINT BOUNDARY VALUE PROBLEMS FOR FOURTH ORDER NONLINEAR DIFFERENTIAL EQUATIONS

In this paper, the two and three-point boundsry problems (with nonlinear boundary conditions)for the genaral noniinear equations of fourth order are discussed.We have set some grups of the assurnpion coditions and proved the existence of solutins for corresponding boundary value problems under these conditons.     

Integral global optimization method for solution of nonlinear complementarity problems

The mapping in a nonlinear complementarity problem may be discontinuous. The integral global optimization algorithm is proposed to solve a nonlinear complementarity problem with a robust piecewise continuous mapping. Numerical examples are given to illustrate the effectiveness of the algorithm.     

Semilinear evolution equations with nonlinear constraints and applications

This paper deals with a general class of evolution problems for semilinear equations coupled with nonlinear constraints. Those constraints may contain compositions of nonlinear operators and unbounded linear operators, and hence the associated operators are not necessarily formulated in the form of continuous perturbations of linear operators. Accordingly, a family of equivalent norms is introduced to discuss 'quasidissipativity' in a local sense of the operators and a generation theory for nonlinear semigroups is employed to construct solution operators on bounded sets. It is a feature of our treatment that the resultant solution operators are obtained as nonlinear semigroups on the whole space which are not 'quasicontractive' but locally equi-Lipschitz continuous.     

Synchronization of nonlinear systems with delays via periodically nonlinear intermittent control

In this paper, the synchronization for a class of nonlinear chaotic systems with delays is proposed by using periodically intermittent nonlinear feedback control. Some synchronization criteria are derived based on Lyapunov functional theory and several differential inequalities such as Halanay inequality. As a special case, some sufficient conditions are obtained to ensure the synchronization of nonlinear systems without delays. Finally, some numerical simulations are presented to verify the theoretical results.     

Asymptotic linearization and stabilization for a class of nonlinear systems

This paper presents a nonlinear output feedback which asymptotically linearizes the class of nonlinear, continuous-time, affine in the control systems having relative degree equal to the state space dimension. Moreover, we show that any set of eigenvalues can be assigned for the asymptotic closed-loop linear system. The controller is based on a nonlinear observer, presented in a previous paper, and on the linearizing state feedback proposed by Isidori and computed in the estimated state. The main result obtained is equivalent to the separation theorem in the linear case.This work was supported by the Italian Ministry for University and Scientific and Technological Research.     

非线性抛物方程在非线性边界条件下解的爆破时间的下界

研究了非线性抛物方程在非线性边界条件下的解的爆破问题,通过构造一个能量表达式,运用微分不等式的方法,得到该能量表达式所满足的微分不等式,然后通过积分得到当爆破发生时解在非线性边界条件下的爆破时间的下界.