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

generalized two step maruyama methods for stochastic differential equations abstract in this paper we propose generalized two step maruyama methods for solving ito stochastic differential equations numerical analysis concerning consistency convergence and numerical stability in the mean square sense is presented we derive sufficient and necessary conditions for linear mean square stability of the generalized two step maruyama methods we compare the stability region of the generalized two step maruyama methods of adams type with that of the corresponding two step maruyama methods of adams type and show that our proposed methods have better linear mean square stability a numerical example is given to confirm our theoretical results

mean square stability of second order runge kutta methods for stochastic differential equations in a previous paper we proposed the stochastic generalization of classical second order two stage explicit runge kutta rk methods the obtained stochastic schemes have second order in the weak sense in this paper the numerical stability of these rk schemes is studied the study focuses on stability with respect to the second moment ms stability figures of the stability domains of the numerical schemes are shown numerical examples that confirm the theoretical results are also presented

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

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

universal resolutions for np complete problems abstract agrawal and biswas 1992 define a notion stronger than np completeness with every language x in np is associated a polynomial time verifiable binary relation y called a resolution so that x is in x if and only if there exists y which size is a polynomial function of the size of x and x y is in y such an y is called a solution of x for x if y and y are resolutions associated with x and x a solution preserving reduction of y to y is a reduction of x to x so that the solutions of any instance for x can be quickly recovered from the solutions of the image of the instance under the reduction a resolution y is called universal if there exists a solution preserving reduction from every resolution to y then manindra agrawal and somenath biswas give a theorem that help us to show that a resolution is universal without searching for reduction we generalize this definition and this theorem for languages over an arbitrary structure and in particular over the reals as it was defined by blum et al 1989 we then study examples with neural networks

numerical solution of stochastic differential problems in the biosciences stochastic differential equations sdes models play a prominent role in many application areas including biology epidemiology and population dynamics mostly because they can offer a more sophisticated insight through physical phenomena than their deterministic counterparts do so suitable numerical methods must be introduced to simulate the solutions of the resulting stochastic differential systems in this work we take into account both euler taylor expansion and runge kutta type methods for stochastic ordinary differential equations sodes and the euler maruyama method for stochastic delay differential equations sddes focusing on the most relevant implementation issues the corresponding matlab codes for both sodes and sddes problems are tested on mathematical models arising in the biosciences math comput simulation numerical analysis of a campbell like stochastic delay model preprint

the αth moment stability for the stochastic pantograph equation in this paper we investigate the ath moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the razumikhin technique especially the linear stochastic pantograph equations and the semi implicit euler method applying them are considered the convergence result of the semi implicit euler method is obtained the stability conditions of the analytic solution of those equations and the numerical method are given finally some experiments are given

stabilization of hybrid stochastic differential equations by feedback control based on discrete time state observations recently mao 2013 discusses the mean square exponential stabilization of continuous time hybrid stochastic differential equations by feedback controls based on discrete time state observations mao 2013 also obtains an upper bound on the duration between two consecutive state observations however it is due to the general technique used there that the bound on is not very sharp in this paper we will consider a couple of important classes of hybrid sdes making full use of their special features we will be able to establish a better bound on our new theory enables us to observe the system state less frequently so costs less but still to be able to design the feedback control based on the discrete time state observations to stabilize the given hybrid sdes in the sense of mean square exponential stability

numerical solutions of stochastic differential delay equations under the generalized khasminskii type conditions the classical existence and uniqueness theorem of the solution to a stochastic differential delay equation sdde requires the local lipschitz condition and the linear growth condition see e g 11 12 and 20 the numerical solutions under these conditions have also been discussed intensively see e g 4 10 13 16 17 18 21 22 and 24 recently mao and rassias 14 and 15 established the generalized khasminskii type existence and uniqueness theorems for sddes where the linear growth condition is no longer imposed these generalized khasminskii type theorems cover a wide class of highly nonlinear sddes but these nonlinear sddes do not have explicit solutions whence numerical solutions are required in practice however there is so far little numerical theory on sddes under these generalized khasminskii type conditions the key aim of this paper is to close this gap

almost sure asymptotic stability of drift implicit θ methods for bilinear ordinary stochastic differential equations in r 1 global almost sure asymptotic stability of stochastic methods with nonrandom variable step sizes when applied to bilinear nonautonomous homogeneous test systems of ordinary stochastic differential equations sdes is investigated sufficient conditions for almost sure asymptotic stability are proved for both analytical and numerical solutions in r1 the results of saito and mitsui world sci ser appl math 2 1993 333 siam j numer anal 33 1996 2254 higham siam j numer anal 38 2001 753 and schurz stochastic anal appl 14 1996 313 handbook of stochastic analysis and applications 2002 for the constant step sizes are carried over to the case with variable step sizes and nonautonomous linear test equations the investigations indicate that methods with variable step sizes or variable parameter governed by certain conditions can successfully be used to guarantee almost sure asymptotic stability while discretizing nonautonomous sdes

strong convergence rates of modified truncated em methods for neutral stochastic differential delay equations abstract the aim of this paper is to investigate strong convergence of modified truncated euler maruyama method for neutral stochastic differential delay equations introduced in lan 2018 strong convergence rates of the given numerical scheme to the exact solutions at fixed time t are obtained under local lipschitz and khasminskii type conditions moreover convergence rates over a time interval 0 t are also obtained under additional polynomial growth condition on g without the weak monotonicity condition which is usually the standard assumption to obtain the convergence rate two examples are presented to interpret our conclusions

convergence and stability of the compensated split step theta method for stochastic differential equations with piecewise continuous arguments driven by poisson random measure abstract this paper deals with the numerical solutions of stochastic differential equations with piecewise continuous arguments sdepcas driven by poisson random measure in which the coefficients are highly nonlinear it is shown that the compensated split step theta csst method with 0 1 is strongly convergent in p th p 2 moment under some polynomially lipschitz continuous conditions it is also obtained that the convergence order is close to 1 p in terms of the stability it is proved that the csst method with 1 2 1 reproduces the exponential mean square stability of the underlying system under the monotone condition and some restrictions on the step size without any restriction on the step size there exists 1 2 1 such that the csst method with 1 is exponentially stable in mean square moreover if the drift and jump coefficients satisfy the linear growth condition the csst method with 0 1 2 also preserves the exponential mean square stability some numerical simulations are presented to verify the conclusions

pp is as hard as the polynomial time hierarchy in this paper two interesting complexity classes pp and oplus text p are compared with ph the polynomial time hierarchy it is shown that every set in ph is polynomial time turing reducible to a set in pp and ph is included in text bp cdot oplus text p as a consequence of the results it follows that text pp subseteq text ph or oplus text p subseteq text ph implies a collapse of ph a stronger result is also shown every set in pp ph is polynomial time turing reducible to a set in pp

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

convergence and stability of the split step θ method for stochastic differential equations in this paper we construct a new split step method for solving stochastic differential equations namely the split step q method under lipschitz and linear growth conditions we establish a mean square convergence theory of split step q approximate solutions moreover the mean square stability of the method for a linear test equation with real parameters is considered and the real mean square stability region is plotted finally numerical results are presented to demonstrate the efficiency of the split step q method

the existence of quantum fields for local nets of algebras of observables a necessary and sufficient criterion for the existence of quantum fields related to a given local net of algebras of observables is proved this criterion is written in terms of the conjugation operators so of the local algebras a o it shows that the modular structure of the local algebras essentially determines the corresponding quantum field theory in the case where the local net of algebras is conformally covariant the criterion gives the result that the existence of quantum fields depends only on the representation of the conformal group and does not depend on the concrete form of the algebras for the proof of the criterion the fact that a quantum field related to a local net of algebras can be essentially characterized by one vector is used

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 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

delay dependent stability analysis of numerical methods for stochastic delay differential equations this paper is concerned with the numerical solution of stochastic delay differential equations the focus is on the delay dependent stability of numerical methods for a linear scalar test equation with real coefficients by using the so called root locus technique the full asymptotic stability region in mean square of stochastic theta methods is obtained which is characterized by a sufficient and necessary condition in terms of the drift and diffusion coefficients as well as time stepsize and method parameter theta then this condition is compared with the analytical stability condition it is proved that the backward euler method completely preserves the asymptotic mean square stability of the underlying system and the euler maruyama method preserves the instability of the system our investigation also shows that not all theta methods with q 12 preserve this delay dependent stability some numerical examples are presented to confirm the theoretical results

mean square stability of two classes of theta methods for numerical computation and simulation of delayed stochastic hopfield neural networks abstract recently the investigation on the stability of the numerical solutions to delayed stochastic differential equations has received an increasing attention but there has been little work on the stability analysis of the numerical solutions to delayed stochastic hopfield neural networks dshnns so far the aim in this paper is to study the mean square exponential stability of the split step theta sst method and the stochastic linear theta slt method for the underlying model it is proved that for any 0 1 2 there exists a constant 0 depending on such that the numerical schemes produced by the sst method and the slt method are mean square exponentially stable for 0 under the same assumptions as those to guarantee the mean square exponential stability of the underlying continuous model for the case 1 2 1 we show the same stability conclusion for all 0 to carry out the required conclusion a novel technique for the stability analysis of discrete numerical schemes with multi time delays namely the weighted sum lyapunov functional method is proposed finally a numerical example is given to illustrate the application of the suggested methods and to verify the stability conclusions obtained

diagonally drift implicit runge kutta methods of strong order one for stiff stochastic differential systems the class of strong stochastic runge kutta methods for stochastic differential equations sdes due to ro sler is considered based on the order conditions of this class we design a family of strong diagonally drift implicit stochastic runge kutta ddisrk methods of order one for solving ito sde systems with an m dimensional standard wiener process we then explicitly provide the structure of the mean square stability matrix of the new ddisrk methods for the general form of linear sde systems with some specific values of the coefficients for this family particular ddisrk methods that have good stability properties are proposed in order to check their convergence order and stability properties some numerical experiments are performed

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

numerical solutions of stochastic differential equations implementation and stability issues stochastic differential equations sdes arise from physical systems where the parameters describing the system can only be estimated or are subject to noise there has been much work done recently on developing numerical methods for solving sdes this paper will focus on stability issues and variable stepsize implementation techniques for numerically solving sdes effectively

theta schemes for sddes with non globally lipschitz continuous coefficients this paper establishes the boundedness convergence and stability of the two classes of theta schemes namely split step theta sst scheme and stochastic linear theta slt scheme for stochastic differential delay equations sddes with non globally lipschitz continuous coefficients when the drift f x y satisfies one sided lipschitz condition with respect to the present state x and the diffusion g x y obeys the global lipschitz condition with respect to the present term x but the delay terms y in the drift and diffusion may be highly nonlinear this paper first examines the strong convergence rates of the theta schemes for sddes it is also proved that the two classes of theta schemes for 1 2 1 converge strongly to the exact solution with the order 1 2 but for 0 1 2 the linear growth condition on drift f x y in x is needed for the strong convergence rates the exponential mean square stability of the theta schemes with 1 2 1 is also investigated for highly nonlinear sddes

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

technical communique stabilization of continuous time hybrid stochastic differential equations by discrete time feedback control in this paper we are concerned with the mean square exponential stabilization of continuous time hybrid stochastic differential equations also known as stochastic differential equations with the markovian switching by discrete time feedback controls although the stabilization by continuous time feedback controls for such equations has been discussed by several authors see e g ji and chizeck 1990 mao lam and huang 2008 mao yin and yuan 2007 wu shi and gao 2010 wu su and shi 2012 there is so far no result on the stabilization by discrete time feedback controls our aim here is to initiate the study in this area by establishing some new results

runge kutta methods for numerical solution of stochastic differential equations the way to obtain deterministic runge kutta methods from taylor approximations is generalized for stochastic differential equations now by means of stochastic truncated expansions about a point for sufficiently smooth functions of an ito process a class of explicit runge kutta schemes of second order in the weak sense for systems of stochastic differential equations with multiplicative noise is developed also two runge kutta schemes of third order have been obtained for scalar equations with constant diffusion coefficients numerical examples that compare the proposed schemes to standard ones are presented

mean square stability and almost sure exponential stability of two step maruyama methods of stochastic delay hopfield neural networks abstract in this paper the two step maruyama methods of stochastic delay hopfield neural networks are studied we have found that under what choices of step size the two step maruyama methods of stochastic delay hopfield networks maintain the stability of exact solutions the mean square stability of two step maruyama methods of stochastic delay hopfield neural networks is investigated under suitable conditions also the almost sure exponential stability of two step maruyama methods of stochastic delay hopfield networks is proved using the semi martingale convergence theorem further the comparisons of stability conditions to the previous results in liu and zhu 2015 rathinasamy 2012 and ronghua et al 2010 are given numerical experiments are provided to illustrate our theoretical results

asymptotic stability of runge kutta methods for nonlinear differential equations with piecewise continuous arguments this paper deals with the asymptotic stability of numerical solutions for differential equations with piecewise continuous arguments epcas the necessary and sufficient condition is given for non confluent runge kutta methods to preserve the stability of nonlinear scalar epcas as for systems we prove that some algebraically stable methods can preserve the asymptotic stability

on the connection between quantum fields and von neumann algebras of local operators the relationship between a standard local quantum field and a net of local von neumann algebras is discussed two natural possibilities for such an association are identified and conditions for these to obtain are found it is shown that the local net can naturally be so chosen that it satisfies the special condition of duality the notion of an intrinsically local field operator is introduced and it is shown that such an operator defines a local net with which the field is locally associated a regularity condition on the field is formulated and it is shown that if this condition holds then there exists a unique local net with which the field is locally associated if and only if the field algebra contains at least one intrinsically local operator conditions under which a field and other fields in its borchers class are associated with the same local net are found in terms of the regularity condition mentioned

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

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

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

asymptotic moment boundedness of the numerical solutions of stochastic differential equations few papers look at the asymptotic boundedness of numerical solutions of stochastic differential equations sdes one of the open questions is whether numerical approximations can reproduce the boundedness property of the underlying sdes in this paper we give positive answer to this question firstly we discuss the asymptotic moment upper bound of the ito type sdes and show that the euler maruyama em method is capable to preserve the boundedness property for sdes with the linear growth condition on both drift and diffusion coefficients but under the weaker assumption the one sided lipschitz on the drift coefficient the em method fails to work we then show that the backward em method can work in this situation

mean square stability properties of an adaptive time stepping sde solver we consider stability properties of a class of adaptive time stepping schemes based upon the milstein method for stochastic differential equations with a single scalar forcing in particular we focus upon mean square stability for a class of linear test problems with multiplicative noise we demonstrate that desirable stability properties can be induced in the numerical solution by the use of two realistic local error controls one for the drift term and one for the diffusion

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

neutral stochastic functional differential equations with additive perturbations the paper deals with the solution to the neutral stochastic functional differential equation whose coefficients depend on small perturbations by comparing it with the solution to the corresponding unperturbed equation of the equal type we give conditions under which these solutions are close in the 2m th mean on finite time intervals and on intervals whose length tends to infinity as small perturbations tend to zero

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

t stability of the semi implicit euler method for delay differential equations with multiplicative noise the paper deals with the t stability of the semi implicit euler method for delay differential equations with multiplicative noise a difference equation is obtained by applying the numerical method to a linear test equation in which the wiener increment is approximated by a discrete random variable with two point distribution the conditions under which the method is t stable are considered and the numerical experiments are given

ms stability of the euler maruyama method for stochastic differential delay equations the aim of this paper is to investigate the stability of the euler maruyama method for the stochastic differential equations with time delay the definition of ms stability of numerical methods is established the condition of the mean square stability of analytical solution is obtained for a linear scalar system with multiplicative noise and the ms stability of the numerical scheme is proved

strong convergence of split step backward euler method for stochastic differential equations with non smooth drift in this paper we are concerned with the numerical approximation of stochastic differential equations with discontinuous nondifferentiable drifts we show that under one sided lipschitz and general growth conditions on the drift and global lipschitz condition on the diffusion a variant of the implicit euler method known as the split step backward euler ssbe method converges with strong order of one half to the true solution our analysis relies on the framework developed in d j higham x mao and a m stuart strong convergence of euler type methods for nonlinear stochastic differential equations siam journal on numerical analysis 40 2002 1041 1063 and exploits the relationship which exists between explicit and implicit euler methods to establish the convergence rate results

almost sure and moment exponential stability in the numerical simulation of stochastic differential equations relatively little is known about the ability of numerical methods for stochastic differential equations sdes to reproduce almost sure and small moment stability here we focus on these stability properties in the limit as the timestep tends to zero our analysis is motivated by an example of an exponentially almost surely stable nonlinear sde for which the euler maruyama em method fails to reproduce this behavior for any nonzero timestep we begin by showing that em correctly reproduces almost sure and small moment exponential stability for sufficiently small timesteps on scalar linear sdes we then generalize our results to multidimensional nonlinear sdes we show that when the sde obeys a linear growth condition em recovers almost surely exponential stability very well under the less restrictive condition that the drift coefficient of the sde obeys a one sided lipschitz condition where em may break down we show that the backward euler method maintains almost surely exponential stability

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

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

mean square exponential stability for stochastic control systems with discrete time state feedbacks and their numerical schemes in simulation aiming at the qualitative numerical simulation this article concerns the mean square exponential stability mses of the stochastic systems with discrete time state feedbacks and their numerical schemes a time delay dependent stability criterion is obtained for the underlying system the euler maruyama and backward euler maruyama schemes are established and their mses are proved under the same criterion which means that the numerical schemes preserve the stability of the continuous model the illustrative example at the end of this letter shows that the theoretical upper bound for the sampling period limited in the stability criterion is much greater than that obtained by the stability criteria from the literature

strong and weak divergence in finite time of euler s method for stochastic differential equations with non globally lipschitz continuous coefficients the stochastic euler scheme is known to converge to the exact solution of a stochastic differential equation sde with globally lipschitz continuous drift and diffusion coefficients recent results extend this convergence to coefficients that grow at most linearly for superlinearly growing coefficients finite time convergence in the strong mean square sense remains in this article we answer this question to the negative and prove for a large class of sdes with non globally lipschitz continuous coefficients that euler s approximation converges neither in the strong mean square sense nor in the numerically weak sense to the exact solution at a finite time point even worse the difference of the exact solution and of the numerical approximation at a finite time point diverges to infinity in the strong mean square sense and in the numerically weak sense

stability of analytical and numerical solutions of nonlinear stochastic delay differential equations this paper concerns the stability of analytical and numerical solutions of nonlinear stochastic delay differential equations sddes we derive sufficient conditions for the stability contractivity and asymptotic contractivity in mean square of the solutions for nonlinear sddes the results provide a unified theoretical treatment for sddes with constant delay and variable delay including bounded and unbounded variable delays then the stability contractivity and asymptotic contractivity in mean square are investigated for the backward euler method it is shown that the backward euler method preserves the properties of the underlying sddes the main results obtained in this work are different from those of razumikhin type theorems indeed our results hold without the necessity of constructing or finding an appropriate lyapunov functional

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

local algebras of observables and pointlike localized fields we present a method to recover wightman fields from a haag kastler theory of local observables this may provide a basis for the comparison of different theories and for an algebraic description of high energy behaviour

high weak order methods for stochastic differential equations based on modified equations inspired by recent advances in the theory of modified differential equations we propose a new methodology for constructing numerical integrators with high weak order for the time integration of stochastic differential equations this approach is illustrated with the constructions of new methods of weak order two in particular semi implicit integrators well suited for stiff mean square stable stochastic problems and implicit integrators that exactly conserve all quadratic first integrals of a stochastic dynamical system numerical examples confirm the theoretical results and show the versatility of our methodology

stabilization of hybrid systems by feedback control based on discrete time state observations recently mao automatica j ifac 49 2013 pp 3677 3681 initiated the study the mean square exponential stabilization of continuous time hybrid stochastic differential equations by feedback controls based on discrete time state observations in the same paper mao also obtains an upper bound on the duration tau between two consecutive state observations however it is due to the general technique used there that the bound on tau is not very sharp in this paper we will be able to establish a better bound on tau making use of lyapunov functionals we will discuss the stabilization not only in the sense of exponential stability as mao does in automatica j ifac 49 2013 pp 3677 3681 but also in other sense that of h infty stability or asymptotic stability we will consider not only the mean square stability but also the almost sure stability

strong convergence and stability of implicit numerical methods for stochastic differential equations with non globally lipschitz continuous coefficients we are interested in the strong convergence and almost sure stability of euler maruyama em type approximations to the solutions of stochastic differential equations sdes with non linear and non lipschitzian coefficients motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence in addition we examine the globally almost surely asymptotic stability in this non linear setting for em type schemes in particular we present a stochastic counterpart of the discrete lasalle principle from which we deduce stability properties for numerical methods

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

towards a systematic linear stability analysis of numerical methods for systems of stochastic differential equations we develop two classes of test equations for the linear stability analysis of numerical methods applied to systems of stochastic ordinary differential equations of ito type sodes motivated by the theory of stochastic stabilization and destabilization these test equations capture certain fundamental effects of stochastic perturbation in systems of sodes while remaining amenable to analysis before and after discretization we then carry out a linear stability analysis of the theta maruyama method applied to these test equations investigating mean square and almost sure asymptotic stability of the test equilibria we discuss the implications of our work for the notion of a stability of the theta maruyama method and use numerical simulation to suggest extensions of our results to test systems with nonnormal drift coefficients

asymptotic mean square stability of weak second order balanced stochastic runge kutta methods for multi dimensional ito stochastic differential systems abstract in this paper the linear asymptotic mean square stability of the weak second order stochastic runge kutta methods for multi dimensional ito stochastic differential equations due to rosler 2009 and tang and xiao 2017 are obtained further we have developed the stochastic runge kutta methods for multi dimensional ito stochastic differential equations to the balanced stochastic runge kutta methods by using the control functions we carry out a linear stability analysis of the class of stochastic runge kutta methods for the linear test equations with multiplicative noise thereby providing an explicit structure of stability matrices some comparisons and illustrations shows that there is an improvement in the stability and error analysis of these new balanced stochastic runge kutta methods comparatively to stochastic runge kutta methods and thus conforming the obtained theoretical results

the euler maruyama approximation of solutions to stochastic differential equations with piecewise constant arguments abstract in this paper we consider stochastic differential equations with piecewise constant arguments these equations describe hybrid dynamical systems that is combinations of continuous and discrete systems our aim is to establish the mean square convergence of the euler maruyama approximate solution under quite general conditions that is the global lipschitz condition and the linear growth condition which guarantee the existence and uniqueness of the true solution then we show that the initial equation is exponentially stable in mean square if and only if for some sufficiently small step size the euler maruyama method is exponentially stable in mean square the stability study does not involve the lyapunov functions nor functionals it should be pointed out that stochastic differential equations with piecewise constant arguments can be regarded as a class of stochastic differential equations with multiple time dependent delays which can be found in the literature however in the convergence analysis of the euler maruyama method of these equations it is required that the delay functions satisfy lipschitz continuity condition in the present paper the delay functions do not satisfy that condition so it represents an extension of results from the papers mao 2003 2007 using the technique similar to that from the first paper

convergence and stability of the split step theta method for stochastic differential equations with piecewise continuous arguments abstract in this paper we consider the stochastic differential equations with piecewise continuous arguments sdepcas in which both the drift and the diffusion coefficients do not satisfy the global lipschitz and linear growth conditions especially the diffusion coefficients are highly non linear growing it is proved that the split step theta sst method with 1 2 1 is strongly convergent to sdepcas under the local lipschitz monotone and one sided lipschitz conditions it is also obtained that the sst method with 1 2 1 preserves the exponential mean square stability of sdepcas under the monotone condition and some condition on the step size without any restriction on the step size there exists 1 2 1 such that the sst method with 1 is exponentially stable in mean square moreover for sufficiently small step size the rate constant can be reproduced some numerical simulations are presented to illustrate the analytical theory

almost sure and moment exponential stability of euler maruyama discretizations for hybrid stochastic differential equations positive results are derived concerning the long time dynamics of numerical simulations of stochastic differential equation systems with markovian switching euler maruyama discretizations are shown to capture almost sure and moment exponential stability for all sufficiently small timesteps under appropriate conditions

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

exponential stability of the exact solutions and θ em approximations to neutral sddes with markov switching exponential stability of the exact solutions and em 1 2 1 approximations to neutral stochastic differential delay equations with markov switching are investigated in this paper sufficient conditions are obtained to ensure the p th moment p 1 and almost sure exponential stability of the exact solutions as well as em approximations p 2 an example is provided to support our conclusions

universal relations and p completeness this paper follows the methodology introduced by agrawal and biswas in manindra agrawal somenath biswas universal relations in structure in complexity theory conference 1992 pp 207 220 based on a notion of universality for the relations associated with np complete problems the purpose was to study np complete problems by examining the effects of reductions on the solution sets of the associated witnessing relations this provided a useful criterion for np completeness while suggesting structural similarities between natural np complete problems we extend these ideas to the class p the notion we find also yields a practical criterion for p completeness as illustrated by a varied set of examples and strengthens the argument for structural homogeneity of natural complete problems