stepsize control for mean square numerical methods for stochastic differential equations with small noise a strategy for controlling the stepsize in the numerical integration of stochastic differential equations sdes is presented it is based on estimating the pth mean of local errors the strategy leads to stepsize sequences that are identical for all computed paths for the family of euler schemes for sdes with small noise we derive computable estimates for the dominating term of the pth mean of local errors and show that the strategy becomes efficient for reasonable stepsizes numerical experience is reported for test examples including scalar sdes and a stochastic circuit model

convergence rates of the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2002 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 recently we developed a new explicit method in mao 2015 called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and established 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 however due to the page limit there we did not study the convergence rates for the method which is the aim of this paper we will under some additional conditions discuss the rates of l q convergence of the truncated em method for 2 q p and show that the order of l q convergence can be arbitrarily close to q 2

bispectrum estimation a parametric approach higher order spectra contain information about random processes that is not contained in the ordinary power spectrum such as the degree of nonlinearity and deviations from normality estimation of the bispectrum which is a third order spectrum has been applied in various fields to obtain information regarding quadratic phase coupling among harmonic components and non gaussianness of processes existing methods of bispectrum estimation are patterned after the conventional methods of power spectrum estimation which are known to possess certain limitations the paper proposes a parametric approach to bispectrum estimation based on a non gaussian white noise driven autoregressive ar model the ar parameter estimates are obtained by solving the third order recursion equations which may be toeplitz in form but not symmetric it is shown that the method provides bispectral estimates that are far superior to the conventional estimates in terms of bispectral fidelity and that in the case of detecting phase coupling among sinusoids the method provides significantly better resolution

approximable sets much structural work on np complete sets has exploited sat s d self reducibility in this paper we exploit the additional fact that sat is a d cylinder to show that np complete sets are p superterse unless p np in fact every set that is np hard under polynomial time no 1 tt reductions is p superterse unless p np in particular no p selective set is np hard under polynomial time no 1 tt reductions unless p np in addition no easily countable set is np hard under turing reductions unless p np self reducibility does not seem to suffice far our main result in a relativized world we construct a d self reducible set in np p that is polynomial time 2 tt reducible to a p selective set

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

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

a new numerical method for sdes and its application in circuit simulation abstract in the context of computer aided design of electric circuits the modeling of circuits under the influence of electronic noise leads to stochastic differential algebraic equations sdaes a new numerical integration scheme for the pathwise approximative solution of stochastic differential equations sdes is presented the available numerical schemes for sdes cannot solve these equations efficiently the new discretization scheme is constructed by the means of ito taylor expansions a fortran77 implementation of the presented integration scheme reduces the simulation time for a ring oscillator to about 85 compared with common methods

asymptotic properties of monte carlo estimators of diffusion processes abstract this paper studies the limit distributions of monte carlo estimators of diffusion processes we examine two types of estimators based on the euler scheme one applied to the original processes the other to a doss transformation of the processes we show that the transformation increases the speed of convergence of the euler scheme we also study estimators of conditional expectations of diffusions after characterizing expected approximation errors we construct second order bias corrected estimators we also derive new convergence results for the mihlstein scheme illustrations of the results are provided in the context of simulation based estimation of diffusion processes

adaptive schemes for the numerical solution of sdes a comparison the efficient numerical solution of stochastic differential equations is important for applications in many fields adaptive schemes well developed in the deterministic setting may be one possible way to reduce computational cost we review the two main step size control algorithms that have been proposed in recent years for stochastic differential systems and compare their efficiency in a simulation study

a new adaptive runge kutta method for stochastic differential equations in this paper we will present a new adaptive time stepping algorithm for strong approximation of stochastic ordinary differential equations we will employ two different error estimation criteria for drift and diffusion terms of the equation both of them based on forward and backward moves along the same time step we will use step size selection mechanisms suitable for each of the two main regimes in the solution behavior which correspond to domination of the drift based local error estimator or diffusion based one numerical experiments will show the effectiveness of this approach in the pathwise approximation of several standard test problems

random generation of stochastic area integrals the authors describe a method of random generation of the integrals a 1 2 t t h int t t h int t s dw 1 r dw 2 s int t t h int t s dw 2 r dw 1 s together with the increments w 1 t h w 1 t and w 2 t h w 2 t of a two dimensional brownian path w 1 t w 2 t the method chosen is based on marsaglia s rectangle wedge tail method generalised to higher dimensions the motivation is the need for a numerical scheme for simulation of strong solutions of general multidimensional stochastic differential equations with an order of convergence o h where h is the stepsize previously no method has obtained an order of convergence better than o sqrt h in the general case

a transition to sharp timing in stochastic leaky integrate and fire neurons driven by frozen noisy input the firing activity of intracellularly stimulated neurons in cortical slices has been demonstrated to be profoundly affected by the temporal structure of the injected current mainen sejnowski 1995 this suggests that the timing features of the neural response may be controlled as much by its own biophysical characteristics as by how a neuron is wired within a circuit modeling studies have shown that the interplay between internal noise and the fluctuations of the driving input controls the reliability and the precision of neuronal spiking cecchi et al 2000 tiesinga 2002 fellous rudolph destexhe sejnowski 2003 in order to investigate this interplay we focus on the stochastic leaky integrate and fire neuron and identify the holder exponent h of the integrated input as the key mathematical property dictating the regime of firing of a single unit neuron we have recently provided numerical evidence taillefumier magnasco 2013 for the existence of a phase transition when becomes less than the statistical holder exponent associated with internal gaussian white noise h 1 2 here we describe the theoretical and numerical framework devised for the study of a neuron that is periodically driven by frozen noisy inputs with exponent h 0 in doing so we account for the existence of a transition between two regimes of firing when h 1 2 and we show that spiking times have a continuous density when the holder exponent satisfies h 1 2 the transition at h 1 2 formally separates rate codes for which the neural firing probability varies smoothly from temporal codes for which the neuron fires at sharply defined times regardless of the intensity of internal noise

gaussian mixture density modeling of non gaussian source for autoregressive process a new approach is taken to model non gaussian sources of ar processes using gaussian mixture densities that are known to be effective for approximating wide varieties of probability distributions a maximum likelihood estimation algorithm is derived for estimating the ar parameters by solving a generalized normal equation and a clustering algorithm is used for estimating the parameters of gaussian mixture density of the source signals the correlation matrix of the generalized normal equation is not toeplitz but is symmetric and in general positive definite higher order statistics of skewness and kurtosis are used for identifying the source distribution as being gaussian or non gaussian and consequently determining the parameter estimation technique between the conventional method and the proposed method experiments on non gaussian source ar processes demonstrate that under high snr conditions snr spl ges 20 db the proposed algorithm outperforms the conventional ar estimation algorithm and the cumulant based algorithm by an order of magnitude reduction of average estimation errors the proposed algorithm also has very low estimation errors with short data records finally a maximum likelihood prediction method is formulated for non gaussian source ar processes that has shown potential in achieving higher efficiency signal coding than linear predictive coding

strong convergence of the stopped euler maruyama method for nonlinear stochastic differential equations in this paper numerical methods for the nonlinear stochastic differential equations sdes with non global lipschitz drift coefficient are discussed the existing known results have only so far shown that the classical explicit euler maruyama em approximate solutions converge to the true solution in probability 22 23 more recently the authors in 16 proved that the classical em method will diverge in l 2 sense for the underlying sdes in this paper and those sdes with superlinearly growing coefficients these strongly indicate that the classical em method is not good enough for the highly nonlinear sdes however in this paper we introduce a modified em method using stopping time and show successfully that the discrete version of the modified em approximate solution converges to the true solution in the strong sense namely in l 2 with a order arbitrarily close to a half

stochastic lie group integrators we present lie group integrators for nonlinear stochastic differential equations with noncommutative vector fields whose solution evolves on a smooth finite dimensional manifold given a lie group action that generates transport along the manifold we pull back the stochastic flow on the manifold to the lie group via the action and subsequently pull back the flow to the corresponding lie algebra via the exponential map we construct an approximation to the stochastic flow in the lie algebra via closed operations and then push back to the lie group and then to the manifold thus ensuring that our approximation lies in the manifold we call such schemes stochastic munthe kaas methods after their deterministic counterparts we also present stochastic lie group integration schemes based on castell gaines methods these involve using an underlying ordinary differential integrator to approximate the flow generated by a truncated stochastic exponential lie series they become stochastic lie group integrator schemes if we use munthe kaas methods as the underlying ordinary differential integrator further we show that some castell gaines methods are uniformly more accurate than the corresponding stochastic taylor schemes lastly we demonstrate our methods by simulating the dynamics of a free rigid body such as a satellite and an autonomous underwater vehicle both perturbed by two independent multiplicative stochastic noise processes

on index 2 linear implicit difference equations this paper deals with an index 2 notion for linear implicit difference equations lides and with the solvability of initial value problems ivps for index 2 lides besides the cocycle property as well as the multiplicative ergodic theorem of oseledets type are also proved

step size control in the numerical solution of stochastic differential equations abstract we introduce a variable step size algorithm for the pathwise numerical approximation of solutions to stochastic ordinary differential equations the algorithm is based on a new pair of embedded explicit runge kutta methods of strong order 1 5 1 0 where the method of strong order 1 5 advances the numerical computation and the difference between approximations defined by the two methods is used for control of the local error we show that convergence of our method is preserved though the discretization times are not stopping times any more and further we present numerical results which demonstrate the effectiveness of the variable step size implementation compared to a fixed step size implementation

convergence rate of weak local linearization schemes for stochastic differential equations with additive noise there exists a diversity of weak local linearization ll schemes for the integration of stochastic differential equations with additive noise which differ in the algorithms employed for the numerical implementation of the weak local linear discretizations despite convergence results for these discretizations have been already developed the convergence of the weak ll schemes has not been considered up to date in this work a general result concerning the convergence rate of the weak ll schemes is presented as well as specificities for a number of particular schemes as an application the convergence of weak ll schemes for equations driven by poisson processes is presented in addition

variable step size control in the numerical solution of stochastic differential equations we introduce a variable step size method for the numerical approximation of pathwise solutions to stochastic differential equations sdes the method which is dependent on a representation of bro

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

flow of diffeomorphisms induced by a geometric multiplicative functional in this paper it is shown that the unique multiplicative functional solution to a differential equation driven by a geometric multiplicative functional consitutes a flow of local diffeomorphisms in the case where the driving geometric multiplicative functional is generated by a brownian motion the result in particular presents an answer to an open problem proposed in ikeda and watanabe 4

on parameter and state estimation for linear differential algebraic equations the current demand for more complex models has initiated a shift away from state space models towards models described by differential algebraic equations daes these models arise as the natural product of object oriented modeling languages such as modelica however the mathematics of daes is somewhat more involved than the standard state space theory the aim of this work is to present a well posed description of a linear stochastic differential algebraic equation and more importantly explain how well posed estimation problems can be formed we will consider both the system identification problem and the state estimation problem besides providing the necessary theory we will also explain how the procedures can be implemented by means of efficient numerical methods

time bounded frequency computations 1 we obtain two new results concerning the inclusion problem of polynomial time frequency classes with equal numbers of errors 1 m m d p m 1 m d 1 pform 2d 2 m m d p m 1 m d 1 pform c d wherec d is large enough this disproves a conjecture of kinber 2 we give a transparent proof of a generalization of kinber s result that there exist arbitrarily complex problems admitting a polynomial time frequency computation several corollaries provide more insight into the structure of the hierarchy of polynomial time frequency classes 3 the relationships between polynomial time frequency classes and selectivity classes are studied

improved linear multi step methods for stochastic ordinary differential equations we consider linear multi step methods for stochastic ordinary differential equations and study their convergence properties for problems with small noise or additive noise we present schemes where the drift part is approximated by well known methods for deterministic ordinary differential equations in previous work we considered maruyama type schemes where only the increments of the driving wiener process are used to discretize the diffusion part here we suggest the improvement of the discretization of the diffusion part by also taking into account mixed classical stochastic integrals we show that the relation of the applied step sizes to the smallness of the noise is essential in deciding whether the new methods are worthwhile simulation results illustrate the theoretical findings

numerical methods for stochastic differential equations stochastic differential equations sde s play an important role in physics but existing numerical methods for solving such equations are of low accuracy and poor stability a general strategy for developing accurate and efficient schemes for solving stochastic equations is outlined here high order numerical methods are developed for the integration of stochastic differential equations with strong solutions we demonstrate the accuracy of the resulting integration schemes by computing the errors in approximate solutions for sde s which have known exact solutions

comment on numerical methods for stochastic differential equations wilkie phys rev e 70 017701 2004 used a heuristic approach to derive runge kutta based numerical methods for stochastic differential equations based on methods used for solving ordinary differential equations the aim was to follow solution paths with high order we point out that this approach is invalid in the general case and does not lead to high order methods we warn readers against the inappropriate use of deterministic calculus in a stochastic setting

stability analysis of high order runge kutta methods for index 1 stochastic differential algebraic equations with scalar noise abstract in this paper a new class of implicit stochastic runge kutta srk methods is constructed for numerically solving systems of index 1 stochastic differential algebraic equations sdaes with scalar multiplicative noise by applying rooted tree theory analysis the family of coefficients of the proposed methods of order 1 5 are calculated in the mean square sense in particular we derive some four stage stiffly accurate semi implicit srk methods for approximating index 1 sdaes with scalar noise for these methods first ms stability functions applied to a scalar linear test equation with multiplicative noise are calculated then their regions of ms stability are compared with the corresponding ms stability region of the original sde accordingly we illustrate that the proposed schemes seems to have good stability properties numerical results will be presented to check the convergence order and computational efficiency of the new methods

numerical methods for stochastic delay differential equations via the wong zakai approximation we use the wong zakai approximation as an intermediate step to derive numerical schemes for stochastic delay differential equations by approximating the brownian motion with its truncated spectral expansion and then using different discretizations in time we present three schemes a predictor corrector scheme a midpoint scheme and a milstein like scheme we prove that the predictor corrector scheme converges with order half in the mean square sense while the milstein like scheme converges with order one numerical tests confirm the theoretical prediction and demonstrate that the midpoint scheme is of half order convergence numerical results also show that the predictor corrector and midpoint schemes can be of first order convergence under commutative noises when there is no delay in the diffusion coefficients

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

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

stability of singular stochastic systems with markovian switching in recent years singular hybrid systems have received considerable attention however few results are concerned with stability of singular hybrid stochastic systems based on a stability result for singular hybrid systems this note studies mean square stability of singular hybrid stochastic systems it is found that the diffusions and the markov chain may play important roles in system analysis in mean square sense as a special case a criterion for mean square stability of hybrid stochastic systems is developed by the frequency domain approach

approximation of continuous time stochastic processes by a local linearization method this paper investigates the rate of convergence of an alternative approximation method for stochastic differential equations the rates of convergence of the one step and multi step approximation errors are proved to be o t 2 and o t in the l p sense respectively where t is discrete time interval the rate of convergence of the one step approximation error is improved as compared with methods assuming the value of brownian motion to be known only at discrete time through numerical experiments the rate of convergence of the multi step approximation error is seen to be much faster than in the conventional method

parameter estimation for autoregressive gaussian mixture processes the emax algorithm the problem of estimating parameters of discrete time non gaussian autoregressive ar processes is addressed the subclass of such processes considered is restricted to those whose driving noise samples are statistically independent and identically distributed according to a gaussian mixture probability density function pdf because the likelihood function for this problem is typically unbounded in the vicinity of undesirable degenerate parameter estimates the maximum likelihood approach is not fruitful hence an alternative approach is taken whereby a finite local maximum of the likelihood surface is sought this approach which is termed the quasimaximum likelihood qml approach is used to obtain estimates of the ar parameters as well as the means variances and weighting coefficients that define the gaussian mixture pdf a technique for generating solutions to the qml problem is derived using a generalized version of the expectation maximization principle this technique which is referred to as the emax algorithm is applied in four illustrative examples its performance is compared directly with that of previously proposed algorithms based on the same data model and that of conventional least squares techniques

multistep methods for sdes and their application to problems with small noise in this article the numerical approximation of solutions of ito stochastic differential equations is considered in particular for equations with a small parameter epsilon in the noise coefficient we construct stochastic linear multistep methods and develop the fundamental numerical analysis concerning their mean square consistency numerical stability in the mean square sense and mean square convergence for the special case of two step maruyama schemes we derive conditions guaranteeing their mean square consistency further for the small noise case we obtain expansions of the local error in terms of the step size and the small parameter epsilon simulation results using several explicit and implicit stochastic linear k step schemes k 1 2 illustrate the theoretical findings

numerical solution of stochastic differential algebraic equations with applications to transient noise simulation of microelectronic circuits the transient simulation of noise in electronic circuits leads to differential algebraic equations additively disturbed by white noise for these systems we present a mathematical model based on the theory of stochastic differential equations along with an implicit two step method for their numerical treatment this numerical scheme works directly on the given structure of the equations which makes very efficient implementations possible the order of convergence is preserved the theoretical results are verified by numerical noise simulations of benchmark circuits

efficient blind system identification of non gaussian autoregressive models with hmm modeling of the excitation we have previously proposed a blind system identification method that exploits the underlying dynamics of non gaussian signals in li and andersen blind identification of non gaussian autoregressive models for efficient analysis of speech signals proceedings of the international conference on acoustics speech and signal processing icassp may 2006 vol 1 pp i 1205 i 1208 the signal model being identified is an autoregressive ar model driven by a discrete state hidden markov process an exact expectation maximization em algorithm was derived for the joint estimation of the ar parameters and the hidden markov model hmm parameters in this paper we extend the system model by introducing an additive measurement noise the identification of the extended system model becomes much more complicated since the system output is now hidden we propose an exact em algorithm that incorporates a novel switching kalman smoother which obtains nonlinear minimum mean square error mmse estimates of the system output based on the state information given by the hmm filter the exact em algorithms for both models are obtainable only by appropriate constraints in the model design and have better convergence properties than algorithms employing generalized em algorithm or empirical iterative schemes the proposed methods also enjoy good data efficiency since only second order statistics are involved in the computation the signal models are general and suitable to numerous signals such as speech and baseband communication signals this paper describes the two system identification algorithms in an integrated form and provides supplementary results to the noise free model and new results to the extended model with applications in speech analysis and channel equalization

stochastic optimal control of state constrained systems in this article we consider the problem of stochastic optimal control in continuous time and state action space of systems with state constraints these systems typically appear in the area of robotics where hard obstacles constrain the state space of the robot a common approach is to solve the problem locally using a linear quadratic gaussian lqg method we take a different approach and apply path integral control as introduced by kappen kappen h j 2005a path integrals and symmetry breaking for optimal control theory journal of statistical mechanics theory and experiment 2005 p11011 kappen h j 2005b linear theory for control of nonlinear stochastic systems physical review letters 95 200201 we use hybrid monte carlo sampling to infer the control we introduce an adaptive time discretisation scheme for the simulation of the controlled dynamics we demonstrate our approach on two examples a simple particle in a halfspace and a more complex two joint manipulator and we show tha

regular frequency computations an m n computation of a function f is given by a deterministic turing machine which on n pairwise different inputs produces n output values where at least m of the n values are in accordance with f in such a case we say that the turing machine computes f with frequency m n the most prominent result for frequency computations is due to trakhtenbrot the class of m n computable functions equals the class of computable functions if and only if 2m n via characteristic functions the definition of m n computability carries over to sets here trakhtenbrot s result reads as the class of m n computable sets equals the class of recursive sets if and only if 2m n the notion of frequency computation can be extended to other models of computation for resource bounded computations the behavior is completely different for e g whenever n m n m it is known that under any reasonable resource bound there are sets m n computable but not m n computable however scaling down to finite automata the analogue of trakhtenbrot s result holds again we show here that the class of languages m n recognizable by deterministic finite automata equals the class of regular languages if and only if 2m n this was originally stated by kinber but his proof has a flaw as pointed out by tantau conversely for 2m n the class of languages m n recognizable by deterministic finite automata is uncountable for a two letter alphabet when restricted to a one letter alphabet then every m n recognizable language is regular this was also shown by kinber we give a new and more direct proof for this result

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

polynomial terse sets abstract let a be a set and k n be such that we wish to know the answers to x1 a x2 a xk a for various k tuples x1 x2 xk if this problem requires k queries to a in order to be solved in polynomial time then a is called polynomial terse or pterse we show the existence of both arbitrarily complex pterse and non pterse sets and that p np iff every np complete set is pterse we also show connections with p immunity p selective p generic sets and the boolean hierarchy in our framework unique satisfiability and a variation of it called ksat is in some sense close to satisfiability

simultaneous time and chance discretization for stochastic differential equations abstract the paper deals with weak approximations of stochastic differential equations of ito type where convergence rates of the approximate solutions are shown using e c t 0 t p p 2 the rates can also be interpreted as rates for the lp wasserstein metrics p 1 between the distributions of exact and approximate solutions the two approximation schemes considered are a combination of the time discretization methods of euler and milshtein with a chance discretization based on the invariance principle and they work on a grid constructed to tune both discretizations

application of cumulant method in performance evaluation of turbo like codes in this paper a new method for performance evaluation of turbo like codes is presented this is based on estimating the probability density function of the bit log likelihood ratio using higher order statistics we do not restrict ourselves to any specific model for the pdf and try to estimate it directly using a cumulant matching method numerical results show a close agreement between the proposed method and the simulations the complexity of this method is similar to the monte carlo simulation with the advantage of providing similar accuracy using significantly fewer samples

runge kutta methods for stratonovich stochastic differential equation systems with commutative noise a class of explicit stochastic runge kutta srk methods for stratonovich stochastic differential equation systems w r t m dimensional wiener processes satisfying a commutativity condition is developed general conditions for the coefficients of the srk method assuring convergence with order two in the weak sense are presented due to the commutativity condition no correlated random variables have to be generated for the considered runge kutta methods

order conditions of stochastic runge kutta methods by b series in this paper general order conditions and a global convergence proof are given for stochastic runge kutta methods applied to stochastic ordinary differential equations sodes of stratonovich type this work generalizes the ideas of b series as applied to deterministic ordinary differential equations odes to the stochastic case and allows a completely general formalism for constructing high order stochastic methods either explicit or implicit some numerical results will be given to illustrate this theory

adaptive weak approximation of stochastic differential equations adaptive time stepping methods based on the monte carlo euler method for weak approximation of ito stochastic differential equations are developed the main result is new expansions of the computational error with computable leading order term in a posteriori form based on stochastic flows and discrete dual backward problems the expansions lead to efficient and accurate computation of error estimates adaptive algorithms for either stochastic time steps or deterministic time steps are described numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps stochastic time steps use brownian bridges and require more work for a given number of time steps deterministic time steps may yield more time steps but require less work for example in the limit of vanishing error tolerance the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps c 2001 john wiley sons inc

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

a variable stepsize implementation for stochastic differential equations stochastic differential equations sdes arise from physical systems where the parameters describing the system can only be estimated or are subject to noise much work has been done recently on developing higher order runge kutta methods for solving sdes numerically fixed stepsize implementations of numerical methods have limitations when for example the sde being solved is stiff as this forces the stepsize to be very small this paper presents a completely general variable stepsize implementation of an embedded runge kutta pair for solving sdes numerically in this implementation there is no restriction on the value used for the stepsize and it is demonstrated that the integration remains on the correct brownian path

higher order time integration of coulomb collisions in a plasma using langevin equations the extension of langevin equation monte carlo algorithms for coulomb collisions from the conventional euler maruyama time integration to the next higher order of accuracy the milstein scheme has been developed implemented and tested this extension proceeds via a formulation of the angular scattering directly as stochastic differential equations in the fixed frame spherical coordinate velocity variables results from the numerical implementation show the expected improvement o dt vs o dt 1 2 in the strong convergence rate both for the speed v and angular components of the scattering an important result is that this improved convergence is achieved for the angular component of the scattering if and only if the area integral terms in the milstein scheme are included the resulting milstein scheme is of value as a step towards algorithms with both improved accuracy and efficiency these include both algorithms with improved convergence in the averages weak convergence and multi time level schemes the latter have been shown to give a greatly reduced cost for a given overall error level when compared with conventional monte carlo schemes and their performance is improved considerably when the milstein algorithm is used for the underlying time advance versus the euler maruyama algorithm a new method for sampling the area integrals is given which is a simplification of an earlier direct method and which retains high accuracy this method while being useful in its own right because of its relative simplicity is also expected to considerably reduce the computational requirements for the direct conditional sampling of the area integrals that is needed for adaptive strong integration

parameter estimation for partially observed hypoelliptic diffusions hypoelliptic diffusion processes can be used to model a variety of phenomena in applications ranging from molecular dynamics to audio signal analysis we study parameter estimation for such processes in situations where we observe some components of the solution at discrete times since exact likelihoods for the transition densities are typically not known approximations are used that are expected to work well in the limit of small intersample times t and large total observation times n t hypoellipticity together with partial observation leads to ill conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated we combine these in a deterministic scan gibbs sampler alternating between missing data in the unobserved solution components and parameters numerical experiments illustrate asymptotic consistency of the method when applied to simulated data the paper concludes with an application of the gibbs sampler to molecular dynamics data

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

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

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

stochastic differential algebraic equations of index 1 and applications in circuit simulation we discuss differential algebraic equations driven by gaussian white noise which are assumed to have noise free constraints and to be uniformly of dae index 1 we first provide a rigorous mathematical foundation of the existence and uniqueness of strong solutions our theory is based upon the theory of stochastic differential equations sdes and the theory of differential algebraic equations daes to each of which our problem reduces on making appropriate simplifications we then consider discretization methods implicit methods are necessary because of the differential algebraic structure and we consider adaptations of such methods used for sdes the consequences of an inexact solution of the implicit equations roundoff and truncation errors are analysed by means of the mean square numerical stability of general drift implicit discretization schemes for sdes we prove that the convergence properties of our drift implicit euler scheme split step backward euler scheme trapezoidal scheme and drift implicit milstein scheme carry over from the corresponding properties of these methods applied to sdes finally we show how the theory applies to the transient noise simulation of electronic circuits

convergence and stability of numerical solutions to a class of index 1 stochastic differential algebraic equations with time delay in this paper we study the convergence and stability of the stochastic theta method stm for a class of index 1 stochastic delay differential algebraic equations first in the case of constrained mesh i e the stepsize is a submultiple of the delay it is proved that the method is strongly consistent and convergent with order 1 2 in the mean square sense then the result is further extended to the case of non constrained mesh where we employ linear interpolation to approximate the delay argument later under a sufficient condition for mean square stability of the analytical solution it is proved that when the stepsizes are sufficiently small the stm approximations reproduce the stability of the analytical solution finally some numerical experiments are presented to illustrate the theoretical findings

numerical methods for strong solutions of stochastic differential equations an overview this paper gives a review of recent progress in the design of numerical methods for computing the trajectories sample paths of solutions to stochastic differential equations we give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important with a particular focus on computational biology applications and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes we present the stochastic taylor series expansion as the fundamental mechanism for constructing effective numerical methods give general results that relate local and global order of convergence and mention the magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem we also present various classes of explicit and implicit methods for strong solutions based on the underlying structure of the problem finally we discuss implementation issues relating to maintaining the brownian path efficient simulation of stochastic integrals and variable step size implementations based on various types of control

efficient strong integrators for linear stochastic systems we present numerical schemes for the strong solution of linear stochastic differential equations driven by an arbitrary number of wiener processes these schemes are based on the neumann stochastic taylor and magnus expansions first we consider the case when the governing linear diffusion vector fields commute with each other but not with the linear drift vector field we prove that numerical methods based on the magnus expansion are more accurate in the mean square sense than corresponding stochastic taylor integration schemes second we derive the maximal rate of convergence for arbitrary multidimensional stochastic integrals approximated by their conditional expectations consequently for general nonlinear stochastic differential equations with noncommuting vector fields we deduce explicit formulae for the relation between error and computational costs for methods of arbitrary order third we consider the consequences in two numerical studies one of which is an application arising in stochastic linear quadratic optimal control

stability of random attractors under perturbation and approximation abstract the comparison of the long time behaviour of dynamical systems and their numerical approximations is not straightforward since in general such methods only converge on bounded time intervals however one can still compare their asymptotic behaviour using the global attractor and this is now standard in the deterministic autonomous case for random dynamical systems there is an additional problem since the convergence of numerical methods for such systems is usually given only on average in this paper the deterministic approach is extended to cover stochastic differential equations giving necessary and sufficient conditions for the random attractor arising from a random dynamical system to be upper semi continuous with respect to a given family of perturbations or approximations

optimal approximation of stochastic differential equations by adaptive step size control we study the pathwise strong approximation of scalar stochastic differential equations with respect to the global error in the l2 norm for equations with additive noise we establish a sharp lower error bound in the class of arbitrary methods that use a fixed number of observations of the driving brownian motion as a consequence higher order methods do not exist if the global error is analyzed we introduce an adaptive step size control for the euler scheme which performs asymptotically optimally in particular the new method is more efficient than an equidistant discretization this superiority is confirmed in simulation experiments for equations with additive noise as well as for general scalar equations

adaptive stepsize based on control theory for stochastic differential equations the numerical solution of stochastic differential equations sdes has been focussed recently on the development of numerical methods with good stability and order properties these numerical implementations have been made with fixed stepsize but there are many situations when a fixed stepsize is not appropriate in the numerical solution of ordinary differential equations much work has been carried out on developing robust implementation techniques using variable stepsize it has been necessary in the deterministic case to consider the best choice for an initial stepsize as well as developing effective strategies for stepsize control the same of course must be carried out in the stochastic case in this paper proportional integral pi control is applied to a variable stepsize implementation of an embedded pair of stochastic runge kutta methods used to obtain numerical solutions of nonstiff sdes for stiff sdes the embedded pair of the balanced milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change the extension of these stepsize controllers from a digital filter theory point of view via pi with derivative pid control will also be implemented the implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy

on the complexity of stochastic integration we study the complexity of approximating stochastic integrals with error e for various classes of functions for ito integration we show that the complexity is of order e 1 even for classes of very smooth functions the lower bound is obtained by showing that ito integration is not easier than lebesgue integration in the average case setting with the wiener measure the upper bound is obtained by the milstein algorithm which is almost optimal in the considered classes of functions the milstein algorithm uses the values of the brownian motion and the integrand it is bilinear in these values and is very easy to implement for stratonovich integration we show that the complexity depends on the smoothness of the integrand and may be much smaller than the complexity of ito integration

a fast algorithm for the first passage times of gauss markov processes with holder continuous boundaries even for simple diffusion processes treating first passage problems analytically proves intractable for generic barriers and existing numerical methods are inaccurate and computationally costly here we present a novel numerical method that is faster and has more tightly controlled accuracy our algorithm is a probabilistic variant of dichotomic search for the computation of first passage times through non negative homogeneously holder continuous boundaries by gauss markov processes these include the ornstein uhlenbeck process underlying the ubiquitous leaky integrate and fire model of neuronal excitation our method evaluates discrete points in a sample path exactly and refines this representation recursively only in regions where a passage is rigorously estimated to be probable e g when close to the boundary

runge kutta methods for jump diffusion differential equations in this paper we consider runge kutta methods for jump diffusion differential equations we present a study of their mean square convergence properties for problems with multiplicative noise we are concerned with two classes of runge kutta methods first we analyse schemes where the drift is approximated by a runge kutta ansatz and the diffusion and jump part by a maruyama term and second we discuss improved methods where mixed stochastic integrals are incorporated in the approximation of the next time step as well as the stage values of the runge kutta ansatz for the drift the second class of methods are specifically developed to improve the accuracy behaviour of problems with small noise we present results showing when the implicit stochastic equations defining the stage values of the runge kutta methods are uniquely solvable finally simulation results illustrate the theoretical findings

an adaptive timestepping algorithm for stochastic differential equations we introduce a variable timestepping procedure using local error control for the pathwise strong numerical integration of a system of stochastic differential equations forced by a single wiener process the milstein method is used to advance the numerical solution and the stepsizes are determined via two local error estimates that roughly correspond to leading order deterministic and stochastic local error components one advantage of using two error controls is an increased flexibility that allows for the treatment of both drift and diffusion dominated regimes in a consistent manner numerical results are presented and the generalization of this approach to wider classes of problems and methods is discussed

feasibility study on the least square method for fitting non gaussian noise data abstract this study is to investigate the feasibility of least square method in fitting non gaussian noise data we add different levels of the two typical non gaussian noises levy and stretched gaussian noises to exact value of the selected functions including linear equations polynomial and exponential equations and the maximum absolute and the mean square errors are calculated for the different cases levy and stretched gaussian distributions have many applications in fractional and fractal calculus it is observed that the non gaussian noises are less accurately fitted than the gaussian noise but the stretched gaussian cases appear to perform better than the levy noise cases it is stressed that the least squares method is inapplicable to the non gaussian noise cases when the noise level is larger than 5

mean square convergence of stochastic multi step methods with variable step size we study mean square consistency stability in the mean square sense and mean square convergence of drift implicit linear multi step methods with variable step size for the approximation of the solution of ito stochastic differential equations we obtain conditions that depend on the step size ratios and that ensure mean square convergence for the special case of adaptive two step maruyama schemes further in the case of small noise we develop a local error analysis with respect to the h e approach and we construct some stochastic linear multi step methods with variable step size that have order 2 behaviour if the noise is small enough