convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay this paper can be regarded as the continuation of the work contained in papers milosevic 2011 2013 at the same time it represents the extension of the paper wu et al 2010 in this paper the one sided lipschitz condition is employed in the context of the backward euler method for a class of neutral stochastic differential equations with constant delay sufficient conditions for this method to be well defined are revealed under certain nonlinear growth conditions the convergence in probability is established for the continuous forward backward euler method as well as for the discrete backward euler method additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without requiring for the drift coefficient to satisfy the linear growth condition this manuscript represents the continuation of the previous work of the author related to a class of highly nonlinear neutral stochastic differential equations with time dependent delay main results of this manuscript are convergence in probability and almost sure exponential stability of the backward euler approximate solution for a class of stochastic differential equations with constant delay conditions under which both the exact and approximate solutions share the property of the almost sure exponential stability are revealed this paper also illustrates that in some cases stochastic differential equations with constant delay should be studied separately from those with time dependent delay in that sense it should be stressed that results of this paper are obtained under weaker conditions comparing to those which would be obtained by replacing the time dependent delay by the constant delay

an explicit analytic approximation of solutions for a class of neutral stochastic differential equations with time dependent delay based on taylor expansion this paper represents a contribution to the analysis of approximate methods for stochastic differential equations based on the application of taylor expansion under the lipschitz and linear growth conditions the lp and almost sure convergence of the appropriate approximate solutions are considered for a class of neutral stochastic differential equations with time dependent delay coefficients of the approximate equations including the neutral term are taylor approximations of the coefficients of the initial equation up to the first derivatives for p 2 the rate of the lp convergence of the sequence of approximate solutions to the exact solution is estimated as 2 l p 1 2 l where l 1 is an integer the presence of the neutral term in the equation reflected to the rate of convergence

exponential stability of equidistant euler maruyama approximations of stochastic differential delay equations our aim is to study under what conditions the exact and numerical solution based on equidistant nonrandom partitions of integration time intervals to a stochastic differential delay equation sdde share the property of mean square exponential stability our approach is trying to avoid the use of lyapunov functions or functionals we show that under a global lipschitz assumption an sdde is exponentially stable in mean square if and only if for some sufficiently small stepsize d the euler maruyama em method is exponentially stable in mean square we then replace the global lipschitz condition with a finite time convergence condition and establish the same if and only if result the important feature of this result is that it transfers the asymptotic problem into a finite time convergence problem replacing the exact and em numerical solution with stochastic processes we also discuss whether a family of stochastic processes share the stability property this new approach allows us to discuss i whether a family of sddes share the stability property and ii whether an sdde with variable time lag shares stability property with the modified em method as another application of this general approach we consider a linear sdde with variable time lag and establish an if and only if result it should also be mentioned that the questions addressed results proved as well as style of analysis borrow heavily from 14 but the computations involved to cope with time delay are nontrivial

an analytic approximate method for solving stochastic integrodifferential equations abstract in this paper we compare the solution of a general stochastic integrodifferential equation of the ito type with the solutions of a sequence of appropriate equations of the same type whose coefficients are taylor series of the coefficients of the original equation the approximate solutions are defined on a partition of the time interval the rate of the closeness between the original and approximate solutions is measured in the sense of the l p norm so that it decreases if the degrees of these taylor series increase analogously to real analysis the convergence with probability one is also proved

discrete time approximations of stochastic delay equations the milstein scheme in this paper we develop a strong milstein approximation scheme for solving stochastic delay differential equations sddes the scheme has convergence order 1 in order to establish the scheme we prove an infinite dimensional ito formula for tame functions acting on the segment process of the solution of an sdde it is interesting to note that the presence of the memory in the sdde requires the use of the malliavin calculus and the anticipating stochastic analysis of nualart and pardoux given the nonanticipating nature of the sdde the use of anticipating calculus methods in the context of strong approximation schemes appears to be novel

profile quasi maximum likelihood estimation of partially linear spatial autoregressive models abstract we propose profile quasi maximum likelihood estimation of spatial autoregressive models that are partially linear the rate of convergence of the spatial parameter estimator depends on some general features of the spatial weight matrix of the model the estimators of other finite dimensional parameters in the model have the regular n rate of convergence and the estimator of the nonparametric component is consistent but with different restrictions on the choice of bandwidth parameter associated with different natures of the spatial weights monte carlo simulations verify our theory and indicate that our estimators perform reasonably well in finite samples

formulation and estimation of dynamic models using panel data abstract this paper presents a statistical analysis of time series regression models for longitudinal data with and without lagged dependent variables under a variety of assumptions about the initial conditions of the processes being analyzed the analysis demonstrates how the asymptotic properties of estimators of longitudinal models are critically dependent on the manner in which samples become large by expanding the number of observations per person holding the number of people fixed or by expanding the number of persons holding the number of observations per person fixed the paper demonstrates which parameters can and cannot be identified from data produced by different sampling plans

implicit numerical methods for highly nonlinear neutral stochastic differential equations with time dependent delay abstract this paper represents the continuation of the analysis from papers milosevic 2011 10 and milosevic 2013 11 the main aim of this paper is to establish certain results for the backward euler method for a class of neutral stochastic differential equations with time dependent delay for that purpose the split step backward euler method which represents an extension of the backward euler method is introduced for this class of equations conditions under which the split step backward euler method and thus the backward euler method is well defined are revealed moreover the convergence in probability of the backward euler method is proved under certain nonlinear growth conditions including the one sided lipschitz condition this result is proved using the technique which is based on the application of the continuous time approximation for this reason the discrete forward backward euler method is involved since it allows its continuous version to be well defined from the aspect of measurability the convergence in probability is established for the continuous forward backward euler solution which is essential for proving the same result for both discrete forward backward and backward euler methods additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without the linear growth condition on the drift coefficient of the equation as usual the whole consideration is affected by the presence and properties of the delay function

an approximate method via taylor series for stochastic functional differential equations abstract the subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals sufficiently smooth in the sense of frechet derivatives the approximate equations are defined on equidistant partitions of the time interval and their coefficients are general taylor expansions of the coefficients of the initial equation it will be shown that the approximate solutions converge in the l p norm and with probability one to the solution of the initial equation and also that the rate of convergence increases when degrees in taylor expansions increase analogously to real analysis

multirate generalized additive runge kutta methods this work constructs a new class of multirate schemes based on the recently developed generalized additive runge kutta gark methods sandu and gunther siam j numer anal 53 1 17 42 2015 multirate schemes use different step sizes for different components and for different partitions of the right hand side based on the local activity levels we show that the new multirate gark family includes many well known multirate schemes as special cases the order conditions theory follows directly from the gark accuracy theory nonlinear stability and monotonicity investigations show that these properties are inherited from the base schemes provided that additional coupling conditions hold

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

estimation of spatial autoregressive panel data models with fixed effects this paper establishes asymptotic properties of quasi maximum likelihood estimators for sar panel data models with fixed effects and sar disturbances a direct approach is to estimate all the parameters including the fixed effects because of the incidental parameter problem some parameter estimators may be inconsistent or their distributions are not properly centered we propose an alternative estimation method based on transformation which yields consistent estimators with properly centered distributions for the model with individual effects only the direct approach does not yield a consistent estimator of the variance parameter unless t is large but the estimators for other common parameters are the same as those of the transformation approach we also consider the estimation of the model with both individual and time effects

convex splitting rungekutta methods for phase field models in this paper we present the convex splitting rungekutta csrk methods which provide a simple unified framework to solve phase field models such as the allencahn cahnhilliard and phase field crystal equations the core idea of the csrk methods is the combination of convex splitting methods and multi stage implicitexplicit rungekutta methods our csrk methods are high order accurate in time and we investigate the energy stability numerically we present numerical experiments to show the accuracy and efficiency of the proposed methods up to the third order accuracy

unconditionally stable methods for gradient flow using convex splitting runge kutta scheme abstract we propose a convex splitting runge kutta csrk scheme which provides a simple unified framework to solve a gradient flow in an unconditionally gradient stable manner the key feature of the scheme is a combination of a convex splitting method and a specially designed multi stage two additive runge kutta method our methods are high order accurate in time and assure the gradient energy stability for any time step size we provide detailed proof of the unconditional energy stability and present issues on the practical implementations we demonstrate the accuracy and stability of the proposed methods using numerical experiments of the cahn hilliard equation

computing the ground state solution of bose einstein condensates by a normalized gradient flow in this paper we present a continuous normalized gradient flow cngf and prove its energy diminishing property which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of bose einstein condensates bec we also investigate the energy diminishing property for the discretization of the cngf two numerical methods are proposed for such discretizations one is the backward euler centered finite difference befd method the other is an explicit time splitting sine spectral tssp method energy diminishing for befd and tssp for the linear case and monotonicity for befd for both linear and nonlinear cases are proven comparison between the two methods and existing methods e g crank nicolson finite difference cnfd or forward euler finite difference fefd shows that befd and tssp are much better in terms of preserving the energy diminishing property of the cngf numerical results in one two and three dimensions with magnetic trap confinement potential as well as a potential of a stirrer corresponding to a far blue detuned gaussian laser beam are reported to demonstrate the effectiveness of befd and tssp methods furthermore we observe that the cngf and its befd discretization can also be applied directly to compute the first excited state solution in bec when the initial data is chosen as an odd function

existence uniqueness almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the euler maruyama approximation abstract in the first part of this paper the existence uniqueness and almost sure polynomial stability of solutions for pantograph stochastic differential equations are considered under nonlinear growth conditions the results are obtained using the idea from mao and rassias 2005 10 where stochastic differential equation with constant delay are considered however the presence of the unbounded delay in stochastic pantograph differential equations required certain modification of that idea moreover the convergence in probability of the appropriate euler maruyama solution is proved under the same nonlinear growth conditions adding the linear growth condition we show that the almost sure polynomial stability of the euler maruyama solution implies the almost sure polynomial stability of the exact solution this part of the paper represents the extension of the idea from wu et al 2010 17 the main novelty in this part of the paper is also related to the treatment of the unbounded delay in pantograph stochastic differential equations

an analytic approximation of solutions of stochastic differential equations abstract this paper is devoted to the construction of an approximate solution of the stochastic differential equation of the ito type defined on a partition of the time interval the coefficients of the equation by their taylor series up to arbitrary derivatives are approximated the closeness of the original and approximate solutions is measured in the sense of the lp norm and with probability one

a new semiparametric spatial model for panel time series abstract this paper presents a semiparametric model for large dimension vector time series whose elements correspond to economic agents dependence between agents variables is characterized using a spatial model functions of agents economic distances provide restrictions that enable estimation of a vector autoregressive specification we present sufficient conditions for our model to generate stationary mixing series with finite higher order moments we estimate the model using a simple two step sieve least squares procedure where the sieve estimators are constructed to preserve shape restrictions on the functions of economic distance e g positive definiteness of a covariance function we provide rates of convergence for the sieve estimators t limiting distributions for the model s finite dimensional parameters and a bootstrap method for inference in an illustrative application we use this model to characterize how the comovement in output growth across us industrial sectors depends on the similarity of sectors technologies we also present a small monte carlo evaluation of our estimators

qml estimation of dynamic panel data models with spatial errors abstract we propose quasi maximum likelihood qml estimation of dynamic panel models with spatial errors when the cross sectional dimension n is large and the time dimension t is fixed we consider both the random effects and fixed effects models and prove consistency and derive the limiting distributions of the qml estimators under different assumptions on the initial observations we propose a residual based bootstrap method for estimating the standard errors of the qml estimators monte carlo simulation shows that both the qml estimators and the bootstrap standard errors perform well in finite samples under a correct assumption on initial observations but may perform poorly when this assumption is not met

an application of taylor series in the approximation of solutions to stochastic differential equations with time dependent delay the subject of this paper is the analytic approximation method for solving stochastic differential equations with time dependent delay approximate equations are defined on equidistant partitions of the time interval and their coefficients are taylor approximations of the coefficients of the initial equation it will be shown without making any restrictive assumption for the delay function that the approximate solutions converge in l p norm and with probability 1 to the solution of the initial equation also the rate of the l p convergence increases when the degrees in the taylor approximations increase analogously to what is found in real analysis at the end a procedure will be presented which allows the application of this method with the assumption of continuity of the delay function

implicit numerical methods for neutral stochastic differential equations with unbounded delay and markovian switching abstract this paper contains results on the backward euler method for a class of neutral stochastic differential equations with both unbounded and bounded delays and markovian switching the convergence in probability of the backward euler method is proved under nonlinear growth conditions including the one sided lipschitz condition in order for the backward euler method to be well defined the presence of the neutral term which is hybrid that is it depends on the markov chain is essential for consideration of these equations it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without the linear growth condition on the drift coefficient

an energy stable runge kutta method for convex gradient problems abstract we propose a class of runge kutta methods which provide a simple unified framework to solve the gradient flow of a convex functional in an unconditionally energy stable manner stiffly accurate runge kutta methods are high order accurate in terms of time and also assure the energy stability for any time step size when they satisfy the positive definite condition we provide a detailed proof of the unconditional energy stability as well as unique solvability of the proposed scheme we demonstrate the accuracy and stability of the proposed methods using numerical experiments for a specific example