convergence and almost sure exponential stability of implicit numerical methods for a class of highly nonlinear neutral stochastic differential equations with constant delay this paper can be regarded as the continuation of the work contained in papers milosevic 2011 2013 at the same time it represents the extension of the paper wu et al 2010 in this paper the one sided lipschitz condition is employed in the context of the backward euler method for a class of neutral stochastic differential equations with constant delay sufficient conditions for this method to be well defined are revealed under certain nonlinear growth conditions the convergence in probability is established for the continuous forward backward euler method as well as for the discrete backward euler method additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without requiring for the drift coefficient to satisfy the linear growth condition this manuscript represents the continuation of the previous work of the author related to a class of highly nonlinear neutral stochastic differential equations with time dependent delay main results of this manuscript are convergence in probability and almost sure exponential stability of the backward euler approximate solution for a class of stochastic differential equations with constant delay conditions under which both the exact and approximate solutions share the property of the almost sure exponential stability are revealed this paper also illustrates that in some cases stochastic differential equations with constant delay should be studied separately from those with time dependent delay in that sense it should be stressed that results of this paper are obtained under weaker conditions comparing to those which would be obtained by replacing the time dependent delay by the constant delay

higher order implicit strong numerical schemes for stochastic differential equations higher order implicit numerical methods which are suitable for stiff stochastic differential equations are proposed these are based on a stochastic taylor expansion and converge strongly to the corresponding solution of the stochastic differential equation as the time step size converges to zero the regions of absolute stability of these implicit and related explicit methods are also examined

investigation of the logarithmic kdv equation involving mittag leffler type kernel with atangana baleanu derivative abstract this work presents analysis of the logarithmic kdv equation involving new fractional operator called atangana baleanu ab fractional derivative with mittag leffler ml type kernel the existence and uniqueness of the governing equation having ab fractional derivative with ml type kernel is proved with the aid of a fixed point theorem we present numerical simulations by using iterative algorithm the effectiveness of various parameters and variables on the displacement are presented in figures 1 and 2

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

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

variational problems with fractional derivatives invariance conditions and nother s theorem abstract a variational principle for lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases necessary and sufficient conditions for an infinitesimal transformation group basic nother s identity are obtained these conditions extend the classical results valid for integer order derivatives a generalization of nother s theorem leading to conservation laws for fractional euler lagrangian equation is obtained as well results are illustrated by several concrete examples finally an approximation of a fractional euler lagrangian equation by a system of integer order equations is used for the formulation of an approximated invariance condition and corresponding conservation laws

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

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

lie symmetries group invariant solutions and conservation laws of the vasicek pricing equation of mathematical finance abstract the one factor term structure model by vasicek is analysed from the point of view of lie symmetry analysis its one parameter lie point symmetries and corresponding group of adjoint representations are obtained an optimal system of one dimensional subalgebras is derived and is then used to obtain symmetry reductions and group invariant solutions the group invariant solutions presented here are new and have not appeared in the literature moreover we derive conservation laws for the vasicek equation by employing the theorem due to ibragimov

mean square stability of two classes of theta method for neutral stochastic differential delay equations in this paper a stochastic linear theta slt method is introduced and analyzed for neutral stochastic differential delay equations nsddes we give some conditions on neutral item drift and diffusion coefficients which admit that the diffusion coefficient can be highly nonlinear and does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the slt method with 1 2 1 is asymptotically mean stable and so is 0 1 2 under a stronger assumption furthermore we consider the split step theta sst method and obtain a similar but better result that is the sst method with 1 2 1 is exponentially mean stable and so is 0 1 2 finally two numerical examples are given to show the efficiency of the obtained results

on invariant analysis of some time fractional nonlinear systems of partial differential equations i an investigation of lie point symmetries for systems of time fractional partial differential equations including ito system coupled burgers equations coupled korteweg de vries equations hirota satsuma coupled kdv equations and coupled nonlinear hirota equations has been done using the obtained symmetries each one of the systems is reduced to the nonlinear system of fractional ordinary differential equations involving erdelyi kober fractional differential operator depending on a parameter

approximation of upper lyapunov exponents of bilinear stochastic differential systems a bilinear stochastic differential system is considered whose solution starting from x is denoted by x t x it is supposed that its upper lyapunov exponent lambda exists the purpose of the paper is to propose an efficient algorithm to approximate lambda it is shown that for markov chains bar x p h x p in mathbb n defined by several approximation schemes the lyapunov exponents bar lambda h are well defined and can be computed estimates are given in terms of the discretization step h of the theoretical error lambda bar lambda h then these results are applied to an engineering problem the stability of the motion of helicopter rotor blades in a turbulent wind

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

a wave function for stock market returns the instantaneous return on the financial times stock exchange ftse all share index is viewed as a frictionless particle moving in a one dimensional square well but where there is a non trivial probability of the particle tunneling into the well s retaining walls our analysis demonstrates how the complementarity principle from quantum mechanics applies to stock market prices and of how the wave function presented by it leads to a probability density which exhibits strong compatibility with returns earned on the ftse all share index in particular our analysis shows that the probability density for stock market returns is highly leptokurtic with slight though not significant negative skewness moreover the moments of the probability density determined under the complementarity principle employed here are all convergent in contrast to many of the probability density functions on which the received theory of finance is based

almost sure exponential stability of numerical solutions for stochastic delay differential equations using techniques based on the continuous and discrete semimartingale convergence theorems this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations sddes the important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of sddes directly this is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the chebyshev inequality and the borel cantelli lemma

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

mean square stability and dissipativity of two classes of theta methods for systems of stochastic delay differential equations in this paper we first study the mean square stability of numerical methods for stochastic delay differential equations under a coupled condition on the drift and diffusion coefficients this condition admits that the diffusion coefficient can be highly nonlinear i e it does not necessarily satisfy a linear growth or global lipschitz condition it is proved that for all positive stepsizes the classical stochastic theta method with q 0 5 is asymptotically mean square stable and the split step theta method with q 0 5 is exponentially mean square stable conditional stability results for the methods with q 0 5 and prove that the method possesses a bounded absorbing set in mean square independent of initial data

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

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

lie symmetry analysis explicit solutions and conservation laws for the space time fractional nonlinear evolution equations abstract this paper studies the symmetry analysis explicit solutions convergence analysis and conservation laws cls for two different space time fractional nonlinear evolution equations with riemann liouville rl derivative the governing equations are reduced to nonlinear ordinary differential equation ode of fractional order using their lie point symmetries in the reduced equations the derivative is in erdelyi kober ek sense power series technique is applied to derive an explicit solutions for the reduced fractional odes the convergence of the obtained power series solutions is also presented moreover the new conservation theorem and the generalization of the noether operators are developed to construct the nonlocal cls for the equations some interesting figures for the obtained explicit solutions are presented

implicit numerical methods for highly nonlinear neutral stochastic differential equations with time dependent delay abstract this paper represents the continuation of the analysis from papers milosevic 2011 10 and milosevic 2013 11 the main aim of this paper is to establish certain results for the backward euler method for a class of neutral stochastic differential equations with time dependent delay for that purpose the split step backward euler method which represents an extension of the backward euler method is introduced for this class of equations conditions under which the split step backward euler method and thus the backward euler method is well defined are revealed moreover the convergence in probability of the backward euler method is proved under certain nonlinear growth conditions including the one sided lipschitz condition this result is proved using the technique which is based on the application of the continuous time approximation for this reason the discrete forward backward euler method is involved since it allows its continuous version to be well defined from the aspect of measurability the convergence in probability is established for the continuous forward backward euler solution which is essential for proving the same result for both discrete forward backward and backward euler methods additionally it is proved that the discrete backward euler equilibrium solution is globally a s asymptotically exponentially stable without the linear growth condition on the drift coefficient of the equation as usual the whole consideration is affected by the presence and properties of the delay function

an approximate method via taylor series for stochastic functional differential equations abstract the subject of this paper is an analytic approximate method for stochastic functional differential equations whose coefficients are functionals sufficiently smooth in the sense of frechet derivatives the approximate equations are defined on equidistant partitions of the time interval and their coefficients are general taylor expansions of the coefficients of the initial equation it will be shown that the approximate solutions converge in the l p norm and with probability one to the solution of the initial equation and also that the rate of convergence increases when degrees in taylor expansions increase analogously to real analysis

invariance of a partial differential equation of fractional order under the lie group of scaling transformations abstract in this article a symmetry group of scaling transformations is determined for a partial differential equation of fractional order containing among particular cases the diffusion equation the wave equation and the fractional diffusion wave equation for its group invariant solutions an ordinary differential equation of fractional order with the new independent variable z xt 2 is derived the derivative then is an erdelyi kober derivative depending on a parameter its complete solution is given in terms of the wright and the generalized wright functions

exponential mean square stability of numerical methods for systems of stochastic differential equations this paper is concerned with exponential mean square stability of the classical stochastic theta method and the so called split step theta method for stochastic systems first we consider linear autonomous systems under a sufficient and necessary condition for exponential mean square stability of the exact solution it is proved that the two classes of theta methods with q 0 5 are exponentially mean square stable for all positive step sizes and the methods with q 0 5 still unconditionally preserves the exponential mean square stability of the underlying systems but the stochastic theta method does not have this property finally we consider stochastic differential equations with jumps some similar results are derived

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

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

new criteria on exponential stability of neutral stochastic differential delay equations neutral stochastic differential delay equations nsddes have recently been studied intensively see e g v b kolmanovskii v r nosov stability and periodic modes of control systems with aftereffect nauka moscow 1981 x mao exponential stability in mean square of neutral stochastic differential functional equations systems control lett 26 1995 245 251 x mao razumikhin type theorems on exponential stability of neutral stochastic functional differential equations siam j math anal 28 2 1997 389 401 x mao stochastic differential equations and their applications horwood publishing chichester 1997 more recently mao asymptotic properties of neutral stochastic differential delay equations stochastics and stochastics rep 68 2000 273 295 provided with some useful criteria on the exponential stability for nsddes however the criteria there require not only the coefficients of the nsddes to obey the linear growth condition but also the time delay to be a constant one of our aims in this paper is to remove these two restrictive conditions moreover the key condition on the diffusion operator associated with the underlying nsdde will take a much more general form our new stability criteria not only cover many highly non linear nsddes with variable time delays but they can also be verified much more easily than the known criteria

modified riemann liouville derivative and fractional taylor series of nondifferentiable functions further results the paper gives some results and improves the derivation of the fractional taylor s series of nondifferentiable functions obtained recently in the form f g h e a h ad g a f g 0 a 1 where e a is the mittag leffier function here one defines fractional derivative as the limit of fractional difference and by this way one can circumvent the problem which arises with the definition of the fractional derivative of constant using riemann liouville definition as a result a modified riemann liouville definition is proposed which is fully consistent with the fractional difference definition and avoids any reference to the derivative of order greater than the considered one s in order to support this f taylor series one shows how its first term can be obtained directly in the form of a mean value formula the fractional derivative of the dirac delta function is obtained together with the fractional taylor s series of multivariate functions the relation with irreversibility of time and symmetry breaking is exhibited and to some extent this f taylor s series generalizes the fractional mean value formula obtained a few years ago by kolwantar

numerical integration of multiplicative noise stochastic differential equations a two stage runge kutta algorithm for vector ito and by transform also stratonovich stochastic differential equations with multiplicative noise has been developed the method is second order accurate but for vanishing drift the algorithm yields a martingale independent of step size several examples are included to illustrate our method a discussion of errors shows that sample size can be as important as truncation

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

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

a momentum trading approach to technical analysis of dow jones industrials abstract a momentum trading approach is presented to examine the dow jones industrial components for a period of about past 10 years 1992 2002 an analogy between the classical dynamics in physics and the stock trade dynamics is used with the momentum p mv where the velocity v is a relative price change in a period and the inertial mass m is a normalized trade volume extrema in the momentum time series i e the singularities in the driving force provide the signals for executing trades minima with negative momentum to buy and maxima with positive momentum to sell trades are implemented using a momentum threshold pc a range of periodic cycles 5 240 days in time series and trading momentum thresholds pc 0 01 0 5 are considered and returns maximum minimum accumulative and average are examined in detail on the historical dji data for about a decade 1992 2002 frequency of trade is generally higher with smaller periods with the high probability of higher returns at pc 0 02 0 1 for nearly all stocks in dji

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

numerical approximations to the stationary solutions of stochastic differential equations in this paper we investigate the possibility of approximating the stationary solution of a stochastic differential equation sde we start with the random dynamical system generated by the sde with the multiplicative noise we prove that the pullback flow has a stationary point however the stationary point is not constructible explicitly therefore we look at the numerical approximation we prove that the discrete time random dynamical system also has a stationary point finally we prove mean square convergence of the approximate stationary solution to the exact stationary solution as the time step diminishes as well as almost surely convergence when the time step is rational

time fractional cahn allen and time fractional klein gordon equations lie symmetry analysis explicit solutions and convergence analysis abstract this research analyzes the symmetry analysis explicit solutions and convergence analysis to the time fractional cahn allen ca and time fractional klein gordon kg equations with riemann liouville rl derivative the time fractional ca and time fractional kg are reduced to respective nonlinear ordinary differential equation of fractional order we solve the reduced fractional odes using an explicit power series method the convergence analysis for the obtained explicit solutions are investigated some figures for the obtained explicit solutions are also presented

choice of θ and mean square exponential stability in the stochastic theta method of stochastic differential equations this paper examines the relationship of choice of and mean square exponential stability in the stochastic theta method stm of stochastic differential equations sdes and mainly includes the following three results i under the linear growth condition for the drift term when 0 1 2 the stm may preserve the mean square exponential stability of the exact solution but the counterexample shows that the stm cannot reproduce this stability without this linear growth condition ii when 1 2 1 without the linear growth condition for the drift term the stm may reproduce the mean square exponential stability of the exact solution but the bound of the lyapunov exponent cannot be preserved iii when 1 this stm is called as the backward euler maruyama bem method the stm can reproduce not only the mean square exponential stability but also the bound of the lyapunov exponent this paper also gives the sufficient and necessary conditions of the mean square exponential stability of the stm for the linear sde when 0 1 2 and 1 2 1 respectively and the simulations also illustrate these theoretical results

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

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

exponential stability of the split step θ method for neutral stochastic delay differential equations with jumps abstract the exponential mean square stability of the split step method for neutral stochastic delay differential equations nsddes with jumps is considered new conditions for jumps are proposed to ensure the exponential mean square stability of the trivial solution if the drift coefficient satisfies the linear growth condition it is shown that the split step method can reproduce the exponential mean square stability of the trivial solution for the constrained stepsize then by applying the chebyshev inequality and the borel cantelli lemma the almost sure exponential stability of both the trivial solution and the numerical solution can be obtained since split step method covers euler maruyama em method and split step backward euler ssbe method the conclusions are valid for these two methods moreover they can adapt to the nsddes and the sddes with jumps finally a numerical example illustrates the effectiveness of the theoretical results

a new conservation theorem a general theorem on conservation laws for arbitrary differential equations is proved the theorem is valid also for any system of differential equations where the number of equations is equal to the number of dependent variables the new theorem does not require existence of a lagrangian and is based on a concept of an adjoint equation for non linear equations suggested recently by the author it is proved that the adjoint equation inherits all symmetries of the original equation accordingly one can associate a conservation law with any group of lie lie backlund or non local symmetries and find conservation laws for differential equations without classical lagrangians

equity prices as a simple harmonic oscillator with noise the centred return on the london stock exchange s ftse all share index is modelled as a simple harmonic oscillator with noise over the period from 1 january 1994 until 30 june 2006 our empirical results are compatible with the hypothesis that there is a period in the ftse all share index of between two and two and one half years this means the centred return will on average continue to increase for about a year after reaching the minimum in its oscillatory cycle alternatively it will continue on average to decline for about a year after reaching a maximum our analysis also shows that there is potential to exploit the harmonic nature of the returns process to earn abnormal profits extending our analysis to the low energy states of a quantum harmonic oscillator is also suggested

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

thermodynamics of relation based systems with applications in econophysics sociophysics and music abstract a methodology was developed to analyze relation based systems evolving in time by using the fundamental concepts of thermodynamics the behavior of such systems can be tracked from the scattering matrix which is actually a network of directed vectors or pathways connecting subsequent values which characterize an event such as the index values in stock markets a system behaves in a rigid elastic way to an external effect and resists permanent deformation or it behaves in a viscous or soft way and deforms in an irreversible way it was shown in the past that a formula derived using the slope of paths gives a measure about the extent of viscoelastic behavior of relation based systems gunduz 2009 5 gunduz and gunduz 2010 6 in this research the work associated with elastic component and heat associated with viscous component were discussed and elaborated in a simple two subsequent pathway system in a scattering diagram the first vector represents the cause and the second the effect by using work and heat energy relations that involve force and also storage and loss modulus terms respectively one can calculate the energy involved in relation based systems the modulus values can be found from the parallel and vertical components of the second vector with respect to the first vector once work like and heat like terms were determined the internal energy is also easily found from their summation the parallel and vertical components can also be used to calculate the magnitude of torque and torque energy in the system three cases i the behavior of the nasdaq 100 index ii a social revolt and iii the structure of a melody were analyzed for their work like heat like and torque like energies in the course of their evolution nasdaq 100 exhibits highly dissipative behavior and its work terms are very small but heat terms are of large magnitude its internal energy highly fluctuates in time in the social revolt studied work and heat terms are of comparable magnitude the melody depicts highly organized structure and usually has larger work terms than heat terms but at some intervals heat terms burst out and attain very large magnitudes torque terms reach high values when the system is recovering from a minimum value

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

numerical treatment of stochastic differential equations we define general runge kutta approximations for the solution of stochastic differential equations sde these approximations are proved to converge in quadratic mean to the solution of an sde with a corrected drift the explicit form of the correction term is given concerning the order of convergence we show that in general it is impossible for the quadratic mean of the one step error to be of an order greater than o h 3 this order is attained e g by the stochastic analogue of heun s method in the n dimensional case the highest order of convergence is in general only o h 2 attained by euler s method the order o h 3 can only be reached if nabla x sigma r sigma s nabla x sigma s sigma r for the diffusion matrix

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

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

dynamical analysis of s p500 momentum abstract the dynamics of the s p500 price signal is studied using a moving average technique particular attention is paid to intersections of two moving averages with different time horizons the distributions of the slopes and angle between two moving averages at intersections is analyzed as well as that of the waiting times between intersections in addition the distribution of maxima and minima in the moving average signal is investigated in all cases persistent patterns are observed in these probability measures and it is suggested that such variables be considered for better analysis and possible prediction of the trends of the signal

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

an application of taylor series in the approximation of solutions to stochastic differential equations with time dependent delay the subject of this paper is the analytic approximation method for solving stochastic differential equations with time dependent delay approximate equations are defined on equidistant partitions of the time interval and their coefficients are taylor approximations of the coefficients of the initial equation it will be shown without making any restrictive assumption for the delay function that the approximate solutions converge in l p norm and with probability 1 to the solution of the initial equation also the rate of the l p convergence increases when the degrees in the taylor approximations increase analogously to what is found in real analysis at the end a procedure will be presented which allows the application of this method with the assumption of continuity of the delay function

a formulation of noether s theorem for fractional problems of the calculus of variations abstract fractional or non integer differentiation is an important concept both from theoretical and applicational points of view the study of problems of the calculus of variations with fractional derivatives is a rather recent subject the main result being the fractional necessary optimality condition of euler lagrange obtained in 2002 here we use the notion of euler lagrange fractional extremal to prove a noether type theorem for that we propose a generalization of the classical concept of conservation law introducing an appropriate fractional operator

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