higher order implicit strong numerical schemes for stochastic differential equations higher order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed these are based on a stochastic taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero the regions of absolute stability of these implicit and related explicit methods are also examined

strong convergence of a tamed theta scheme for nsddes with one sided lipschitz drift abstract this paper is concerned with strong convergence of a tamed theta scheme for neutral stochastic differential delay equations with one sided lipschitz drift strong convergence rate is revealed under a global one sided lipschitz condition while for a local one sided lipschitz condition the tamed theta scheme is modified to ensure the well posedness of implicit numerical schemes then we show the convergence of the numerical solutions

the composite euler method for stiff stochastic differential equations in this paper we present the composite euler method for the strong solution of stochastic differential equations driven by d dimensional wiener processes this method is a combination of the semi implicit euler method and the implicit euler method at each step either the semi implicit euler method or the implicit euler method is used in order to obtain better stability properties we give criteria for selecting the semi implicit euler method or the implicit euler method for the linear test equation the convergence properties of the composite euler method depend on the criteria for selecting the methods numerical results suggest that the convergence properties of the composite euler method applied to nonlinear sdes is the same as those applied to linear equations the stability properties of the composite euler method are shown to be far superior to those of the euler methods and numerical results show that the composite euler method is a very promising method

a manifold independent approach to understanding transport in stochastic dynamical systems abstract we develop a new collection of tools aimed at studying stochastically perturbed dynamical systems specifically in the setting of bi stability that is a two attractor system it has previously been numerically observed that a small noise volume is sufficient to destroy would be zero noise case barriers in the phase space pseudo barriers thus creating a pre heteroclinic tangency chaos like behavior the stochastic dynamical system has a corresponding frobenius perron operator with a stochastic kernel which describes how densities of initial conditions move under the noisy map thus in studying the action of the frobenius perron operator we learn about the transport of the map we have employed a galerkin ulam like method to project the frobenius perron operator onto a discrete basis set of characteristic functions to highlight this action localized in specified regions of the phase space graph theoretic methods allow us to re order the resulting finite dimensional markov operator approximation so as to highlight the regions of the original phase space which are particularly active pseudo barriers of the stochastic dynamics our toolbox allows us to find 1 regions of high activity of transport 2 flux across pseudo barriers and also 3 expected time of escape from pseudo basins some of these quantities are also possible via the manifold dependent stochastic melnikov method but melnikov only applies to a very special class of models for which the unperturbed homoclinic orbit is available our methods are unique in that they can essentially be considered as a black box of tools which can be applied to a wide range of stochastic dynamical systems in the absence of a priori knowledge of manifold structures we use here a model of childhood diseases to showcase our methods our tools will allow us to make specific observations of 1 loss of reducibility between basins with increasing noise 2 identification in the phase space of active regions of stochastic transport 3 stochastic flux which essentially completes the heteroclinic tangle

convergence of numerical solutions to neutral stochastic delay differential equations with markovian switching recently numerical solutions of stochastic differential equations have received a great deal of attention it is surprising that there are not any numerical methods established for neutral stochastic delay differential equations yet in the paper the euler maruyama method for neutral stochastic delay differential equations is developed the key aim is to show that the numerical solutions will converge to the true solutions under the local lipschitz condition

mean square stability of two classes of theta method for neutral stochastic differential delay equations in this paper a stochastic linear theta slt method is introduced and analyzed for neutral stochastic differential delay equations nsddes we give some conditions on neutral item drift and diffusion coefficients which admit that the diffusion coefficient can be highly nonlinear and does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the slt method with 1 2 1 is asymptotically mean stable and so is 0 1 2 under a stronger assumption furthermore we consider the split step theta sst method and obtain a similar but better result that is the sst method with 1 2 1 is exponentially mean stable and so is 0 1 2 finally two numerical examples are given to show the efficiency of the obtained results

approximation of upper lyapunov exponents of bilinear stochastic differential systems a bilinear stochastic differential system is considered whose solution starting from x is denoted by x t x it is supposed that its upper lyapunov exponent lambda exists the purpose of the paper is to propose an efficient algorithm to approximate lambda it is shown that for markov chains bar x p h x p in mathbb n defined by several approximation schemes the lyapunov exponents bar lambda h are well defined and can be computed estimates are given in terms of the discretization step h of the theoretical error lambda bar lambda h then these results are applied to an engineering problem the stability of the motion of helicopter rotor blades in a turbulent wind

stability in distribution of neutral stochastic functional differential equations with markovian switching abstract stability in distribution of stochastic differential equations with markovian switching and stochastic differential delay equations with markovian switching have been studied by several authors and this kind of stability is an important property for stochastic systems there are several papers which study this stability for stochastic differential equations with markovian switching and stochastic differential delay equations with markovian switching technically in our paper we are concerned with the general neutral stochastic functional differential equations with markovian switching and we derive the sufficient conditions for stability in distribution at the end of our paper one example is established to illustrate the theory of our work

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

on the rate of convergence to equilibrium in one dimensional systems we determine the essential spectral radius of the perron frobenius operator for piecewise expanding transformations considered as an operator on the space of functions of bounded variation and relate the speed of convergence to equilibrium in such one dimensional systems to the greatest eigenvalues of generalized perron frobenius operators of the transformations operators which yield singular invariant measures

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

response operators for markov processes in a finite state space radius of convergence and link to the response theory for axiom a systems using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state markov processes the results can be used for studying empirically constructed e g from observations or through coarse graining of model simulations finite state approximation of statistical mechanical systems recent results concerning the convergence of the statistical properties of finite state markov approximation of the full asymptotic dynamics on the srb measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of axiom a system our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language plus providing bounds on the radius of convergence of the perturbative theory in particular we relate the convergence of the response theory to the rate of mixing of the unperturbed system one can use the formulas derived for finite state markov processes to recover previous findings obtained on the response of continuous time axiom a dynamical systems to perturbations by considering the generator of time evolution for the measure and for the observables a very basic low tech and computationally cheap analysis of the response of the lorenz 63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state markov processes to compute the system s response

code construction for the t user noiseless adder channel a general method for constructing codes for the t user noiseless adder channel is proposed which achieves the largest sum rate known the authors consider a multiple access communication system in this system t statistically independent sources which use binary block codes c sub 1 c sub t of equal length n simultaneously transmit information via one common channel maintaining synchronization with respect to words and bits the output of the adder channel is the t l ary vector which is the component wise arithmetic sum of the transmitted binary vectors the task of the decoder is to uniquely determine the messages of all users

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

almost sure exponential stability of numerical solutions for stochastic delay differential equations using techniques based on the continuous and discrete semimartingale convergence theorems this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations sddes the important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of sddes directly this is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the chebyshev inequality and the borel cantelli lemma

disjoint uniquely decodable codebooks for noiseless synchronized multiple access adder channels generated by integer sets disjoint uniquely decodable codebooks for noiseless synchronized multiple access adder channels are studied codebooks are applicable to multiple access systems in which the number of transmitting users varies over the life of the system several designs of codebooks based on certain integer sets from 0 1 2 3 2 sup n 1 are presented they are restricted to codebooks which have the rate sum above one and simple and efficient decoding algorithms

exponential stability in mean square of neutral stochastic differential functional equations abstract the aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d x t g xt f t x t g t xt dt t xt dw t where xt x t s s 0 with 0 is the past history of the solution several interesting examples are a given for illustration

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

an analysis of stability of milstein method for stochastic differential equations with delay this paper deals with the adapted milstein method for solving linear stochastic delay differential equations it is proved that the numerical method is mean square ms stable under suitable conditions the obtained result shows that the method preserves the stability property of a class of linear constant coefficient problems this is also verified by several numerical examples

superimposed codes in r sup n the authors introduce the concept of superimposed codes in euclidean n space r sup n an asymptotic existence bound is derived for such codes the proof uses the idea of random coding in particular the asymptotic properties of long codes are studied it is shown that the derived existence bound differs only by a factor of four from a nonexistence bound obtained by a simple sphere packing argument

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

response and sensitivity using markov chains dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties a prominent real life example of this class of problems is the investigation of climate response to perturbations in this respect it is crucial to determine what the linear response of a system is as a quantification of sensitivity alongside previous work here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations by projecting the transfer operator onto a suitable finite dimensional vector space one is able to obtain matrix representations which determine finite markov processes further using perturbation theory for markov matrices it is possible to determine the linear and nonlinear response of the system given a prescribed forcing here we suggest a methodology which puts the scope on the evolution law of densities the liouville fokker planck equation allowing to effectively calculate the sensitivity and response of two representative dynamical systems

new criteria on exponential stability of neutral stochastic differential delay equations neutral stochastic differential delay equations nsddes have recently been studied intensively see e g v b kolmanovskii v r nosov stability and periodic modes of control systems with aftereffect nauka moscow 1981 x mao exponential stability in mean square of neutral stochastic differential functional equations systems control lett 26 1995 245 251 x mao razumikhin type theorems on exponential stability of neutral stochastic functional differential equations siam j math anal 28 2 1997 389 401 x mao stochastic differential equations and their applications horwood publishing chichester 1997 more recently mao asymptotic properties of neutral stochastic differential delay equations stochastics and stochastics rep 68 2000 273 295 provided with some useful criteria on the exponential stability for nsddes however the criteria there require not only the coefficients of the nsddes to obey the linear growth condition but also the time delay to be a constant one of our aims in this paper is to remove these two restrictive conditions moreover the key condition on the diffusion operator associated with the underlying nsdde will take a much more general form our new stability criteria not only cover many highly non linear nsddes with variable time delays but they can also be verified much more easily than the known criteria

asymptotically efficient runge kutta methods for a class of itoˆ and stratonovich equations in certain applications of stochastic differential equations approximate solutions must be found that depend only on samples of the driving process it is known that the order of convergence of such approximations is limited and that some are asymptotically efficient in the sense that they minimize the leading coefficient in the expansion of mean square errors as power series in the sample step size this article develops asymptotically efficient runge kt tta methods that involve evaluations either of the coefficients of an ito equation or of the coefficients of the corresponding stratonovich equation simpler approximations which converge with the maximum possible order but which are not asymptotically efficient are also defined the runge kutta methods for ito equations differ from those designed for ordinary differential equations in that they involve terms in the square root of the sample step size the approximations are tested along with the classical euler method on five examples the simulatio

exponential mean square stability of numerical methods for systems of stochastic differential equations this paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split step theta method for stochastic systems first we consider linear autonomous systems under a sufficient and necessary condition for exponential mean square stability of the exact solution it is proved that the two classes of theta methods with q 0 5 are exponentially mean square stable for all positive step sizes and the methods with q 0 5 still unconditionally preserves the exponential mean square stability of the underlying systems but the stochastic theta method does not have this property finally we consider stochastic differential equations with jumps some similar results are derived

numerical integration of multiplicative noise stochastic differential equations a two stage runge kutta algorithm for vector ito and by transform also stratonovich stochastic differential equations with multiplicative noise has been developed the method is second order accurate but for vanishing drift the algorithm yields a martingale independent of step size several examples are included to illustrate our method a discussion of errors shows that sample size can be as important as truncation

superimposed codes for the multiaccess binary adder channel superimposed codes for multiple access communication in a binary adder channel are analyzed the superposition mechanism used in this correspondence is ordinary addition each user is assigned a codeword from a superimposed code it is proved that every constant weight code c of weight w and maximal correlation c corresponds to a subclass of a disjunctive code d of order m

numerical solutions of neutral stochastic functional differential equations this paper examines the numerical solutions of neutral stochastic functional differential equations nsfdes d x t u x t f x t dt g x t dw t t geq 0 the key contribution is to establish the strong mean square convergence theory of the euler maruyama approximate solution under the local lipschitz condition the linear growth condition and contractive mapping these conditions are generally imposed to guarantee the existence and uniqueness of the true solution so the numerical results given here are obtained under quite general conditions although the way of analysis borrows from x mao lms j comput math 6 2003 pp 141 161 to cope with u x t several new techniques have been developed

discretization of the frobenius perron operator using a sparse haar tensor basis the sparse ulam method the global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated frobenius perron operator for systems with low dimensional long term dynamics efficient techniques exist for a numerical approximation of the most important eigenfunctions cf m dellnitz and o junge siam j numer anal 36 1999 pp 491 515 they are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor ulam s method in this paper we develop a numerical technique which makes ulam s approach applicable to systems with higher dimensional long term dynamics it is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids c zenger sparse grids in parallel algorithms for partial differential equations kiel 1990 vieweg braunschweig 1991 pp 241 251 h j bungartz and m griebel acta numer 13 2004 pp 147 269 here we use a sparse haar tensor basis as the underlying approximation space we develop the technique establish statements about its complexity and convergence and present two numerical examples

a posteriori error estimates for markov approximations of frobenius perron operators abstract we present lower and upper error bounds under the bv norm for piecewise linear markov finite approximations to frobenius perron operators in terms of the bv norm of the residuals so that a posteriori error estimates can be given numerical results using the monte carlo implementation are presented to support the theoretical analysis

numerical approximations to the stationary solutions of stochastic differential equations in this paper we investigate the possibility of approximating the stationary solution of a stochastic differential equation sde we start with the random dynamical system generated by the sde with the multiplicative noise we prove that the pullback flow has a stationary point however the stationary point is not constructible explicitly therefore we look at the numerical approximation we prove that the discrete time random dynamical system also has a stationary point finally we prove mean square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes as well as almost surely convergence when the time step is rational

razumikhin type exponential stability criteria of neutral stochastic functional differential equations the paper discusses both pth moment and almost sure exponential stability of solutions to neutral stochastic functional differential equations and neutral stochastic differential delay equations by using the razumikhin type technique the main goal is to find sufficient stability conditions that could be verified more easily then by using the usual method with lyapunov functionals the analysis is based on paper x mao razumikhin type theorems on exponential stability of neutral stochastic functional differential equations siam j math anal 28 2 1997 389 401 referring to mean square and almost sure exponential stability

choice of θ and mean square exponential stability in the stochastic theta method of stochastic differential equations this paper examines the relationship of choice of and mean square exponential stability in the stochastic theta method stm of stochastic differential equations sdes and mainly includes the following three results i under the linear growth condition for the drift term when 0 1 2 the stm may preserve the mean square exponential stability of the exact solution but the counterexample shows that the stm cannot reproduce this stability without this linear growth condition ii when 1 2 1 without the linear growth condition for the drift term the stm may reproduce the mean square exponential stability of the exact solution but the bound of the lyapunov exponent cannot be preserved iii when 1 this stm is called as the backward euler maruyama bem method the stm can reproduce not only the mean square exponential stability but also the bound of the lyapunov exponent this paper also gives the sufficient and necessary conditions of the mean square exponential stability of the stm for the linear sde when 0 1 2 and 1 2 1 respectively and the simulations also illustrate these theoretical results

almost sure convergence rate of θ em scheme for neutral sddes abstract this paper is concerned with almost sure convergence of em scheme for neutral stochastic differential delay equations under local lipschitz conditions with 1 2 1 it is revealed that the strong convergence rate under local lipschitz condition is 1 2 while almost sure convergence rate is less than 1 2 provided the local lipschitz constant valid on balls of radius r grows not faster than log r

a multiuser k ary code for the noisy multiple access adder channel multiuser k ary coding is proposed for a noisy multiple access adder channel it is shown that when a t user spl delta decodable k ary code c is given a priori a qt user spl lambda spl delta decodable k ary code c sup is obtained by using a matrix such as a hadamard matrix or a conference matrix of order q where spl lambda is a positive integer depending on the matrix more noteworthy is that the code c is an arbitrary spl delta decodable k ary code and that the coding scheme preserves the total rate i e the total rate of c sup is equal to that of c

on the approximation of invariant measures given a discrete dynamical system defined by the map x x the density of the absolutely continuous a c invariant measure if it exists is the fixed point of the frobenius perron operator defined on l1 x ulam proposed a numerical method for approximating such densities based on the computation of a fixed point of a matrix approximation of the operator t y li proved the convergence of the scheme for expanding maps of the interval g keller and m blank extended this result to piecewise expanding maps of the cube in n we show convergence of a variation of ulam s scheme for maps of the cube for which the frobenius perron operator is quasicompact we also give sufficient conditions on for the existence of a unique fixed point of the matrix approximation and if the fixed point of the operator is a function of bounded variation we estimate the convergence rate

exponential mean square stability of the theta approximations for neutral stochastic differential delay equations in this paper a split step theta sst method is introduced and analyzed for neutral stochastic differential delay equations nsddes it is proved that the sst method with 0 1 2 can recover the exponential mean square stability of the exact solution with some restrictive conditions on stepsize and the drift coefficient but for 1 2 1 the sst can reproduce the exponential mean square stability unconditionally then based on the stability results of sst scheme we examine the exponential mean square stability of the stochastic linear theta slt approximation for nsddes and obtain the similar stability results to that of the sst method moreover for sufficiently small stepsize we show that the decay rate as measured by the lyapunov exponent can be reproduced arbitrarily accurately these results show that different values of theta will drastically affect the exponential mean square stability of the two classes of theta approximations for nsddes

a family of k 1 ary signature codes for noisy multiple access adder channel a coding scheme of k 1 ary error correcting signature codes for a noisy multiple access adder channel is proposed given a signature matrix a and a difference matrix d d d a priori a larger signature matrix is obtained by replacing each element in the hadamard matrix with a or d or d depending on the values of the elements and their locations in the hadamard matrix the set of rows in the proposed matrix gives an error correcting signature code introducing a difference matrix makes it possible to construct an error correcting signature code whose sum rate is increased with an increase in the order of the hadamard matrix either binary or non binary signature codes are constructed when the pairs of matrices a and d are given

exponential stability of the split step θ method for neutral stochastic delay differential equations with jumps abstract the exponential mean square stability of the split step method for neutral stochastic delay differential equations nsddes with jumps is considered new conditions for jumps are proposed to ensure the exponential mean square stability of the trivial solution if the drift coefficient satisfies the linear growth condition it is shown that the split step method can reproduce the exponential mean square stability of the trivial solution for the constrained stepsize then by applying the chebyshev inequality and the borel cantelli lemma the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained since split step method covers euler maruyama em method and split step backward euler ssbe method the conclusions are valid for these two methods moreover they can adapt to the nsddes and the sddes with jumps finally a numerical example illustrates the effectiveness of the theoretical results

computing the invariant measure and the lyapunov exponent for one dimensional maps using a measure preserving polynomial basis we consider a generalisation of ulam s method for approximating invariant densities of one dimensional maps rather than use piecewise constant polynomials to approximate the density we use polynomials of degree n which are defined by the requirement that they preserve the measure on n 1 neighbouring subintervals over the whole interval this results in a discontinuous piecewise polynomial approximation to the density we prove error results where this approach is used to approximate smooth densities we also consider the computation of the lyapunov exponent using the polynomial density and show that the order of convergence is one order better than for the density itself together with using cubic polynomials in the density approximation this yields a very efficient method for computing highly accurate estimates of the lyapunov exponent we illustrate the theoretical findings with some examples

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

finite approximations of markov operators in this paper we survey some recent developments in the numerical analysis of markov operators and in particular frobenius perron operators associated with chaotic discrete dynamical systems

statistically optimal almost invariant sets abstract chaotic dynamical systems are often transitive although this transitivity is sometimes very weak it is of interest to divide the phase space into large regions between which there is relatively little communication of trajectories we present fast simple algorithms to find such divisions the present work builds on the results of froyland and dellnitz g froyland m dellnitz detecting and locating near optimal almost invariant sets and cycles siam j sci comput 24 6 2003 1839 1863 focussing on a statistical description of transitivity that takes into account the fact that trajectories tend to visit different regions of phase space with different frequencies the new work takes advantage of theoretical results from the theory of reversible markov chains a new adaptive algorithm is put forward to efficiently deal with situations where the boundaries of the weakly communicating regions are complicated this algorithm is illustrated with the standard map relevant convergence results are proven

strong convergence of compensated split step theta methods for sdes with jumps under monotone condition abstract this paper is concerned with strong convergence of the compensated split step theta csst method for autonomous stochastic differential equations sdes with jumps under weaker conditions more precisely the diffusion coefficient and the drift coefficient are both locally lipschitz and the jump diffusion coefficient is globally lipschitz while they all satisfy the monotone condition it is proved that the csst method is strongly convergent of order 1 2 finally the obtained results are supported by numerical experiments

detecting and locating near optimal almost invariant sets and cycles the behaviors of trajectories of nonlinear dynamical systems are notoriously hard to characterize and predict rather than characterizing dynamical behavior at the level of trajectories we consider following the evolution of sets there are often collections of sets that behave in a very predictable way in spite of the fact that individual trajectories are entirely unpredictable such special collections of sets are invisible to studies of long trajectories we describe a global set oriented method to detect and locate these large dynamical structures our approach is a marriage of new ideas in modern dynamical systems theory and the novel application of graph dissection algorithms

spreading set with error correction for multiple access adder channel the necessary and sufficient condition for constructing a spreading set with decodability is investigated it is proved that for a given delta decodable spreading set and a qtimesq square matrix h with components 1 or 1 a qdelta decodable spreading set s is obtained if and only if h is a hadamard matrix in addition a decoding rule with error correction and message data detection is provided

estimating long term behavior of flows without trajectory integration the infinitesimal generator approach the long term distributions of trajectories of a flow are described by invariant densities i e fixed points of an associated transfer operator in addition global slowly mixing structures such as almost invariant sets which partition phase space into regions that are almost dynamically disconnected can also be identified by certain eigenfunctions of this operator indeed these structures are often hard to obtain by brute force trajectory based analyses in a wide variety of applications transfer operators have proven to be very efficient tools for an analysis of the global behavior of a dynamical system the computationally most expensive step in the construction of an approximate transfer operator is the numerical integration of many short term trajectories in this paper we propose to directly work with the infinitesimal generator instead of the operator completely avoiding trajectory integration we propose two different discretization schemes a cell based discretization and a spectral coll

approximations of solutions to neutral functional differential equations with nonlocal history conditions abstract this work is concerned with a class of neutral functional differential equations with nonlocal history conditions in a hilbert space the approximation of solution to a class of such problems is studied moreover the convergence of faedo galerkin approximation of solution is shown for illustration an example is worked out

the projection method for computing multidimensional absolutely continuous invariant measures we present an algorithm for numerically computing an absolutely continuous invariant measure associated with a piecewisec2 expanding mappings on a bounded region rn the method is based on the galerkin projection principle for solving an operator equation in a banach space with the help of the modern notion of functions of bounded variation in multidimension we prove the convergence of the algorithm