split step forward methods for stochastic differential equations in this paper we discuss split step forward methods for solving ito stochastic differential equations sdes eight fully explicit methods the drifting split step euler drsse method the diffused split step euler disse method and the three stage milstein tsm 1a tsm 1f methods are constructed based on euler maruyama method and milstein method respectively in this paper their order of strong convergence is proved the analysis of stability shows that the mean square stability properties of the methods derived in this paper are improved on the original methods the numerical results show the effectiveness of these methods in the pathwise approximation of ito sdes

finite element solution for advection dominated thermal flows abstract a comparative study between an upwinding method supg with a discontinuity capturing term and the method of characteristics is presented in the case of advection diffusion problems numerical examples are given that demonstrate the superiority of the second method in terms of accuracy

an euler type method for the strong approximation of the cox ingersoll ross process we analyse the strong approximation of the cox ingersoll ross cir process in the regime where the process does not hit zero by a positivity preserving drift implicit euler type method as an error criterion we use the p th mean of the maximum distance between the cir process and its approximation on a finite time interval we show that under mild assumptions on the parameters of the cir process the proposed method attains up to a logarithmic term the convergence of order 1 2 this agrees with the standard rate of the strong convergence for global approximations of stochastic differential equations with lipschitz coefficients despite the fact that the cir process has a non lipschitz diffusion coefficient

gauss jacobi quadratures for weakly strongly hyper and nearly singular integrals in boundary integral equation methods for domains with sharp edges and corners abstract we present gauss jacobi quadrature rules in terms of hypergeometric functions for the discretization of weakly singular strongly singular hypersingular and nearly singular integrals that arise in integral equation formulations of potential problems for domains with sharp edges and corners the rules are tailored to weight functions with algebraic endpoint singularities of a fairly general form thus allowing one to easily incorporate a wide class of domains into the analysis numerical examples illustrate the accuracy and stability of the proposed algorithms it is shown that the same level of high accuracy can be achieved for any choice of the external variable the usefulness of the method is exemplified by application to the solution of a singular integral equation that arises in time harmonic electromagnetic scattering by either closed or open perfectly conducting cylindrical objects with edges and corners such as polygon cylinders and bent strips some practical aspects concerning the role of nearby singularities in achieving a highly accurate solution of singular integral equations are also discussed

quadratic streamline upwinding for finite element method solutions to 2 d convective transport problems abstract during the past decade finite element methods fems have been recognized to be more powerful tools in the solution of various flow problems as opposed to their predecessors finite difference methods on a fundamental level the fem involves approximations to the solution over a given trial space on a discrete mesh rather than finite difference approximations to the differential operators on a discrete mesh in this way the mechanics operators of the fem remain closer to the physical principles associated with a given problem in addition to this the fem incorporates boundary conditions very efficiently into the numerical formulation much current research is aimed at making these fems viable options for convection dominated flows in 1985 mizukami et al published two papers advancing the approach of streamline upwinding within linear elements for fems the purpose of this work is to demonstrate that streamline upwinding of higher ordered elements is a more accurate option the work goes on to show exactly how streamline upwinding effects the stability of iteratively solving the global algebraic system resulting from the fem

a new upwind function in stabilized finite element formulations using linear and quadratic elements for scalar convection diffusion problems abstract this article presents a study on the function of the peclet number that appears in the upwind function of the gls cau and saupg methods using linear and quadratic triangular elements as well as bilinear and quadratic quadrilateral elements the numerical experiments indicate that in the case of elements with faces on the outflow boundary the upwind function is strongly dependent on the geometry and on the degree of the interpolate polynomial a new function of the peclet number is therefore proposed to take these effects into consideration creating a new method of stabilization for higher order elements this new stabilized and accurate finite element formulation for convection dominated problems based on the idea of the saupg method uses the gls formulation as starting point

numerical solutions of sdes with markovian switching and jumps under non lipschitz conditions abstract in this paper the numerical solutions to stochastic differential equations with markovian switching and jumps under non lipschitz conditions are considered firstly we study the existence and uniqueness of solutions to this class of equations then we present the moment bounds of the exact and numerical solutions it is also proved that the strong rate of convergence of the euler scheme is equal to 1 2 with the drift coefficient satisfying one sided lipschitz condition while the diffusion and jump coefficients satisfy global lipschitz conditions some numerical examples are given to confirm the theoretical results

semi implicit split step numerical methods for a class of nonlinear stochastic differential equations with non lipschitz drift terms abstract in this paper we discuss numerical solutions of a class of nonlinear stochastic differential equations using semi implicit split step methods under some monotonicity conditions on the drift term we study moment estimates and strong convergence properties of the numerical solutions with a focus on stochastic ginzburg landau equations moreover we compare the performance of various numerical methods including the tamed euler truncated euler implicit euler and split step procedures in particular we discuss the empirical rate of convergence and the computational cost of these methods for certain parameter values of the models used

locally corrected multidimensional quadrature rules for singular functions accurate numerical integration of singular functions usually requires either adaptivity or product integration both interfere with fast summation techniques and thus hamper large scale computations this paper presents a method for computing highly accurate quadrature formulas for singular functions which combine well with fast summation methods given the singularity and the n nodes we first construct weights which integrate smooth functions with order k accuracy then we locally correct a small number of weights near the singularity to achieve order k accuracy on singular functions as well the method is highly efficient and runs in o nk 2d n log 2 n time and o k 2d n space we derive precise error bounds and time estimates and confirm them with numerical results which demonstrate the accuracy and efficiency of the method in large scale computations as part of our implementation we also construct a new adaptive multidimensional product gauss quadrature routine with an effective error

the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2003 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc in this paper we will develop a new explicit method called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and establish the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 the type of convergence specifically addressed in this paper is strong l q convergence for 2 q p and p is a parameter in the khasminskii type condition

exponentially converging nystrom methods in scattering from infinite curved smooth strips part 1 tm case low order subdomain basis methods of moments provide little help when highly accurate electromagnetic computations are required high order modeling on the other hand can fill the need for enhanced accuracy but at the expense of greater implementation cost to supplement such methods this paper presents nystrom techniques both easily implemented and highly accurate relevant to te scattering by arbitrarily shaped smooth infinite curved strips the analysis takes full account of both the singular nature of the kernels and the singularities of the solution at the edges as a result the proposed solutions are exponentially converging in addition by eliminating inner product integrals closed form analytical expressions are obtained for all matrix elements thus our algorithms have very low implementation and computational cost detailed numerical examples and case studies amply demonstrate the efficiency stability and extremely high accuracy of the algorithms these algorithms apply uniformly from electrically small to electrically large conducting screens with only slight modifications the present analysis can be also used to obtain exponentially converging galerkin solutions

the euler maruyama approximation for the asset price in the mean reverting theta stochastic volatility model stochastic differential equations sdes have been used to model an asset price and its volatility in finance lewis 2000 10 developed the mean reverting theta processes which can not only model the volatility but also the asset price in this paper we will consider the following mean reverting theta stochastic volatility model dx t a 1 m 1 x t dt s 1v t x t qdw 1 t dv t a 2 m 2 v t dt s 2v t bdw 2 t we will first develop a technique to prove the non negativity of solutions to the model we will then show that the em numerical solutions will converge to the true solution in probability we will also show that the em solutions can be used to compute some financial quantities related to the sde model including the option value for example

sdelab a package for solving stochastic differential equations in matlab we introduce sdelab a package for solving stochastic differential equations sdes within matlab sdelab features explicit and implicit integrators for a general class of ito and stratonovich sdes including milstein s method sophisticated algorithms for iterated stochastic integrals and flexible plotting facilities

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

convergence rate of the truncated milstein method of stochastic differential delay equations abstract this paper is concerned with the strong convergence of highly nonlinear stochastic differential delay equations sddes without the linear growth condition on the one hand these nonlinear sddes do not have explicit solutions therefore implementable numerical methods for such sddes are required on the other hand the implicit euler methods are known to converge strongly to the exact solution of such sddes however they require additional computational efforts in this article we propose the truncated milstein method which is an explicit method under the local lipschitz condition plus khasminskii type condition study its pth monent boundedness p is a parameter in khasminskii type condition and show that its rate of strong convergence is close to one

an implementation of the streamline upwind petrovgalerkin method for linear triangular elements abstract the weighting function of the streamline upwind petrov galerkin method contains a coefficient which has the dimension of time and is a function of element parameters e g element dimension element peclet number etc in this paper a formula for the coefficient applicable to linear triangular elements is presented the formula does not consist of the individual element parameters but rather of the element matrices only and so it is not only simple but also cost effective the validity of the formula is shown by several test problems and numerical examples

moment method analysis of large axially symmetric reflector antennas using entire domain functions the moment method technique utilizing entire domain basis functions is applied to the analysis of large axially symmetric reflector antennas the electric surface current is modeled as a finite series of sinusoids whose domain consists of the entire generating curve this expansion results in a matrix size of less than 5 of that produced with subdomain basis functions only a slight increase in the cpu requirements occurs from this analysis the results from this technique show good agreement when compared to both physical optics and a subdomain based moment method formulation on small axially fed paraboloidal and hyperboloidal reflector antennas extension to a large 100 lambda paraboloidal reflector with f d 0 4 produces results comparable to that obtained using physical optics convergence is obtained with as few as two expansion terms per wavelength discretization of the generating curve with four points per wavelength leads to results which agree within 0 5 db over data from a more densely defined curve

guaranteed convergence method of analysis of the scattering by an arbitrarily oriented zero thickness pec disk buried in a lossy half space in this paper a guaranteed convergence method for the accurate and efficient analysis of the electromagnetic scattering by an arbitrarily oriented zero thickness perfectly electrically conducting pec disk buried in a lossy half space is presented an electric field integral equation efie in the vector hankel transform domain is obtained by expanding the unknown surface current density in cylindrical harmonics a convergence of exponential type is achieved by means of galerkin s method with expansion functions reconstructing the expected physical behavior of the bm n th harmonic at the center and the edge of the disk the obtained coefficient matrix can be quickly computed since its elements can be reviewed as the sum of single integrals efficiently evaluable by means of an analytical acceleration technique and double integrals of exponentially decaying functions

the pod deim reduced order method for stochastic allen cahn equations with multiplicative noise abstract in this paper we propose a reduced order method rom based on the monte carlo finite difference method fdm or monte carlo finite element method fem for the stochastic allen cahn sac equation with multiplicative noise the reduction method is a combination of proper orthogonal decomposition pod and discrete empirical interpolation method deim in the theoretical part the error bounds of full order method fom and pod deim are provided as well as the computational complexity of fom and pod deim in the numerical experiments section several two dimensional sac examples with multiplicative noise are considered using the monte carlo fdm and monte carlo fem as the reference methods the pod deim has obvious advantage in the computational efficiency and with the encryption in the space or time direction this advantage is more prominent

preserving exponential mean square stability in the simulation of hybrid stochastic differential equations positive results are derived concerning the long time dynamics of fixed step size numerical simulations of stochastic differential equation systems with markovian switching euler maruyama and implicit theta method discretisations are shown to capture exponential mean square stability for all sufficiently small time steps under appropriate conditions moreover the decay rate as measured by the second moment lyapunov exponent can be reproduced arbitrarily accurately new finite time convergence results are derived as an intermediate step in this analysis we also show however that the mean square a stability of the theta method does not carry through to this switching scenario the proof techniques are quite general and hence have the potential to be applied to other numerical methods

full wave analysis and optimization of a tara like shield assisted paraboloidal reflector antenna using a nystrom type method using the recently developed method based on the rigorous theory of singular and hypersingular integral equations ies and nystrom type discretization a paraboloidal reflector assisted with a conical shield is investigated to decrease the computation time we propose a new method of calculation of the modal green s functions in the quasi optical range the far and near fields of the studied reflector are analyzed in the transmission and reception cases and reveal fine features that escape asymptotic analysis elementary numerical optimization of the shield assisted paraboloidal antenna is performed

postprocessing for stochastic parabolic partial differential equations we investigate the strong approximation of stochastic parabolic partial differential equations with additive noise we introduce postprocessing in the context of a standard galerkin approximation although other spatial discretizations are possible in time we follow g j lord and j rougemont ima j numer anal 24 2004 pp 587 604 and use an exponential integrator we prove strong error estimates and discuss the best number of postprocessing terms to take numerically we evaluate the efficiency of the methods and observe rates of convergence some experiments with the implicit euler maruyama method are described

data driven non markovian closure models abstract this paper has two interrelated foci i obtaining stable and efficient data driven closure models by using a multivariate time series of partial observations from a large dimensional system and ii comparing these closure models with the optimal closures predicted by the mori zwanzig mz formalism of statistical physics multilayer stochastic models msms are introduced as both a generalization and a time continuous limit of existing multilevel regression based approaches to closure in a data driven setting these approaches include empirical model reduction emr as well as more recent multi layer modeling it is shown that the multilayer structure of msms can provide a natural markov approximation to the generalized langevin equation gle of the mz formalism a simple correlation based stopping criterion for an emr msm model is derived to assess how well it approximates the gle solution sufficient conditions are derived on the structure of the nonlinear cross interactions between the constitutive layers of a given msm to guarantee the existence of a global random attractor this existence ensures that no blow up can occur for a broad class of msm applications a class that includes non polynomial predictors and nonlinearities that do not necessarily preserve quadratic energy invariants the emr msm methodology is first applied to a conceptual nonlinear stochastic climate model of coupled slow and fast variables in which only slow variables are observed it is shown that the resulting closure model with energy conserving nonlinearities efficiently captures the main statistical features of the slow variables even when there is no formal scale separation and the fast variables are quite energetic second an msm is shown to successfully reproduce the statistics of a partially observed generalized lotka volterra model of population dynamics in its chaotic regime the challenges here include the rarity of strange attractors in the model s parameter space and the existence of multiple attractor basins with fractal boundaries the positivity constraint on the solutions components replaces here the quadratic energy preserving constraint of fluid flow problems and it successfully prevents blow up

study of mixed order basis functions for the locally corrected nystro spl uml m method a high order locally corrected nystro spl uml m lcn method employing the mixed order basis functions proposed by c 80al spl iota s 80kan and peterson is presented for the electromagnetic scattering by targets composed of both dielectric and conducting bodies an integral operator based on a combined field formulation for conducting surfaces and a mu spl uml ller formulation for dielectric surfaces is used it is found that for general scattering objects mixed order basis functions accelerate the convergence of the lcn solution can eliminate spurious charges and can significantly reduce the condition number of the impedance matrix

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

theoretical and numerical analysis of the euler maruyama method for generalized stochastic volterra integro differential equations abstract in this paper we concern with the theoretical and numerical analysis of the generalized stochastic volterra integro differential equations svides the existence uniqueness boundedness and holder continuity of the analytic solutions for generalized svides are investigated the euler maruyama method for generalized svides is presented the boundedness of the numerical solution is proved and the strong convergence order is obtained the theoretical results are illustrated by some numerical examples

on spurious oscillations at layers diminishing sold methods for convection diffusion equations part i a review abstract an unwelcome feature of the popular streamline upwind petrov galerkin supg stabilization of convection dominated convection diffusion equations is the presence of spurious oscillations at layers since the mid of the 1980s a number of methods have been proposed to remove or at least to diminish these oscillations without leading to excessive smearing of the layers the paper gives a review and state of the art of these methods discusses their derivation proposes some alternative choices of parameters in the methods and categorizes them some numerical studies which supplement this review provide a first insight into the advantages and drawbacks of the methods

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

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

strong convergence rates for backward euler on a class of nonlinear jump diffusion problems we generalise the current theory of optimal strong convergence rates for implicit euler based methods by allowing for poisson driven jumps in a stochastic differential equation sde more precisely we show that under one sided lipschitz and polynomial growth conditions on the drift coefficient and global lipschitz conditions on the diffusion and jump coefficients three variants of backward euler converge with strong order of one half the analysis exploits a relation between the backward and explicit euler methods

on stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions in a recent article jentzen et al 2016 commun math sci 14 1477 1500 doi 10 4310 cms 2016 v14 n6 a1 it has been established that for every arbitrarily slow convergence speed and every nat

a positivity preserving numerical method for stochastic r d model abstract the stochastic research and development r d model plays an important role in the growth rates of technological progress and capital accumulation however we can not obtain the explicit solution of stochastic r d model to explore a numerical approximate method preserving positivity for the r d model with uncertainty from the population growth we first construct an explicit euler maruyama em method and then verify it converges to the true solution with strong order 1 2 1 1 p but the em method may lead to a negative approximation which is unrealistic so we establish the balanced explicit method bem to overcome the defect of em method and give sufficient conditions to preserve the positivity of bem then the convergence order of bem is obtained finally numerical simulations are carried out to verify our theoretical work

extinction time for stochastic population models the analysis of interacting population models is the subject of much interest in mathematical ecology moreover the persistence and extinction of these models is one of the most interesting and important topics because it provides insight into their behavior the mean extinction time for stochastic population models considered in this paper depends on the initial population size and satisfies a stationary partial differential equation related to the backward kolmogorov differential equation a linear second order partial differential equation with variable coefficients in this communication we review several papers where we have proposed some numerical techniques in order to estimate the mean extinction time for stochastic population models besides we will compare the theoretical predictions and numerical simulations for stochastic differential equations sdes this work can be viewed as a unified review of the contributions de la hoz and vadillo 2012 de la hoz et al 2014 and doubova and vadillo 2014

covolume upwind finite volume approximations for linear elliptic partial differential equations in this paper we study covolume upwind finite volume methods on rectangular meshes for solving linear elliptic partial differential equations with mixed boundary conditions to avoid non physical numerical oscillations for convection dominated problems nonstandard control volumes covolumes are generated based on local peclet s numbers and the upwind principle for finite volume approximations two types of discretization schemes with mass lumping are developed with use of bilinear or biquadratic basis functions as the trial space respectively some stability analyses of the schemes are presented for the model problem with constant coefficients various examples are also carried out to numerically demonstrate stability and optimal convergence of the proposed methods

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 projected explicit ito taylor methods for stochastic differential equations under locally lipschitz conditions and polynomial growth conditions abstract although the numerical methods to stochastic differential equations with the coefficients of locally lipschitz and polynomial growth have been discussed commonly by some authors there are few works on the high strong order numerical methods in this paper the mean square convergence of the general projected explicit ito taylor methods is considered under the assumption that drift and diffusion coefficient functions of stochastic differential equations satisfy the global monotonicity condition suitable local lipschitz conditions and polynomial growth conditions our analysis follows the idea of stochastic c stability and stochastic b consistency after giving the selection strategy of optimal parameters we prove that the projected explicit ito taylor methods owning optimal parameters share the same mean square convergence orders with non projected ones finally two numerical experiments are presented to show the effectiveness of theoretical results

convergence and stability of the backward euler method for jump diffusion sdes with super linearly growing diffusion and jump coefficients abstract this paper firstly investigates convergence of the backward euler method for stochastic differential equations sdes driven by brownian motion and compound poisson process different from existing works we do not impose a globally lipschitz condition on the jump coefficient but formulate a more relaxed condition to allow for its super linear growth it is shown that the mean square convergence order of this method can be arbitrarily close to 1 2 under mild assumptions imposed on sdes allowing for possibly super linearly growing drift diffusion and jump coefficients an exact order 1 2 is recovered when further differentiability assumption is put on the coefficients furthermore the considered method is able to inherit the mean square stability of a wider class of levy noise driven sdes for all stepsizes these results are finally supported by some numerical experiments

noise induced changes to the behaviour of semi implicit euler methods for stochastic delay differential equations undergoing bifurcation we are concerned with estimating parameter values at which bifurcations occur in stochastic delay differential equations after a brief review of bifurcation we employ a numerical approach and consider how bifurcation values are influenced by the choice of numerical scheme and the step length and by the level of white noise present in the equation in this paper we provide a formulaic relationship between the estimated bifurcation value the level of noise the choice of numerical scheme and the step length we are able to show that in the presence of noise there may be some loss of order in the accuracy of the approximation to the true bifurcation value compared to the use of the same approach in the absence of noise

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

method of analytical regularization in the analysis of axially symmetric excitation of imperfect circular disk antennas abstract this work presents accurate numerical method and computational results in the modeling of the axisymmetric radiation of resistive and thin dielectric circular disk on substrate antennas to simulate resistive and thin dielectric disk features we use two side generalized boundary conditions gbc the analysis is based on the singular integral equations ies and the method of analytical regularization mar exploiting a galerkin method gm with judicious basis functions which convert ies into the fredholm second kind infinite matrix equations unlike previously known numerical techniques this guarantees convergence and enables us to compute the solution with controlled accuracy even in sharp resonances numerical results demonstrate the effect of disk losses and transparency on the fundamental radiation phenomena and help improve antenna characteristics such as radiation efficiency

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

existence and uniqueness of the solutions and convergence of semi implicit euler methods for stochastic pantograph equations abstract in this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given i e the local lipschitz condition and the linear growth condition under the lipschitz condition and the linear growth condition it is proved that the semi implicit euler method is convergence with strong order 1 2

explicit milstein schemes with truncation for nonlinear stochastic differential equations convergence and its rate abstract although some implicit numerical procedures have been developed to treat high nonlinearity the question whether one can use explicit schemes to achieve convergence rate similar to that of milstein s procedure remained open this brings us to the current work that focuses on numerical solutions of stochastic differential equations using explicit schemes our main goals are to obtain order one convergence in the second moment in a finite time interval in contrast to the implicit schemes explicit schemes are advantageous easily implementable and computationally less intensive to overcome the difficulties due to super linear growth of the coefficients a truncation device is used in our algorithm in addition to reaching aforementioned goals in the analysis part numerical examples are provided to demonstrate our results

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 ergodic limits for langevin equations abstract to evaluate averages with respect to the invariant law for langevin type equations one has to integrate a system over long time intervals especially when dissipation is small this is a challenging problem from the computational point of view since coefficients of the langevin equations are typically not globally lipschitz the difficulties are redoubled in the nonglobal lipschitz case we observe an exploding behavior of some approximate trajectories due to many usual numerical methods in g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 we propose and justify a concept which in principle allows us to apply any numerical method to stochastic differential equations with nonglobally lipschitz coefficients here quasi symplectic integrators from g n milstein m v tretyakov quasi symplectic methods for langevin type equations ima j numer anal 23 2003 593 626 they are the most appropriate methods for solving langevin type equations especially when damping is small and the concept of g n milstein m v tretyakov numerical integration of stochastic differential equations with nonglobally lipschitz coefficients siam j numer anal 43 2005 1139 1154 are applied for calculation of ergodic limits special attention is paid to the case when the invariant measure is known e g when this measure is gibbsian some computer experiments with three model problems van der pol s equation with additive noise one dimensional arrays of oscillators in thermal equilibrium and a physical pendulum with linear friction and additive noise are presented

a note on strong approximation of sdes with smooth coefficients that have at most linearly growing derivatives abstract in the literature on strong approximation of sdes polynomial error rate results for numerical schemes are typically achieved under the assumption that the first order derivatives of the coefficients of the equation satisfy a polynomial growth condition in the present article we show that the latter condition is far from being sufficient for polynomial error rates we construct an sde with smooth coefficients that have first order derivatives of at most linear growth such that the solution at the final time can at best be approximated with a logarithmic rate whatever numerical method based on observations of the driving brownian motion at finitely many fixed times is used most interestingly it furthermore turns out that using a method that properly adjusts the number of evaluations of the driving brownian motion to its actual path the latter sde can in fact be approximated with polynomial rate 1 in terms of the average number of evaluations that are used to the best of our knowledge this is only the second example in the literature of an sde for which there exist adaptive methods that perform superior to non adaptive ones with respect to the convergence rate

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

simulating diffusion processes in discontinuous media in this article we propose new monte carlo techniques for moving a diffusive particle in a discontinuous media in this framework we characterize the stochastic process that governs the positions of the particle the key tool is the reduction of the process to a skew brownian motion sbm in a zone where the coefficients are locally constant on each side of the discontinuity the new position of the particle after a constant time step is sampled from the exact distribution of the sbm process at the considered time to do so we propose two different but equivalent algorithms a two steps simulation with a stop at the discontinuity and a one step direct simulation of the sbm dynamic some benchmark tests illustrate their effectiveness

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

numerical solution of a fuzzy stochastic single species age structure model in a polluted environment this paper presents an investigation of a fuzzy stochastic single species age structure model in a polluted environment both the fuzziness of the initial condition and the stochastic disturbance of the environment are incorporated into the model by using the theory of fuzzy stochastic differential equation fsde and the successive approximation the global existence and uniqueness of solutions of the model are proved in addition the error estimation and stability of the numerical solutions are obtained furthermore making use of euler maruyama em method the convergence of the em numerical approximation is established numerical simulations are carried out to verify the theoretical results our results show that the technique of numerical solution of fsde can be used to estimate the evolution tendency of the population density in a polluted environment

comments on strong convergence rates for backward euler on a class of nonlinear jump diffusion problems j comput appl math 205 2007 949 956 abstract in this note we correct an inaccuracy of eq 13 in the proof of theorem 1 in the paper higham and kloeden 2007 we also give a complete proof of the theorem by applying the burkholder davis gundy inequality

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

pathwise approximation of stochastic differential equations on domains higher order convergence rates without global lipschitz coefficients we study the approximation of stochastic differential equations on domains for this we introduce modified ito taylor schemes which preserve approximately the boundary domain of the equation under consideration assuming the existence of a unique non exploding solution we show that the modified ito taylor scheme of order has pathwise convergence order e for arbitrary e 0 as long as the coefficients of the equation are sufficiently differentiable in particular no global lipschitz conditions for the coefficients and their derivatives are required this applies for example to the so called square root diffusions

a splitting step algorithm for reflected stochastic differential equations in r 1 a new algorithm which is based on the splitting step idea and the penalization method for reflected stochastic differential equation rsde in the upper half space r 1 is presented in this paper after some important estimates about rsdes and penalization odes are obtained the local pathwise error of the suggested algorithm are considered and an estimate of local error in the 2pth moment is proved under the assumptions of lipschitz continuity and linear growth some numerical experiments based on matlab programs are given to support the suggested algorithm

on deriving a locally corrected nystrom scheme from a quadrature sampled moment method a novel high order method of moment procedure with quadrature point based discretization is presented the scheme is equivalent to a moment method employing smooth basis and testing functions applying a fixed point numerical quadrature approximation for the outer integral mapping the current to the quadrature points then leads to a formulation that is equivalent to that derived via the locally corrected nystrom method the convergence properties of the current density and rcs for smooth and singular geometries in two and three dimensions are also studied

the euler scheme with irregular coefficients weak convergence of the euler scheme for stochastic differential equations is established when coefficients are discontinuous on a set of lebesgue measure zero the rate of convergence is presented when coefficients are holder continuous monte carlo simulations are also discussed

nystrom type method in three dimensional electromagnetic diffraction by a finite pec rotationally symmetric surface time harmonic electromagnetic wave diffraction by a perfectly electrically conducting pec finite rotationally symmetric surface located in free space is investigated the problem is split to independent azimuth orders and reduced to the sets of coupled hypersingular and singular integral equations ies for the surface current components these ies are discretized by the nystrom type method of discrete singularities using the interpolation type quadrature formulas from the solutions of corresponding matrix equations the near and the far field patterns are obtained the presented method has guaranteed convergence for arbitrary not axially symmetric primary field

integral equation analysis of radiating structures of revolution a methods for solving radiation problems involving structures of revolution is presented that is based on the direct solutions of one dimensional integral equations for the currents the kernels of these equations are derived explicitly and their singularities are studied efficient methods for their computation are presented the numerical solution is obtained by a method of moments technique using sinusoidal basis functions applications include the determination of the current densities reflection coefficient and far field patterns for several waveguide fed structures numerical checks and comparisons with experiment demonstrate the accuracy of the method

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

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

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

explicit approximations for nonlinear switching diffusion systems in finite and infinite horizons abstract focusing on hybrid diffusion dynamics involving continuous dynamics as well as discrete events this article investigates the explicit approximations for nonlinear switching diffusion systems modulated by a markov chain different kinds of easily implementable explicit schemes have been proposed to approximate the dynamical behaviors of switching diffusion systems with locally lipschitz continuous drift and diffusion coefficients in both finite and infinite intervals without additional restriction conditions except those which guarantee the exact solutions possess their dynamical properties the numerical solutions converge strongly to the exact solutions in finite horizon moreover realize the approximation of long time dynamical properties including the moment boundedness stability and ergodicity some simulations and examples are provided to support the theoretical results and demonstrate the validity of the approach

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

the linear steklov method for sdes with non globally lipschitz coefficients we present an explicit numerical method for solving stochastic differential equations with non globally lipschitz coefficients a linear version of the steklov average under a split step formulation supports our new solver the linear steklov method converges strongly with a standard one half order also we present numerical evidence that the explicit linear steklov reproduces almost surely stability solutions with high accuracy for diverse application models even for stochastic differential systems with super linear diffusion coefficients

a petrov galerkin finite element method for convection dominated flows an accurate upwinding technique for satisfying the maximum principle abstract in this paper we present a new petrov galerkin method for convection dominated flow problems which is conservative and satisfies the discrete maximum principle this method also possesses no spurious crosswind diffusion and gives very accurate solutions the method is representative of a class of methods which may be described as fixed mesh adaptive

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

boundary preserving semianalytic numerical algorithms for stochastic differential equations construction of splitting step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations sdes of ito type are discussed as the crucial assumption we oppose conditions such that one can decompose the original system of sdes into subsystems for which one knows either the exact solution or its conditional transition probability we present convergence proofs for a newly designed splitting step algorithm and simulation studies for numerous well known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance cox ingersoll ross cir and constant elasticity of variance cev models to measure valued diffusion and super brownian motion stochastic pdes spdes as met in biology and physics

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 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

tamed em scheme of neutral stochastic differential delay equations abstract in this paper we investigate the convergence of the tamed euler maruyama em scheme for a class of neutral stochastic differential delay equations the strong convergence results of the tamed em scheme are presented under global and local non lipschitz conditions respectively moreover under the global lipschitz condition we provide the convergence rate of tamed em scheme which could be the same as the convergence rate of classical em scheme one half

theoretical and numerical analysis for volterra integro differential equations with ito integral under polynomially growth conditions abstract in this paper we theoretically and numerically deal with nonlinear volterra integro differential equations with ito integral under a one sided lipschitz condition and polynomially growth conditions it is proved that both the exact solutions and vector fields are bounded and satisfy a holder condition in the pth moment sense analogously the boundedness and holder condition in the pth moment sense are preserved by the semi implicit euler method for sufficiently small step size moreover by the local truncated errors we prove the strong convergence order 1 finally numerical simulations on stochastic control models and stochastic ginzburg landau equation illustrate our results

convergence and stability of the semi tamed euler scheme for stochastic differential equations with non lipschitz continuous coefficients recently explicit tamed schemes were proposed to approximate the sdes with the non lipschitz continuous coefficients this work proposes a semi tamed euler scheme which is also explicit to solve the sdes with the drift coefficient equipped with the lipschitz continuous part and non lipschitz continuous part it is shown that the semi tamed euler converges strongly with the standard order one half to the exact solution of the sde we also investigate the stability inheritance of the semi tamed euler schemes and reveal that this scheme does have advantage in reproducing the exponential mean square stability of the exact solution numerical experiments confirm the theoretical analysis

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

numerical methods for some nonlinear stochastic differential equations in this paper we study the numerical solutions of the stochastic differential equations of the formdu x t f x t u dt g x t u dw t q r n x n anda q r nx 0 t r n x n where a q q 2m is a family of square matrices whose elements are sufficiently smooth functions on r nx 0 t and d q d 1 q 1 d n q n d i x i

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

strong convergence of the partially truncated euler maruyama method for a class of stochastic differential delay equations abstract this paper establishes the convergence of a class of highly nonlinear stochastic differential delay equations without the linear growth condition replacing by khasminskii type condition so the convergence criteria here may cover a wider class of nonlinear systems our aim is to propose the partially truncated euler maruyama method for stochastic differential delay equations d y t f y t y t d t g y t y t d w t and consider the strong l q convergence for 2 q p under the local lipschitz condition plus khasminskii type condition and p is a parameter in khasminskii type condition

numerical approximation of irregular sdes via skorokhod embeddings abstract we provide a new algorithm for approximating the law of a one dimensional diffusion m solving a stochastic differential equation with possibly irregular coefficients the algorithm is based on the construction of markov chains whose laws can be embedded into the diffusion m with a sequence of stopping times the algorithm does not require any regularity or growth assumption in particular it applies to sdes with coefficients that are nowhere continuous and that grow superlinearly we show that if the diffusion coefficient is bounded and bounded away from 0 then our algorithm has a weak convergence rate of order 1 4 finally we illustrate the algorithm s performance with several examples

pathwise accuracy and ergodicity of metropolized integrators for sdes metropolized integrators for ergodic stochastic differential equations sdes are proposed that 1 are ergodic with respect to the known equilibrium distribution of the sdes and 2 approximate pathwise the solutions of the sdes on finitetime intervals both these properties are demonstrated in the paper and precise strong error estimates are obtained it is also shown that the metropolized integrator retains these properties even in situations where the drift in the sde is nonglobally lipschitz and vanilla explicit integrators for sdes typically become unstable and fail to be ergodic 2009 wiley periodicals inc

the split step backward euler method for linear stochastic delay differential equations in this paper the numerical approximation of solutions of linear stochastic delay differential equations sddes in the ito sense is considered we construct split step backward euler ssbe method for solving linear sddes and develop the fundamental numerical analysis concerning its strong convergence and mean square stability it is proved that the ssbe method is convergent with strong order c 12 in the mean square sense the conditions under which the ssbe method is mean square stable ms stable and general mean square stable gms stable are obtained some illustrative numerical examples are presented to demonstrate the order of strong convergence and the mean square stability of the ssbe method

a high order discontinuous galerkin method for ito stochastic ordinary differential equations in this paper we develop a high order discontinuous galerkin dg method for strong solution of ito stochastic ordinary differential equations sdes driven by one dimensional wiener processes motivated by the dg method for deterministic ordinary differential equations odes we first construct an approximate deterministic ode with a random coefficient on each element using the well known wong zakai approximation theorem since the resulting ode converges to the solution of the corresponding stratonovich sde we apply a transformation to the drift term to obtain a deterministic ode which converges to the solution of the original sde the corrected equation is then discretized using the standard dg method for deterministic odes we prove that the proposed stochastic dg sdg method is equivalent to an implicit stochastic runge kutta method then we study the numerical stability of the sdg scheme applied to linear sdes with an additive noise term the method is shown to be numerically stable in the mean sense and also a stable as a result it is suitable for solving stiff sdes moreover the method is proved to be convergent in the mean square sense numerical evidence demonstrates that our proposed dg scheme for sdes with additive noise has a strong convergence order of 2 p 1 when p degree piecewise polynomials are used when applied to sdes with multiplicative noise the sdg method is strongly convergent with order p several linear and nonlinear test problems are presented to show the accuracy and effectiveness of the proposed method in particular we demonstrate that our proposed scheme is suitable for stiff stochastic differential systems