on large deviations in the averaging principle for sdes with a full dependence we establish the large deviation principle for stochastic differential equations with averaging in the case when all coefficients of the fast component depend on the slow one including diffusion

averaging principle for slow fast stochastic differential equations with time dependent locally lipschitz coefficients abstract this paper is devoted to studying the averaging principle for stochastic differential equations with slow and fast time scales where the drift coefficients satisfy local lipschitz conditions with respect to the slow and fast variables and the coefficients in the slow equation depend on time t and making use of the techniques of time discretization and truncation we prove that the slow component strongly converges to the solution of the corresponding averaged equation

a sequential algorithm for sparse support vector classifiers support vector machines svms though accurate are not preferred in applications requiring high classification speed or when deployed in systems of limited computational resources due to the large number of support vectors involved in the model to overcome this problem we have devised a primal svm method with the following properties 1 it solves for the svm representation without the need to invoke the representer theorem 2 forward and backward selections are combined to approach the final globally optimal solution and 3 a criterion is introduced for identification of support vectors leading to a much reduced support vector set in addition to introducing this method the paper analyzes the complexity of the algorithm and presents test results on three public benchmark problems and a human activity recognition application these applications demonstrate the effectiveness and efficiency of the proposed algorithm

averaging principle for multiscale stochastic fractional schrodinger korteweg de vries system this work concerns the averaging principle for multiscale stochastic fractional schrodinger korteweg de vries system stochastic averaging principle is a powerful tool for studying qualitative analysis of stochastic dynamical systems with different time scales to be more precise under suitable conditions the multiscale system can be reduced to a single stochastic fractional korteweg de vries equation averaged equation with a modified coefficient the slow component of multiscale system towards to the solution of the averaged equation in moment in order to prove the averaging principle there are two key points uniform bounds for the solution and holder continuity of time variable for the slow component for this we need a crucial property smoothing effect of the fractional korteweg de vries semigroup but the traditional method can t help us obtain this property such analysis does not follow in a straightforward manner from results already available in the mathematical literature on the contrary it requires the introduction of some new ideas and techniques we try to overcome this difficulty by the duality method the interpolation arguments the energy estimate method and refined inequality technique

a comparison of minimum mse and maximum power for the nearly integrated non gaussian model we study the optimal choice of quasi likelihoods for nearly integrated possibly non normal autoregressive models it turns out that the two most natural candidate criteria minimum mean squared error mse and maximum power against the unit root null give rise to different optimal quasi likelihoods in both cases the functional specification of the optimal quasi likelihood is the same it is a combination of the true likelihood and the gaussian quasi likelihood the latter is required even if the former is known the optimal relative weights however depend on the criterion chosen and are markedly different throughout we base our results on exact limiting distribution theory we derive a new explicit expression for the joint density of the minimal sufficient functionals of ornstein uhlenbeck processes which also has applications in other fields and we characterize its behavior for extreme values of its arguments using these results we derive the asymptotic power functions of statistics which converge weakly to combinations of these sufficient functionals finally we evaluate numerically our computationally efficient formulae then illustrate by means of simulation how our results extend to finite samples and what recommendations they imply for the importance assigned to the gaussian quasi likelihood component 2003 elsevier b v all rights reserved

averaging principle for one dimensional stochastic burgers equation abstract in this paper we consider the averaging principle for one dimensional stochastic burgers equation with slow and fast time scales under some suitable conditions we show that the slow component strongly converges to the solution of the corresponding averaged equation meanwhile when there is no noise in the slow component equation we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence 1 r for any 0 r 1

one dimensional stochastic burgers equation driven by levy processes abstract in this paper we proved the global existence and uniqueness of the strong weak and mild solutions for one dimensional burgers equation perturbed by a poisson form process a poisson form and q wiener process with the dirichlet bounded condition we also proved the existence of the invariant measure of these models

multiclass classification with multi prototype support vector machines winner take all multiclass classifiers are built on the top of a set of prototypes each representing one of the available classes a pattern is then classified with the label associated to the most similar prototype recent proposal of svm extensions to multiclass can be considered instances of the same strategy with one prototype per class the multi prototype svm proposed in this paper extends multiclass svm to multiple prototypes per class it allows to combine several vectors in a principled way to obtain large margin decision functions for this problem we give a compact constrained quadratic formulation and we propose a greedy optimization algorithm able to find locally optimal solutions for the non convex objective function this algorithm proceeds by reducing the overall problem into a series of simpler convex problems for the solution of these reduced problems an efficient optimization algorithm is proposed a number of pattern selection strategies are then discussed to speed up the optimization process in addition given the combinatorial nature of the overall problem stochastic search strategies are suggested to escape from local minima which are not globally optimal finally we report experiments on a number of datasets the performance obtained using few simple linear prototypes is comparable to that obtained by state of the art kernel based methods but with a significant reduction of one or two orders in response time

strong convergence rate in averaging principle for stochastic fitzhugh nagumo system with two time scales abstract this article deals with averaging principle for stochastic fitzhugh nagumo system with different time scales under suitable conditions the existence of an averaging equation eliminating the fast variable for this coupled system is proved and as a consequence the system can be reduced to a single stochastic ordinary equation with a modified coefficient moreover the rate of convergence for the slow component towards the solution of the averaging equation is of order 1 2

stochastic complexity and modeling on demontre un theoreme fondamental qui donne une borne inferieure pour la longueur de code et donc pour les erreurs de prediction on definit les notions d information a priori et d information utile dans les donnees

a sparsity driven kernel machine based on minimizing a generalization error bound a new sparsity driven kernel classifier is presented based on the minimization of a recently derived data dependent generalization error bound the objective function consists of the usual hinge loss function penalizing training errors and a concave penalty function of the expansion coefficients the problem of minimizing the non convex bound is addressed by a successive linearization approach whereby the problem is transformed into a sequence of linear programs the algorithm produced comparable error rates to the standard support vector machine but significantly reduced the number of support vectors and the concomitant classification time

analysis of multiscale methods for stochastic differential equations we analyze a class of numerical schemes proposed in 26 for stochastic differential equations with multiple time scales both advective and diffusive time scales are considered weak as well as strong convergence theorems are proven most of our results are optimal they in turn allow us to provide a thorough discussion on the efficiency as well as optimal strategy for the method c 2005 wiley periodicals inc

the limiting distribution of the autocorrelation coefficient under a unit root the limiting distribution of the normalized autocorrelation coefficient in the case of a unit root is given in a closed form it is found that high order transcendental functions such as the parabolic cylinder functions are indispensable to express this distribution thus departing from the simple standard normal distribution that arises in the case of a stable root using the formulae derived in this paper some numerical results available from previous studies are then extended and refined finally the formulae are manipulated analytically to explain the unusual shape of the distribution

interior point methods for massive support vector machines we investigate the use of interior point methods for solving quadratic programming problems with a small number of linear constraints where the quadratic term consists of a low rank update to a positive semidefinite matrix several formulations of the support vector machine fit into this category an interesting feature of these particular problems is the volume of data which can lead to quadratic programs with between 10 and 100 million variables and if written explicitly a dense q matrix our code is based on ooqp an object oriented interior point code with the linear algebra specialized for the support vector machine application for the targeted massive problems all of the data is stored out of core and we overlap computation and input output to reduce overhead results are reported for several linear support vector machine formulations demonstrating that the method is reliable and scalable

expansions for some confluent hypergeometric functions power series for the parabolic cylinder function dv z are derived from known addition theorems thus enabling a more compact and efficient expression of the function one of the expansions is applied to find an asymptotic approximation for the function as mod v mod to infinity with mod arg v mod pi next a large argument z asymptotic addition theorem is derived from an integral representation of dv z before extending this type of summation theorem to the two general confluent hypergeometric functions motivating mathematical applications to three integration problems are given as examples of the practical usefulness of the formulae

lagrangian support vector machines an implicit lagrangian for the dual of a simple reformulation of the standard quadratic program of a linear support vector machine is proposed this leads to the minimization of an unconstrained differentiable convex function in a space of dimensionality equal to the number of classified points this problem is solvable by an extremely simple linearly convergent lagrangian support vector machine lsvm algorithm lsvm requires the inversion at the outset of a single matrix of the order of the much smaller dimensionality of the original input space plus one the full algorithm is given in this paper in 11 lines of matlab code without any special optimization tools such as linear or quadratic programming solvers this lsvm code can be used as is to solve classification problems with millions of points for example 2 million points in 10 dimensional input space were classified by a linear surface in 82 minutes on a pentium iii 500 mhz notebook with 384 megabytes of memory and additional swap space and in 7 minutes on a 250 mhz ultrasparc ii processor with 2 gigabytes of memory other standard classification test problems were also solved nonlinear kernel classification can also be solved by lsvm although it does not scale up to very large problems it can handle any positive semidefinite kernel and is guaranteed to converge a short matlab code is also given for nonlinear kernels and tested on a number of problems

ergodicity of 2d navier stokes equations with random forcing and large viscosity the stochastically forced two dimensional incompressable navier stokes equations are shown to possess an unique invariant measure if the viscosity is taken large enough this result follows from a stronger result showing that at high viscosity there is a unique stationary solution which attracts solutions started from arbitrary initial conditions that is to say the system has a trivial random attractor along the way results controling the expectation and averaging time of the energy and enstrophy are given

robust linear and support vector regression the robust huber m estimator a differentiable cost function that is quadratic for small errors and linear otherwise is modeled exactly in the original primal space of the problem by an easily solvable simple convex quadratic program for both linear and nonlinear support vector estimators previous models were significantly more complex or formulated in the dual space and most involved specialized numerical algorithms for solving the robust huber linear estimator numerical test comparisons with these algorithms indicate the computational effectiveness of the new quadratic programming model for both linear and nonlinear support vector problems results are shown on problems with as many as 20000 data points with considerably faster running times on larger problems

large deviations and approximations for slow fast stochastic reaction diffusion equations abstract a large deviation principle is derived for a class of stochastic reaction diffusion partial differential equations with slow fast components the result shows that the rate function is exactly that of the averaged equation plus the fluctuating deviation which is a stochastic partial differential equation with small gaussian perturbation this result also confirms the effectiveness of the approximation of the averaged equation plus the fluctuating deviation to the slow fast stochastic partial differential equations

averaging principle for a class of stochastic reaction diffusion equations we consider the averaging principle for stochastic reaction diffusion equations under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component we calculate limiting slow motion the study of solvability of kolmogorov equations in hilbert spaces and the analysis of regularity properties of solutions allow to generalize the classical approach to finite dimensional problems of this type in the case of spde s

autocovariance functions of series and of their transforms abstract we derive a method to link exactly the autocovariance functions of two arbitrary instantaneous transformations of a time series for example this is useful when one wants to derive the autocovariance of the logarithm of a series from the known autocovariance of the original series and more generally when one wishes to describe the time series effects of applying a nonlinear transformation to a process whose properties are known as an illustration we provide two corollaries and three examples the first corollary is on the commonly used logarithmic transformation and is applied to a geometric auto regressive ar process as well as to a positive moving average ma process the second corollary is on the tan 1 transformation which will turn possibly unstable series into stable ones as an illustration we obtain the autocovariance function of the tan 1 of an arithmetic ar process this filter while always producing a bounded process preserves the stability instability distinction of the original series a feature that can be turned to an advantage in the design of tests we then present a probabilistic interpretation of the main features of the new autocovariance function we also provide a mathematical lemma on a general integral which is of independent interest

stochastic burgers equations and their semi classical expansions in this paper we use the hopf cole logarithmic transformation and the stochastic hamilton jacobi theory to study stochastic heat equations and burgers equations before the caustics the stochastic inviscid burgers equation gives the first term of our semi classical expansion i e the inviscid limit of the viscous stochastic burgers equation in order to push our results beyond the inviscid limit we construct solutions for iterated stochastic hamilton jacobi continuity equations then we give the semi classical asymptotic expansions for stochastic heat equations and burgers equations by using nelson s stochastic mechanical processes with drifts given by the solution of the iterated stochastic hamilton jacobi continuity equations the explicit formula for the remainder term is given by a path integral

strong convergence in stochastic averaging principle for two time scales stochastic partial differential equations abstract the theory of stochastic averaging principle provides an effective approach for the qualitative analysis of stochastic systems with different time scales and is relatively mature for stochastic ordinary differential equations in this paper we study the averaging principle for a class of stochastic partial differential equations with two separated time scales driven by scalar noises under suitable assumptions it is shown that the slow component strongly converges to the solution of the corresponding averaged equation

exactly soluble two state quantum models with linear couplings a class of exact analytic solutions of the time dependent schrequation is presented for a two state quantum system coherently driven by a nonresonant external field the coupling is a linear function of time with a finite duration and the detuning is constant four special models are considered in detail namely the shark double shark tent and zigzag models the exact solution is derived by rotation of the landau zener propagator at an angle of 4 and is expressed in terms of weber s parabolic cylinder function approximations for the transition probabilities are derived for all four models by using the asymptotics of the weber function these approximations demonstrate various effects of physical interest for each model

on stochastic diffusion equations and stochastic burgers equations in this paper we construct a strong solution for the stochastic hamilton jacobi equation by using stochastic classical mechanics before the caustics we thereby obtain the viscosity solution for a certain class of inviscid stochastic burgers equations this viscosity solution is not continuous beyond the caustics of the corresponding hamilton jacobi equation the hopf cole transformation is used to identify the stochastic heat equation and the viscous stochastic burgers equation the exact solutions for the above two equations are given in terms of the stochastic hamilton jacobi function under a no caustic condition we construct the heat kernel for the stochastic heat equation for zero potentials in hyperbolic space and for harmonic oscillator potentials in euclidean space thereby obtaining the stochastic mehler formula

an averaging principle for diffusions in foliated spaces consider an sde on a foliated manifold whose trajectories lay on compact leaves we investigate the effective behavior of a small transversal perturbation of order e an average principle is shown to hold such that the component transversal to the leaves converges to the solution of a deterministic ode according to the average of the perturbing vector field with respect to invariant measures on the leaves as e goes to zero an estimate of the rate of convergence is given these results generalize the geometrical scope of previous approaches including completely integrable stochastic hamiltonian system

average and deviation for slow fast stochastic partial differential equations abstract averaging is an important method to extract effective macroscopic dynamics from complex systems with slow modes and fast modes this article derives an averaged equation for a class of stochastic partial differential equations without any lipschitz assumption on the slow modes the rate of convergence in probability is obtained as a byproduct importantly the stochastic deviation between the original equation and the averaged equation is also studied a martingale approach proves that the deviation is described by a gaussian process this gives an approximation to errors of order o instead of order o attained in previous averaging

averaging principle for stochastic korteweg de vries equation abstract averaging principle is a powerful tool for studying qualitative analysis of nonlinear dynamical systems in this paper we will establish an averaging principle for stochastic korteweg de vries equation under a general averaging condition with the help of this averaging principle we can establish an effective approximation for the solution of stochastic korteweg de vries equation this can tell us the asymptotic behavior of the solution and make the interaction between nonlinearity uncertainty and multiple scales more clear in order to obtain this averaging principle we need to establish the smoothing effect for the third order operator

limit behavior of two time scale diffusions revisited abstract this work is concerned with diffusions with two time scales or singularly perturbed diffusions asymptotic expansions of the solution of the associated cauchy problem for parabolic partial differential equation are obtained and the desired error bounds are derived these asymptotic expansions are then used to analyze related limit distributions of normalized integral functionals

minimal kernel classifiers a finite concave minimization algorithm is proposed for constructing kernel classifiers that use a minimal number of data points both in generating and characterizing a classifier the algorithm is theoretically justified on the basis of linear programming perturbation theory and a leave one out error bound as well as effective computational results on seven real world datasets a nonlinear rectangular kernel is generated by systematically utilizing as few of the data as possible both in training and in characterizing a nonlinear separating surface this can result in substantial reduction in kernel data dependence over 94 in six of the seven public datasets tested on and with test set correctness equal to that obtained by using a conventional support vector machine classifier that depends on many more data points this reduction in data dependence results in a much faster classifier that requires less storage to eliminate data points the proposed approach makes use of a novel loss function the pound function which is a linear combination of the 1 norm and the step function that measures both the magnitude and the presence of any error