on the construction of suitable solutions to the navier stokes equations and questions regarding the definition of large eddy simulation abstract this paper discusses the notion of large eddy simulation les as a mathematical concept and proposes to use the concept of suitable weak solutions as a building block for the development of a mathematical theory of les various techniques for constructing suitable weak solutions to the three dimensional navier stokes equations are reviewed and many features of these mathematical constructions are shown to comply with well accepted heuristic characterizations of les

partial regularity of a generalized solution to the navier stokes equations in exterior domain in this note we prove two regularity theorems for solutions to the navier stokes equations of an i b v p in exterior domains namely we prove that the sets of the singular points of a solution if not empty has at most 1 hausdorff measureh1 s 0 moreover the sets is enclosed in a sphere of rayr for anyt 0 these results are obtained as corollaries to the partial regularity results furnished in 2

simulated moments estimation of markov models of asset prices this paper provides a simulated moments estimator sme of the parameters of dynamic models in which the state vector follows a time homogeneous markov process conditions are provided for both weak and strong consistency as well as asymptotic normality various tradeoffs among the regularity conditions underlying the large sample properties of the sme are discussed in the context of an asset pricing model this paper provides conditions for the consistency and asymptotic normality of a simulated moments estimator sme of the parameters of asset pricing models with time homogeneous markov representations of the stochastic forc ing process sme s for economic models have been proposed by mcfadden 1989 and pakes and pollard 1989 for i i d environments and by lee and ingram 1991 for a time series environment the sme for time series models examined in this paper is as follows the state vector yt that determines asset prices is assumed to follow a time homogeneous markov process whose transi tion function depends on an unknown parameter vector 3 0 asset prices and possibly other relevant data are observed as f yt 0 for some given function f of the underlying state and parameter vector in parallel a simulated state process y is generated analytically or numerically from the economic model and corresponding simulated observations f yj3 13 are taken for a given parameter choice f3 the parameter is chosen so as to match moments that is to minimize the distance between sample moments of the data f y 8030 and those of the simulated series f yt f3 in a sense to be made precise the proposed sme extends the generalized method of moments gmm estimator hansen 1982 to a large class of asset pricing models for which the moment restrictions of interest do not have analytic representations in terms of observable variables and the unknown parameter vector we provide conditions on the transition function of yt and the observation function f under which the sme of 030 is consistent and characterize the normalized asymptotic distribu tion of the estimator for two reasons neither the regularity conditions underly ing hansen s 1982 analysis of gmm estimators for time series models without

special solutions from the variable separation approach the davey stewartson equation a variable separation procedure for the davey stewartson ds equation is proposed by using a prior ansatz to its bilinear form the reduced equations for two variable separated fields have the same trilinear form although they possess different independent variables the trilinear equation can be changed to a spacetime symmetric form and can be solved by means of a boussinesq type equation system whenever a pair of solutions of the reduced fields are obtained a corresponding solution of the ds equation can be obtained algebraically the single dromion solution and some kinds of positon solutions are obtained explicitly

a monte carlo approach to nonnormal and nonlinear state space modeling abstract a solution to multivariate state space modeling forecasting and smoothing is discussed we allow for the possibilities of nonnormal errors and nonlinear functionals in the state equation the observational equation or both an adaptive monte carlo integration technique known as the gibbs sampler is proposed as a mechanism for implementing a conceptually and computationally simple solution in such a framework the methodology is a general strategy for obtaining marginal posterior densities of coefficients in the model or of any of the unknown elements of the state space missing data problems including the k step ahead prediction problem also are easily incorporated into this framework we illustrate the broad applicability of our approach with two examples a problem involving nonnormal error distributions in a linear model setting and a one step ahead prediction problem in a situation where both the state and observational equations are nonlinear and involve unknown parameters

boundedness for large x of suitable weak solutions of the navier stokes equations with prescribed velocity at infinity we consider time dependent perturbationsu of r finn s stationary pr solution of the navier stokes equations which converges to a constant vectorv as x for a given time interval t we find a radiusk such thatu is essentially bounded on t x k

a direct method for dromion solutions of the davey stewartson equations and their asymptotic properties a new direct approach to determining a class of solutions including the dromion solutions of the davey stewartson equations is presented the form of solution obtained allows a detailed asymptotic analysis of the dromion solutions and compact expression for the phase shifts and changes of amplitude as a result of interaction of the dromions to be determined a class of solutions describing dromion scattering is discussed and computer plots of the interactions of such solutions are presented

maximum likelihood estimation of time inhomogeneous diffusions we extend the maximum likelihood estimation method of ait sahalia econometrica 70 2002 223 for time homogeneous diffusions to time inhomogeneous ones we derive a closed form approximation of the likelihood function for discretely sampled time inhomogeneous diffusions and prove that this approximation converges to the true likelihood function and yields consistent parameter estimates monte carlo simulations for several financial models reveal that our method largely outperforms other widely used numerical procedures in approximating the likelihood function furthermore parameter estimates produced by our method are very close to the parameter estimates obtained by maximizing the true likelihood function and superior to estimates obtained from the euler approximation

variable separation approach for a differential difference system special toda equation a bilinear variable separation approach is used to construct some special solutions for a differential difference toda equation the semi discrete form of the continuous formula which describes some types of special solutions for many 2 1 dimensional continuous systems is found for a suitable quantity of the differential difference toda equation thus abundant semi discrete localized coherent structures are constructed by appropriately selecting the arbitrary functions

dromion can be remote controlled the one dromion solution of the davey stewartson 1 equations with driving boundaries is analysed numerically it is shown that the dromion can follow the motion of the crosspoint of the mean flows even if it is not of exact solutions moreover it is found that the localized structure keeps its shape when it is exposed to the forced motion of the mean flows in the case where dromion runs on a circle path the decay ratio of the localized structure versus the radius is given several types of movement of mean flows are also studied it is expected that dromion can be controlled arbitrarily by driving mean flows at the boundaries

a new proof of the caffarelli kohn nirenberg theorem here we give a self contained new proof of the partial regularity theorems for solutions of incompressible navier stokes equations in three spatial dimensions these results were originally due to scheffer and caffarelli kohn and nirenberg our proof is much more direct and simpler 1998 john wiley sons inc

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

a quasi maximum likelihood method for estimating the parameters of multivariate diffusions a quasi maximum likelihood procedure for estimating the parameters of multi dimensional diffusions is developed in which the transitional density is a multivariate gaussian density with first and second moments approximating the true moments of the unknown density for affine drift and diffusion functions the moments are exactly those of the true transitional density and for nonlinear drift and diffusion functions the approximation is extremely good and is as effective as alternative methods based on likelihood approximations the estimation procedure generalises to models with latent factors a conditioning procedure is developed that allows parameter estimation in the absence of proxies

extended multilinear variable separation approach and multivalued localized excitations for some 2 1 dimensional integrable systems the multilinear variable separation approach and the related universal formula have been applied to many 2 1 dimensional nonlinear systems starting from the universal formula abundant 2 1 dimensional localized excitations have been found in this paper the universal formula is extended in two different ways one is obtained for the modified nizhnik novikov veselov equation such that two universal terms can be combined linearly and this type of extension is also valid for the 2 1 dimensional symmetric sine gordon system the other is for the dispersive long wave equation the broer kaup kupershmidt system the higher order broer kaup kupershmidt system and the burgers system where arbitrary number of variable separated functions can be involved because of the existence of the arbitrary functions in both the original universal formula and its extended forms the multivalued functions can be used to construct a new type of localized excitations folded solitary waves fsws and foldons the fsws

on solutions of the camassa holm equation the techniques that have been developed for the application of the inverse scattering transform method to the solution of the camassa holm equation principally by constantin are implemented we use this approach first to represent the known solitary wave solution in a simple parametric form and then second to obtain the general two and three soliton solutions these latter two solutions require rather extensive use of mathematical packages mathematica and maple in order to complete the construction of solutions a number of examples are presented the phase shifts evident after an interaction are found and the special limit that recovers the peakon solutions is discussed

some novel nonlinear coherent excitations of the davey stewartson system exact solutions of many integrable 2 1 2 spatial and 1 temporal dimensional systems of nonlinear evolution equations e g the davey stewartson model can be obtained by a special separation of variables procedure by choosing the jacobi elliptic functions as the building blocks exact doubly periodic solutions are obtained analytically here two sets of elliptic functions with two different independent moduli are employed and the resulting wave packets are expressed as rational functions of elliptic functions by taking the long wave limit in one spatial variable peculiar wave patterns localized in one direction but periodic in the other direction will arise by taking the long wave limit in both spatial variables exponentially localized wave patterns which differ from the known dromions will result the boundary conditions relating to these localized structures are studied

maximum likelihood estimation of partially observed diffusion models this paper develops a maximum likelihood ml method to estimate partially observed diffusion models based on data sampled at discrete times the method combines two techniques recently proposed in the literature in two separate steps in the first step the closed form approach of ait sahalia 2008 is used to obtain a highly accurate approximation to the joint transition probability density of the latent and the observed states in the second step the efficient importance sampling technique of richard and zhang 2007 is used to integrate out the latent states thereby yielding the likelihood function using both simulated and real data we show that the proposed ml method works better than alternative methods the new method does not require the underlying diffusion to have an affine structure and does not involve infill simulations therefore the method has a wide range of applicability and its computational cost is moderate

regularity and singularity in solutions of the three dimensional navier stokes equations higher moments of the vorticity field m t in the form of l 2 m norms graphic 1 are used to explore the regularity problem for solutions of the three dimensional incompressible navier stokes equations on the domain graphic 2 it is found that the set of quantities formula 3 provide a natural scaling in the problem resulting in a bounded set of time averages d m t on a finite interval of time 0 t the behaviour of d m 1 d m is studied on what are called good and bad intervals of 0 t which are interspersed with junction points neutral i for large but finite values of m with large initial data m 0 o gr 4 it is found that there is an upper bound formula 4 which is punctured by infinitesimal gaps or windows in the vertical walls between the good bad intervals through which solutions may escape while this result is consistent with that of leray leray 1934 acta math 63 193 248 doi 10 1007 bf02547354 5 and scheffer scheffer 1976 pacific j math 66 535 552 this estimate for m corresponds to a length scale well below the validity of the navier stokes equations 1 embed inline graphic 1 gif 2 embed inline graphic 2 gif 3 embed graphic 1 gif 4 embed graphic 2 gif 5 http dx doi org doi 10 1007 bf02547354

analytic approach to the problem of convergence of truncated levy flights towards the gaussian stochastic process an analytic expression for characteristic function defining a truncated l evy flight is derived it is shown that the characteristic function yields results in agreement with recent simulations of truncated l evy flights by mantegna and stanley phys rev lett 73 2946 1994 with the analytic expression for the characteristic function the convergence of the l evy process towards the gaussian is demonstrated without simulations in the calculation of first return probability the simulations are replaced by numerical integration using simple quadratures

on series representations of infinitely divisible random vectors general results on series representations involving arrival times in a poisson process are established for infinitely divisible banach space valued random vectors without gaussian components applying these results various generalization of lepage s representation are obtained in a unified way certain conditionally gaussian infinitely divisible random vectors are characterized and some problems related to a gaussian randomization method are investigated

revisitation of the localized excitations of the 2 1 dimensional kdv equation in a previous paper lou s y 1995 j phys a math gen 28 7227 a generalized dromion structure was revealed for the 2 1 dimensional kdv equation which was first derived by boiti et al boiti m leon j j p manna m and pempinelli f 1986 inverse problems 2 271 using the idea of the weak lax pair in this paper using a backlund transformation and the variable separation approach we find there exist much more abundant localized structures for the 2 1 dimensional kdv equation the abundance of the localized structures of the model is introduced by the entrance of an arbitrary function of the seed solution some special types of dromion solution lumps breathers instantons and the ring type of soliton are discussed by selecting the arbitrary functions appropriately the dromion solutions can be driven by sets of straight line and curved line ghost solitons the dromion solutions may be located not only at the cross points of the lines but also at the closed points of the curves the breathers may breathe both in amplitude and in shape

cnoidal waves as exact sums of repeated solitary waves new series for elliptic functions via poisson summation of the corresponding fourier series it is shown that the cnoidal wave of the korteweg de vries equation can be written as an infinite sum of periodically repeated solitary waves a result first proved by toda through a different method similar series of hyperbolic secant or tangent functions are also derived for the elliptic dn sn and cn functions which make it possible to express the cnoidal wave of almost any evolution equation in terms of a series of the corresponding solitary waves as a by product new series are also obtained for the complete elliptic integral k and modulus k

on partial regularity results for the navier stokes equations looking at the regularity results of scheffer respectively caffarelli kohn and nirenberg from a new point of view indicates that estimates for the pressure do not play an essential role in partial regularity results for the navier stokes equations

interactions among periodic waves and solitary waves of the n 1 dimensional sine gordon field exact solutions of the n 1 dimensional sine gordon field equation are studied with help of those of the cubic nonlinear klein gordon fields the mapping relations among the sine gordon field equation and the cubic nonlinear klein gordon fields are pure algebraic by solving the cubic nonlinear klein gordon equations many new types of exact explicit solutions such as the periodic periodic interaction waves periodic kink interaction waves periodic perturbed snake shape solitary waves etc are displayed both analytically and graphically

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

on partial regularity for weak solutions to the navier stokes equations in this paper we study the partial regularity of the general weak solution u l 0 t l2 l2 0 t h1 to the navier stokes equations which include the well known leray hopf weak solutions it is shown that there is a absolute constant e such that for the weak solution u if either the scaled local lq 1 q 2 norm of the gradient of the solution or the scaled local lq 1 q 103 norm of u is less than e then u is locally bounded this implies that the one dimensional hausdorff measure is zero for the possible singular point set which extends the corresponding result due to caffarelli et al comm pure appl math 35 1982 717 to more general weak solution

non gaussian ornstein uhlenbeck based models and some of their uses in financial economics non gaussian processes of ornstein uhlenbeck ou type offer the possibility of capturing important distributional deviations from gaussianity and for flexible modelling of dependence structures this paper develops this potential drawing on and extending powerful results from probability theory for applications in statistical analysis their power is illustrated by a sustained application of ou processes within the context of finance and econometrics we construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive ou processes and we study these models in relation to financial data and theory

the euler scheme for levy driven stochastic differential equations in relation with monte carlo methods to solve some integro differential equations we study the approximation problem of ee g x t by ee g bar x t n where x t 0 leq t leq t is the solution of a stochastic differential equation governed by a levy process z t bar x n t is defined by the euler discretization scheme with step fractn with appropriate assumptions we show that the error ee g x t ee g bar x t n can be expanded in powers of frac1n if the levy measure of z has finite moments of order high enough otherwise the rate of convergence is slower and its speed depends on the behavior of the tails of the levy measure the simulation of the increments of z t is also discussed

extremes and local dependence in stationary sequences extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction a a weak mixing condition restricting long range dependence b a local condition restricting the clustering of high level exceedances