strong convergence of a tamed theta scheme for nsddes with one sided lipschitz drift abstract this paper is concerned with strong convergence of a tamed theta scheme for neutral stochastic differential delay equations with one sided lipschitz drift strong convergence rate is revealed under a global one sided lipschitz condition while for a local one sided lipschitz condition the tamed theta scheme is modified to ensure the well posedness of implicit numerical schemes then we show the convergence of the numerical solutions

moderate deviations of dependent random variables related to clt this paper consists of three parts in the first part we find a common condition the c 2 regularity both for clt and for moderate deviations we show that this condition is verified in two important situations the lee yang theorem case and the fkg system case in the second part we apply the previous results to the additive functionals of a markov process by means of feynman kac formula and kato s analytic perturbation theory we show that the lee yang theorem holds under the assumption that 1 is an isolated simple and the only eigenvalue with modulus 1 of the operator p 1 acting on an appropriate banach space be c b e l 2 the last part is devoted to some applications to statistical mechanical systems where the c 2 regularity becomes a property of the pressure functionals and the two situations presented above become exactly the lee yang theorem case and the fkg system case we shall discuss in detail the ferromagnetic model and give some general remarks on some other models

simulation of stopped diffusions in this work we study standard euler updates for simulating stopped diffusions as an immediate application we discuss the computation of first exit times of diffusions from a domain we focus on one dimensional situations and show how the ideas for the simulation of killed diffusions can be adapted to this problem in particular we give a fully implementable algorithm to compute the first exit time from an interval numerically the brownian motion case is treated in detail and extensions to general diffusions are given special emphasis is given to numerical experiments for every ansatz we include numerical experiments confirming the conjectured accuracy of our methods our algorithm is of order one in a weak sense comparisons with other algorithms are shown results that are superior to those obtained with other methods are presented when approximating a first hitting time distribution the results obtained with our algorithm are much better than those achieved with other methods

vertex similarity in networks we consider methods for quantifying the similarity of vertices in networks we propose a measure of similarity based on the concept that two vertices are similar if their immediate neighbors in the network are themselves similar this leads to a self consistent matrix formulation of similarity that can be evaluated iteratively using only a knowledge of the adjacency matrix of the network we test our similarity measure on computer generated networks for which the expected results are known and on a number of real world networks

decoupling inequalities for the tail probabilities of multivariate u statistics in this paper we present a decoupling inequality that shows that multivariate u statistics can be studied as sums of conditionally independent random variables this result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration more precisely we get the following result let x j be a sequence of independent random variables on a measurable space mathscr j s and let x j i j 1 ldots k be k independent copies of x i let f i 1i 2 ldots i k be families of functions of k variables taking s times cdots times s into a banach space b cdots then for all n geq k geq 2 t 0 there exist numerical constants c k depending on k only so that p bigg big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 1 i 2 ldots x 1 i k big geq t bigg leq c kp bigg c k big sum 1 leq i 1 neq i 2 neq cdots neq i k leq n f i 1 cdots i k x 1 i 1 x 2 i 2 ldots x k i k big geq t bigg the reverse bound holds if in addition the following symmetry condition holds almost surely f i 1i 2 cdots i k x i 1 x i 2 ldots x i k f i pi 1 i pi 2 cdots i pi k x i pi 1 x i pi 2 ldots x i pi k for all permutations pi of 1 ldots k

numerical approximation for nonlinear stochastic pantograph equations with markovian switching the main aim of the paper is to prove that the implicit numerical approximation can converge to the true solution to highly nonlinear hybrid stochastic pantograph differential equation after providing the boundedness of the exact solution the paper proves that the backward euler maruyama numerical method can preserve boundedness of moments and the numerical approximation converges strongly to the true solution finally the exponential stability criterion on the backward euler maruyama scheme is given and a high order example is provided to illustrate the main result

stochastic versions of the lasalle theorem the main aim of this paper is to establish stochastic versions of the well known lasalle stability theorem from these stochastic versions follow many classical results on stochastic stability this shows clearly the power of our new results

on the bootstrap of u and v statistics bootstrap distributional limit theorems for u and v statistics are proved they hold a s under weak moment conditions and without restrictions on the bootstrap sample size as long as it tends to infty regardless of the degree of degeneracy of u and v a testing procedure based on these results is outlined

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

the improved lasalle type theorems for stochastic functional differential equations abstract the main aim of this paper is to improve some results obtained by mao x mao the lasalle type theorems for stochastic functional differential equations nonlinear stud 7 2000 307 328 our new theorems give better results while conditions imposed are much weaker than in the paper mentioned above for example we need only the local lipschitz condition but neither the linear growth condition nor the bounded moment condition on the solutions to guarantee the existence and uniqueness of the global solution to the underlying stochastic functional differential equation sfde under the weaker conditions imposed in this paper we establish a generalised existence and uniqueness theorem which covers a wider class of nonlinear sfdes as demonstrated by the examples discussed in this paper moreover from our improved results follow some new criteria on the stochastic asymptotic stability for sfdes

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

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

numerical analysis of the balanced implicit methods for stochastic pantograph equations with jumps this paper deals with a family of balanced implicit methods with linear interpolation for the stochastic pantograph equations with jumps in this paper the strong mean square convergence theory is established for the numerical solutions of the system it is shown that the balanced implicit methods which are fully implicit methods give strong convergence rate of at least 1 2 for a linear scalar test equation the balanced implicit methods are shown to capture the mean square stability for all sufficiently small time steps under appropriate conditions furthermore weak variants are also considered and their mean square stability analyzed several numerical experiments are given for illustration and show that the fully implicit methods are superior to those of the explicit methods in terms of mean square stabilities

exponential stability in p th mean of solutions and of convergent euler type solutions of stochastic delay differential equations one concept of the stability of a solution of an evolutionary equation relates to the sensitivity of the solution to perturbations in the initial data there are other stability concepts notably those concerned with persistent perturbations results are presented on the stability in p th mean of solutions of stochastic delay differential equations with multiplicative noise and of stochastic delay difference equations the difference equations are of a type found in numerical analysis and we employ our results to obtain mean square stability criteria for the solution of the euler maruyama discretization of stochastic delay differential equations the analysis proceeds as follows we show that an inequality of halanay type derivable via comparison theory can be employed to derive conditions for p th mean stability of a solution we then produce a discrete analogue of the halanay type theory that permits us to develop a p th mean stability analysis of analogous stochastic difference equations the application of the theoretical results is illustrated by deriving mean square stability conditions for solutions and numerical solutions of a constant coefficient linear test equation

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

finding minimum weight connected dominating set in stochastic graph based on learning automata finding the minimum weight connected dominating set mcds in an arbitrary graph is an np hard problem and several heuristics and approximation methods have been proposed to solve it forwarding the messages along the virtual backbone induced by the connected dominating set cds significantly reduces the routing overhead as well as the power consumption by reducing the routing nodes to the backbone nodes this paper first defines the stochastic mcds problem where the probability distribution function pdf of the random weight associated with the graph vertices is unknown then it presents several learning automata based algorithms algorithms 1 6 to solve the stochastic mcds problem taking advantage of learning automata the proposed algorithms significantly reduce the number of samples that must be taken from the graph to construct the mcds it is proved that by the proper choice of the learning rate the probability of finding the mcds is close enough to unity the standard sampling method ssm is the baseline with which we compare the performance of the proposed algorithms experimental results show that algorithm 6 significantly outperforms the ssm and the other proposed algorithms in terms of the sampling rate

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

asymptotic behaviour of the stochastic lotka volterra model this paper examines the asymptotic behaviour of the stochastic extension of a fundamentally important population process namely the lotka volterra model the stochastic version of this process appears to have far more intriguing properties than its deterministic counterpart indeed the fact that a potential deterministic population explosion can be prevented by the presence of even a tiny amount of environmental noise shows the high level of difference which exists between these two representations

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

limit theorems for u processes a u process is a collection of u statistics concretely a u process over a family h of kernels h of m variables based on a probability measure p on s s is the collection u n m h p h in mathcal h of u statistics this chapter is devoted to the asymptotic theory of u processes we are interested in finding conditions on h and p ensuring that the law of large numbers or the central limit theorem or the law of the iterated logarithm hold for u n m h p uniformly in h h the theory of empirical processes deals with the same questions for the case m 1 and u process theory is patterned after it this is a relatively new subject at least in the generality presented here h being an arbitrary collection of kernels which are defined on a general measurable space there is therefore a need to indicate its usefulness to this effect a section on applications is added at the end of the chapter section 5 5 there we illustrate the use of each of the main theorems in this chapter by deriving properties of certain multidimensional generalizations of the median and in general of m estimators and by studying estimators of the cumulative hazard and distribution functions of a random variable based on truncated data

approximations of euler maruyama type for stochastic differential equations with markovian switching under non lipschitz conditions we develop the euler maruyama scheme for a class of stochastic differential equations with markovian switching sdewmss under non lipschitz conditions both l 1 and l 2 convergence are discussed under different non lipschitz conditions to overcome the mathematical difficulties arisen from the markovian switching as well as the non lipschitz coefficients several new analytical techniques have been developed in this paper which should prove to be very useful in the numerical analysis of stochastic systems

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

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

an iterative shifted chebyshev method for nonlinear stochastic ito volterra integral equations abstract in the present work we propose an iterative method based on the shifted chebyshev polynomials for the numerical investigation of the nonlinear stochastic volterra integral equations moreover the convergence of the proposed method is studied the proposed iterative method is fast and easy to implement some numerical examples are given to show the applicability and efficiency of the presented method the numerical results of the proposed method are compared with the reported results in the literature and with the exact solutions if they exist the reported comparison confirm the applicability and high accuracy of the proposed method

moderate deviations and associated laplace approximations for sums of independent random vectors let xj be an i i d sequence of banach space valued r v s and let sn z 1 xj for certain positive sequences bn oo we determine the exact asymptotic behavior of e exp b2 n d sn bn where d is a smooth function we also prove a large deviation principle for 2y sn bn

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

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

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

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

an evidential link prediction method and link predictability based on shannon entropy abstract predicting missing links is of both theoretical value and practical interest in network science in this paper we empirically investigate a new link prediction method base on similarity and compare nine well known local similarity measures on nine real networks most of the previous studies focus on the accuracy however it is crucial to consider the link predictability as an initial property of networks itself hence this paper has proposed a new link prediction approach called evidential measure em based on dempster shafer theory moreover this paper proposed a new method to measure link predictability via local information and shannon entropy

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

link prediction based on temporal similarity metrics using continuous action set learning automata abstract link prediction is a social network research area that tries to predict future links using network structure the main approaches in this area are based on predicting future links using network structure at a specific period without considering the links behavior through different periods for example a common traditional approach in link prediction calculates a chosen similarity metric for each non connected link and outputs the links with higher similarity scores as the prediction result in this paper we propose a new link prediction method based on temporal similarity metrics and continuous action set learning automata cala the proposed method takes advantage of using different similarity metrics as well as different time periods in the proposed algorithm we try to model the link prediction problem as a noisy optimization problem and use a team of calas to solve the noisy optimization problem cala is a reinforcement based optimization tool which tries to learn the optimal behavior from the environment feedbacks to determine the importance of different periods and similarity metrics on the prediction result we define a coefficient for each of different periods and similarity metrics and use a cala for each coefficient each cala tries to learn the true value of the corresponding coefficient final link prediction is obtained from a combination of different similarity metrics in different times based on the obtained coefficients the link prediction results reported here show satisfactory of the proposed method for some social network data sets

rank dependent moderate deviations of u empirical measures in strong topologies we prove a rank dependent moderate deviation principle for u empirical mea sures where the underlying i i d random variables take values in a measurable not neces sarily polish space s s the result can be formulated on a suitable subset of all signed measures on s m s m we endow this space with a topology which is stronger than the usual topology a moderate deviation principle for banach space valued u statistics is obtained as a particular application the advantage of our result is that we obtain in the degenerate case moderate deviations in non gaussian situations with non convex rate func tions