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

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

numerical simulation of creeping fluid flow in reconstruction models of porous media abstract in this paper we examine representative examples of realistic three dimensional models for porous media by comparing their geometry and permeability with those of the original experimental specimen the comparison is based on numerically exact evaluations of permeability porosity specific internal surface mean curvature euler number and local percolation probabilities the experimental specimen is a three dimensional computer tomographic image of fontainebleau sandstone the three models are stochastic reconstructions for which many of the geometrical characteristics coincide with those of the experimental specimen we find that in spite of the similarity in the geometrical properties the permeability and formation factor can differ greatly between models and the experiment

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

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

on volterra equations driven by semimartingales on considere des equations stochastiques de la forme x t h t o t f t s x dzs ou z est une semimartingale de dimension finie h est un processus cadlag et f est une fonctionnelle

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

a remark on the approximation of p compact operators by finite rank operators abstract the banach operator ideal k p of p compact operators was introduced in p d sinha a k karn compact operators whose adjoints factor through subspaces of l p studia math 150 2002 17 33 let p 1 and let x be a closed subspace of an l p space we show that x has the approximation property if and only if for every banach space y the linear space f y x of finite rank operators is k p norm dense in k p y x i e x has the k p approximation property

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

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

the dual space of l x y τp and the p approximation property we establish a representation of the dual space of l x y the space of bounded linear operators from a banach space x into a banach space y endowed with the topology p of uniform convergence on p compact subsets of x we apply this representation and solve the duality problem for the p approximation property p ap that is if the dual space x has the p ap then so does x however the converse does not hold in general we show that given 2 2p p 2 this subspace is the davie space in lq davie 1973 5 which does not have the approximation property it follows that for every 2 p there exists a banach space yp such that it has the p ap but its dual space yp fails to have the p ap we study the relation of the p ap with the denseness of finite rank operators in the topology p finally we introduce the p compact approximation property p cap and show for every 2 p that the davie space in c0 fails to have the p cap and also that a variant of the willis space willis 1992 17 has the p cap but it fails to have the p ap

stochastic string models with continuous semimartingales abstract this paper reformulates the stochastic string model of santa clara and sornette using stochastic calculus with continuous semimartingales we present some new results such as a the dynamics of the short term interest rate b the pde that must be satisfied by the bond price and c an analytic expression for the price of a european bond call option additionally we clarify some important features of the stochastic string model and show its relevance to price derivatives and the equivalence with an infinite dimensional hjm model to price european options

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

applications of statistical physics to the oil industry predicting oil recovery using percolation theory in this paper we apply scaling laws from percolation theory to the problem of estimating the time for a uid injected into an oil eld for the purposes of recovering the oil to breakthrough into a production well the main contribution is to show that percolation theory when applied to a realistic model can be used to obtain the same results as calculated in a more conventional way but signicantly more quickly specically we found that a previously proposed scaling form for the breakthrough time distribution when applied to a real oil eld is in good agreement with more time consuming simulation results consequently these methods can be used in practical engineering circumstances to aid decision making for real eld problems c 1999 elsevier science b v all rights reserved

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

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

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

domain decomposition method for parallel lattice boltzmann simulation of incompressible flow in porous media the lattice boltzmann method has proven to be a promising method to simulate flow in porous media its practical application often relies on parallel computation because of the demand for a large domain and fine grid resolution to adequately resolve pore heterogeneity the existing domain decomposition methods for parallel computation usually decompose a domain into a number of subdomains first and then recover the interfaces and perform the load balance normally the interface recovery and the load balance have to be performed iteratively until an acceptable load balance is achieved this costs time in this paper we propose a cell based domain decomposition method for parallel lattice boltzmann simulation of flow in porous media unlike the existing methods the cell based method performs the load balance first to divide the total number of fluid cells into a number of groups or subdomains in which the difference of fluid cells in each group is either 0 or 1 depending on if the total number of fluid cells is a multiple of the processor numbers the interfaces between the subdomains are recovered at last the cell based method is to recover the interfaces rather than the load balance it does not need iteration and gives an exact load balance the performance of the proposed method is analyzed and compared using different computer systems the results indicate that it reaches the theoretical parallel efficiency the method is then applied to simulate flow in a three dimensional porous medium obtained with microfocus x ray computed tomography to calculate the permeability and the result shows good agreement with the experimental data

reconstruction of random media using monte carlo methods a simulated annealing algorithm is applied to the reconstruction of two dimensional porous media with prescribed correlation functions the experimental correlation function of an isotropic sample of fontainebleau sandstone and a synthetic correlation function with damped oscillations are used in the reconstructions to reduce the numerical effort we follow a proposal suggesting the evaluation of the correlation functions only along certain directions the results show that this simplification yields significantly different microstructures as compared to a full evaluation of the correlation function in particular we find that the simplified reconstruction method introduces an artificial anisotropy that is originally not present

on p compact mappings and the p approximation property abstract the notion of p compact sets arises naturally from grothendieck s characterization of compact sets as those contained in the convex hull of a norm null sequence the definition due to sinha and karn 2002 leads to the concepts of p approximation property and p compact operators which form an ideal with its ideal norm p this paper examines the interaction between the p approximation property and certain space of holomorphic functions the p compact analytic functions in order to understand these functions we define a p compact radius of convergence which allows us to give a characterization of the functions in the class we show that p compact holomorphic functions behave more like nuclear than compact maps we use the product of schwartz to characterize the p approximation property of a banach space in terms of p compact homogeneous polynomials and in terms of p compact holomorphic functions with range on the space finally we show that p compact holomorphic functions fit into the framework of holomorphy types which allows us to inspect the p approximation property our approach also allows us to solve several questions posed by aron maestre and rueda 2010

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

density of finite rank operators in the banach space of p compact operators a banach space x is said to have the kp approximation property kp ap if for every banach space y the space f y x of finite rank operators is dense in the space kp y x of p compact operators endowed with its natural ideal norm kp in this paper we study this notion that has been previously treated by sinha and karn 2002 in 15 as application the kp ap of dual banach spaces is characterized via density of finite rank operators in the space of quasi p nuclear operators for the p summing norm this allows to obtain a relation between the kp ap and saphar s approximation property as another application the kp ap is characterized in terms of a trace condition finally we relate the kp ap to the p p approximation property introduced in sinha and karn 2002 15 for subspaces of lp spaces

hydraulic permeability of ordered and disordered single layer arrays of cylinders the hydraulic permeability of single layer fibrous media is studied through two dimensional 2d and three dimensional navier stokes based flow simulations as simple representations of such materials one dimensional arrays of parallel cylinders as well as two dimensional arrays of perpendicularly crossing cylinders were used the distance between the cylinders was either constant ordered layers or variable disordered layers for both 1d and 2d ordered layers we propose a geometrical scaling rule for the hydraulic permeability as a function of cylinder radius and solid volume fraction porosity which is a modification of a scaling rule previously reported by clague and co workers the proposed modification is based on theoretical considerations and leads to significantly improved correspondence with simulation results the hydraulic permeability of unstructured layers is found to be higher than that of structured layers of equal porosity for both 1d and 2d arrays we propose a single parameter that can be easily determined experimentally to characterize the degree of disorder as well as a generally valid correction factor in the proposed geometrical scaling rule to account for the influence of disorder on the hydraulic permeability

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

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

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

lattice boltzmann and finite difference simulations for the permeability for three dimensional porous media numerical micropermeametry is performed on three dimensional porous samples having a linear size of approximately 3 mm and a resolution of 7 5 microm one of the samples is a microtomographic image of fontainebleau sandstone two of the samples are stochastic reconstructions with the same porosity specific surface area and two point correlation function as the fontainebleau sample the fourth sample is a physical model that mimics the processes of sedimentation compaction and diagenesis of fontainebleau sandstone the permeabilities of these samples are determined by numerically solving at low reynolds numbers the appropriate stokes equations in the pore spaces of the samples the physical diagenesis model appears to reproduce the permeability of the real sandstone sample quite accurately while the permeabilities of the stochastic reconstructions deviate from the latter by at least an order of magnitude this finding confirms earlier qualitative predictions based on local porosity theory two numerical algorithms were used in these simulations one is based on the lattice boltzmann method and the other on conventional finite difference techniques the accuracy of these two methods is discussed and compared also with experiment

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

reconstructing random media we formulate a procedure to reconstruct the structure of general random heterogeneous media from limited morphological information by extending the methodology of rintoul and torquato j colloid interface sci bold 186 467 1997 developed for dispersions the procedure has the advantages that it is simple to implement and generally applicable to multidimensional multiphase and anisotropic structures furthermore an extremely useful feature is that it can incorporate any type and number of correlation functions in order to provide as much morphological information as is necessary for accurate reconstruction we consider a variety of one and two dimensional reconstructions including periodic and random arrays of rods various distribution of disks debye random media and a fontainebleau sandstone sample we also use our algorithm to construct heterogeneous media from specified hypothetical correlation functions including an exponentially damped oscillating function as well as physically unrealizable ones copyright ital 1998 ital the american physical society

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

a direct proof of the bichteler dellacherie theorem and connections to arbitrage we give an elementary proof of the celebrated bichteler dellacherie theorem which states that the class of stochastic processes s allowing for a useful integration theory consists precisely of those processes which can be written in the form s m a where m is a local martingale and a is a finite variation process in other words s is a good integrator if and only if it is a semi martingale we obtain this decomposition rather directly from an elementary discrete time doob meyer decomposition by passing to convex combinations we ob tain a direct construction of the continuous time decomposition which then yields the desired decomposition as a by product of our proof we obtain a characterization of semi martingales in terms of a variant of no free lunch thus extending a result from ds94

the central limit theorem for stochastic integrals with respect to levy processes let m be a symmetric independently scattered random measure on lbrack 0 1 rbrack with control measure m which is uniformly in the domain of normal attraction of a stable measure of index p in 0 2 rbrack let f be a non anticipating process with respect to x t m lbrack 0 t rbrack if m is continuous and a previsible process in general satisfying int 1 0 e f p dm infty then the stochastic integral int t 0 f dm can be defined as a process in d lbrack 0 1 rbrack and is in the domain of normal attraction of a stable process of order p in d lbrack 0 1 rbrack in the sense of of weak convergence of probability measures if m is gaussian and continuous in probability then the central limit theorem holds in c lbrack 0 1 rbrack in particular ito and diffusion processes satisfy the clt our main tool is an upper bound for the weak l p norm of sup 0 leq t leq 1 int t 0 f dm in terms of the l p p times m norm of f

a note on p limited sets abstract given p 1 a subset a of a banach space x is said to be p limited if for every weakly p summable sequence x n in x there exists n l p such that x n x n for all x a and n n it is showed that p limited sets are q limited whenever p q and banach spaces enjoying the property that every q limited subset is p limited are characterized we also prove that an operator has p summing adjoint if and only if it maps relatively compact sets to p limited sets

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

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

parallel domain decomposition method with non blocking communication for flow through porous media this paper introduces a domain decomposition method for numerically solving the stokes equation for very large complex geometries examples arise from realistic porous media the computational method is based on the simple semi implicit method for pressure linked equations algorithm which uses a finite differences approach for discretizing the underlying equations it achieves comparable speed and efficiency as lattice boltzmann methods the domain decomposition method splits a large three dimensional region into slices that can be processed in parallel on multi processor computation environments with only minimal communication between the computation nodes with this method the flow through a porous medium with grid sizes up to 20483 voxel has been calculated

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