a natural approach to the numerical integration of riccati differential equations this paper introduces a new class of methods which we call mobius schemes for the numerical solution of matrix riccati differential equations the approach is based on viewing the riccati equation in its natural geometric setting as a flow on the grassmannian of m dimensional subspaces of an n m dimensional vector space since the grassmannians are compact differentiable manifolds and the coefficients of the equation are assumed continuous there are no singularities or intrinsic instabilities in the associated flow the presence of singularities and numerical instabilities is an artifact of the coordinate system but since mobius schemes are based on the natural geometry they are able to deal with numerical instability and pass accurately through the singularities a number of examples are given to demonstrate these properties

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

flots et series de taylor stochastiques we study the expansion of the solution of a stochastic differential equation as an infinite sum of iterated stochastic stratonovitch integrals this enables us to give a universal and explicit formula for any invariant diffusion on a lie group in terms of lie brackets as well as a universal and explicit formula for the brownian motion on a riemannian manifold in terms of derivatives of the curvature tensor the first of these formulae contains and extends to the non nilpotent case the results of doss 6 sussmann 17 yamato 18 fliess and normand cyrot 7 krener and lobry 19 and kunita 11 on the representation of solutions of stochastic differential equations

adaptive lossless steganographic scheme with centralized difference expansion in this paper a novel adaptive lossless data hiding scheme is presented that is capable of offering greater embedding capacity than the existing schemes unlike the fixed hiding capacity each block provides in most of the currently available lossless data hiding approaches the proposed method utilizes a block based lossless data embedding algorithm where the quantity of the hidden information each block bears is variable to both reduce the image distortion and increase the hiding capacity the payload of each block depends on its cover image complexity due to the fact that schemes with difference expansion tend to damage the image quality seriously in the edge areas in the proposed scheme smoother areas are chosen to conceal more secret bits this way a better balance can be reached between the embedding ratio and the stego image quality in addition when recovered the cover image can came back to its old self to the last bit without any distortion at all experimental results as this paper will show have demonstrated that the proposed method is capable of hiding more secret data while maintaining imperceptible stego image quality degradation

a lie algebraic approach to numerical integration of stochastic differential equations in this article composition methods or operator splitting methods for autonomous stochastic differential equations sdes are formulated to produce numerical approximation schemes for the equations in the proposed methods the exponential map which is given by the solution of an sde is approximated by composition of the stochastic flows derived from simpler and exactly integrable vector field operators having stochastic coefficients the local and global errors of the numerical schemes derived from the stochastic composition methods are investigated the new schemes are advantageous to preserve the special character of sdes numerically and are useful for approximations of the solutions to stochastic nonlinear equations to examine their superiority several numerical simulations on the basis of the proposed schemes are carried out for sdes which arise in mathematical finance and stochastic hamiltonian dynamical systems

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

multilevel monte carlo path simulation we show that multigrid ideas can be used to reduce the computational complexity of estimating an expected value arising from a stochastic differential equation using monte carlo path simulations in the simplest case of a lipschitz payoff and a euler discretisation the computational cost to achieve an accuracy of o e is reduced from o e 3 to o e 2 log e 2 the analysis is supported by numerical results showing significant computational savings

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

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

lossless data hiding for color images based on block truncation coding in this paper we present a novel reversible steganographic method which can reconstruct an original image effectively after extracting the embedded secret data the proposed reversible hiding method aims at btc block truncation coding compressed color images conventionally each block of a color image compressed by btc requires three bitmaps and three pairs of quantization levels for reconstruction in order to improve the compression rate a genetic algorithm ga is applied to find an approximate optimal common bitmap to replace the original three the secret data then are embedded in the common bitmap and the quantization levels of each block use the properties of side matching and the order of these quantization levels to achieve reversibility the experimental results demonstrate that the proposed method is practical for btc compressed color images and can embed more than three bits in each btc encoded block on average

multidimensional backward stochastic riccati equations and applications backward stochastic riccati differential equations bsrdes for short arise from the solution of general linear quadratic optimal stochastic control problems with random coefficients the existence and uniqueness question of the global adapted solutions has been open since bismut s pioneering research publication in 1978 seminaire de probabilites xii lecture notes in math 649 c dellacherie p a meyer and m weil eds springer verlag berlin 1978 pp 180 264 one distinguishing difficulty lies in the quadratic nonlinearity of the drift term in the second unknown component in a previous article stochastic process appl 97 2002 pp 255 288 the authors solved the one dimensional case driven by brownian motions in this paper the multidimensional case driven by brownian motions is studied a closeness property for solutions of bsrdes with respect to their coefficients is stated and is proved for general bsrdes which is used to obtain the existence of a global adapted solution to some bsrdes the global existence and uniqueness results are obtained for two classes of bsrdes whose generators contain a quadratic term of l the second unknown component more specifically the two classes of bsrdes are for the regular case n 0 email protected email protected email protected dk displaystyle a k ka q ld n d kd 1 d l dt l dw k t m end array right under the condition d 1 and for the singular case left begin array email protected email protected email protected dk displaystyle a k ka c kc q c l lc displaystyle kb c kd ld d kd 1 kb c kd ld dt displaystyle l dw k t m end array right under the condition d 1 and m n the arguments given in this paper are completely new and they consist of some simple techniques of algebraic matrix transformations and direct applications of the closeness property mentioned above we make full use of the special structure the nonnegativity of the quadratic term for example of the underlying riccati equation applications in optimal stochastic control are exposed

numerical methods for second order stochastic differential equations we seek numerical methods for second order stochastic differential equations that reproduce the stationary density accurately for all values of damping a complete analysis is possible for scalar linear second order equations damped harmonic oscillators with additive noise where the statistics are gaussian and can be calculated exactly in the continuous time and discrete time cases a matrix equation is given for the stationary variances and correlation for methods using one gaussian random variable per timestep the only runge kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule numerical experiments comparing the implicit midpoint rule with heun and leapfrog methods on nonlinear equations with additive or multiplicative noise produce behavior similar to the linear case

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

the campbell baker hausdorff dynkin formula and solutions of differential equations the campbell baker hausdorff dynkin formula is a special case of a simpler and more general formula for the solution of nonautonomous systems of first order ordinary differential equations in terms of autonomous systems specifically suppose u t takes values in a c manifold and satisfies the initial value problem u t a t u t u 0 a where a t is a vector field on the manifold depending continuously on t then u t exp z t a here exp z a means the solution at s 1 to v s z v s v 0 a for z t r 1 pr e r2r 1e tr t x a s 1 a s 2 a s r ds as t 0 where tr t s rr 0 s1 s2 sr t pr is the set of permutations on 1 r e is the number of errors in ordering consecutive terms in 1 r and is the usual commutator of vector fields under appropriate analyticity assumptions the series for z t is convergent for small t this formula gives an explicit formulation of results of k t chen published in 1957 applications are given to problems in sub riemannian geometry and to improving convergence estimates for the campbell baker hausdorff dynkin formula in the context of banach algebras our formula can be thought of as a noncommutative generalization of the familiar formula u t a exp 0t a s ds in the scalar linear case in the same way that the campbell baker hausdorff dynkin formula is a noncommutative generalization of the familiar formula exey ex y

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

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

numerical methods for stochastic systems preserving symplectic structure stochastic hamiltonian systems with multiplicative noise phase flows of which preserve symplectic structure are considered to construct symplectic methods for such systems sufficiently general fully implicit schemes i e schemes with implicitness both in deterministic and stochastic terms are needed a new class of fully implicit methods for stochastic systems is proposed increments of wiener processes in these fully implicit schemes are substituted by some truncated random variables a number of symplectic integrators is constructed special attention is paid to systems with separable hamiltonians some results of numerical experiments are presented they demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones

efficient computation of the matrix exponential by generalized polar decompositions in this paper we explore the computation of the matrix exponential in a manner that is consistent with lie group structure our point of departure is the method of generalized polar decompositions which we modify and combine with similarity transformations that bring the underlying matrix to a form more amenable to efficient computation we develop techniques valid for a range of lie groups the orthogonal group the symplectic group lorentz isotropy and scaling groups however the gpd approach is equally promising in a more general context even when lie group structure is not at issue our algorithm is more efficient in many settings than classical methods for the computation of the matrix exponential

explicit magnus expansions for nonlinear equations in this paper we develop and analyse new explicit magnus expansions for the nonlinear equation y a t y y defined in a matrix lie group in particular integration methods up to order four are presented in terms of integrals which can be either evaluated exactly or replaced by conveniently adapted quadrature rules the structure of the algorithm allows us to change the step size and even the order during the integration process thus improving its efficiency several examples are considered including isospectral flows and highly oscillatory nonlinear differential equations

exponential integrators for stochastic schrodinger equations we present a class of exponential integrators to compute solutions of the stochastic schrodinger equations arising from the modeling of open quantum systems to be able to implement the methods within the same framework as the deterministic counterpart we express the solution using kunita s representation with appropriate truncations the solution operator can be written as matrix exponentials which can be efficiently implemented by the krylov subspace projection the accuracy is examined in terms of the strong convergence by comparing trajectories and in terms of the weak convergence by comparing the density matrix operators we show that the local accuracy can be further improved by introducing third order commutators in the exponential the effectiveness of the proposed methods is tested using the example from di ventra et al j phys condens matter 16 8025 2004 jcomel0953 898410 1088 0953 8984 16 45 024

convex functions on alexandrov surfaces we investigate the topological structure of alexandrov surfaces of curvature bounded below which possess convex functions we do not assume the continuities of these functions nevertheless if the convex functions satisfy a condition of local nonconstancy then the topological structures of alexandrov surfaces and the level sets configurations of these functions in question are determined

cubature on wiener space it is well known that there is a mathematical equivalence between solving parabolic partial differential equations pdes and the integration of certain functionals on wiener space monte carlo simulation of stochastic differential equations sdes is a naive approach based on this underlying principle in finite dimensions it is well known that cubature can be a very effective approach to integration we discuss the appropriate extension of this idea to wiener space in the process we develop high order numerical schemes valid for high dimensional sdes and semi elliptic pdes

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

high strong order methods for non commutative stochastic ordinary differential equation systems and the magnus formula in recent years considerable attention has been paid to the numerical solution of stochastic ordinary differential equations sodes as sodes are often more appropriate than their deterministic counterparts in many modelling situations however unlike the deterministic case numerical methods for sodes are considerably less sophisticated due to the difficulty in representing the possibly large number of random variable approximations to the stochastic integrals although burrage and burrage high strong order explicit runge kutta methods for stochastic ordinary differential equations applied numerical mathematics 22 1996 81 101 were able to construct strong local order 1 5 stochastic runge kutta methods for certain cases it is known that all extant stochastic runge kutta methods suffer an order reduction down to strong order 0 5 if there is non commutativity between the functions associated with the multiple wiener processes this order reduction down to that of the euler maruyama method imposes severe difficulties in obtaining meaningful solutions in a reasonable time frame and this paper attempts to circumvent these difficulties by some new techniques an additional difficulty in solving sodes arises even in the linear case since it is not possible to write the solution analytically in terms of matrix exponentials unless there is a commutativity property between the functions associated with the multiple wiener processes thus in this present paper first the work of magnus on the exponential solution of differential equations for a linear operator communications on pure and applied mathematics 7 1954 649 673 applied to deterministic non commutative linear problems will be applied to non commutative linear sodes and methods of strong order 1 5 for arbitrary linear non commutative sode systems will be constructed hence giving an accurate approximation to the general linear problem secondly for general nonlinear non commutative systems with an arbitrary number d of wiener processes it is shown that strong local order i runge kutta methods with d 1 stages can be constructed by evaluated a set of lie brackets as well as the standard function evaluations a method is then constructed which can be efficiently implemented in a parallel environment for this arbitrary number of wiener processes finally some numerical results are presented which illustrate the efficacy of these approaches c 1999 elsevier science b v all rights reserved

on wong zakai approximations with δ martingales we present a class of families of absolutely continuous stochastic processes approximating the wiener process we estimate the rate of convergence of the solutions of the stochastic differential equations driven by these families

a new framework of lsb steganalysis of digital media we propose a general framework for the detection of the least significant bit lsb steganography using digital media files as cover objects the new framework exploits high order statistics of the samples it can compute a robust estimate of the length of a secret message hidden in the lsbs of samples for a large class of digital media contents such as image video and audio in which the underlying signals consist of correlated samples a case study on the lsb steganalysis of natural grey scale and color images and experimental results are reported

a survey of nonsymmetric riccati equations abstract we survey recent and also older results on nonsymmetric matrix riccati differential equations and in the time invariant case on the corresponding algebraic riccati equations in particular we cite various applications connected with matrix riccati equations

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

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

asymptotic expansion of stochastic flows we study the asymptotic expansion in small time of the solution of a stochastic differential equation we obtain a universal and explicit formula in terms of lie brackets and iterated stochastic stratonovich integrals this formula contains the results of doss 6 sussmann 15 fliess and normand cyrot 7 krener and lobry 10 yamato 17 and kunita 11 in the nilpotent case and extends to general diffusions the representation given by ben arous 3 for invariant diffusions on a lie group the main tool is an asymptotic expansion for deterministic ordinary differential equations given by strichartz 14

stochastic expansions and hopf algebras we study solutions to nonlinear stochastic differential systems driven by a multi dimensional wiener process a useful algorithm for strongly simulating such stochastic systems is the castell gaines method which is based on the exponential lie series when the diffusion vector fields commute it has been proved that at low orders this method is more accurate in the mean square error than corresponding stochastic taylor methods however it has also been shown that when the diffusion vector fields do not commute this is not true for strong order one methods here we prove that when there is no drift and the diffusion vector fields do not commute the exponential lie series is usurped by the sinh log series in other words the mean square error associated with a numerical method based on the sinh log series is always smaller than the corresponding stochastic taylor error in fact to all orders our proof uses the underlying hopf algebra structure of these series and a two alphabet associative algebra of shuffle and concatenation operations we illustrate the benefits of the proposed series in numerical studies

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

algebraic structures and stochastic differential equations driven by levy processes we construct an efficient integrator for stochastic differential systems driven by lvy processes an efficient integrator is a strong approximation that is more accurate than the corresponding stoc

mean variance hedging and stochastic control beyond the brownian setting we show for continuous semimartingales in a general filtration how the mean variance hedging problem can be treated as a linear quadratic stochastic control problem the adjoint equations lead to backward stochastic differential equations for the three coefficients of the quadratic value process and we give necessary and sufficient conditions for the solvability of these generalized stochastic riccati equations motivated from mathematical finance this paper takes a first step toward linear quadratic stochastic control in more general than brownian settings