Efficient transient noise analysis in circuit simulation

Stochastic differential algebraic equations (SDAEs) arise as a mathematical model for electrical network equations that are influenced by additional sources of Gaussian white noise. We discuss adaptive linear multi-step methods for their numerical integration, in particular stochastic analogues of the trapezoidal rule and the two-step backward differentiation formula, and we obtain conditions that ensure mean-square convergence of this methods. For the case of small noise we present a strategy for controlling the step-size in the numerical integration. It is based on estimating the mean-square local errors and leads to step-size sequences that are identical for all computed paths. Test results illustrate the performance of the presented methods. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulations and modeling for stochastic biological systems with jumps

This paper gives a numerical method to simulate sample paths for stochastic differential equations (SDEs) driven by Poisson random measures. It provides us a new approach to simulate systems with jumps from a different angle. The driving Poisson random measures are assumed to be generated by stationary Poisson point processes instead of Lévy processes. Methods provided in this paper can be used to simulate SDEs with Lévy noise approximately. The simulation is divided into two parts: the part of jumping integration is based on definition without approximation while the continuous part is based on some classical approaches. Biological explanations for stochastic integrations with jumps are motivated by several numerical simulations. How to model biological systems with jumps is showed in this paper. Moreover, method of choosing integrands and stationary Poisson point processes in jumping integrations for biological models are obtained. In addition, results are illustrated through some examples and numerical simulations. For some examples, earthquake is chose as a jumping source which causes jumps on the size of biological population.     

Optimal sampling design for global approximation of jump diffusion stochastic differential equations

The paper deals with strong global approximation of stochastic differential equations (SDEs) driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition (see Chapter 6.3 in [21]). We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient, with respect to asymptotic constants, than those based on the equidistant mesh.     

Itô’s stochastic calculus: Its surprising power for applications

We trace Itô’s early work in the 1940s, concerning stochastic integrals, stochastic differential equations (SDEs) and Itô’s formula. Then we study its developments in the 1960s, combining it with martingale theory. Finally, we review a surprising application of Itô’s formula in mathematical finance in the 1970s. Throughout the paper, we treat Itô’s jump SDEs driven by Brownian motions and Poisson random measures, as well as the well-known continuous SDEs driven by Brownian motions.     

Asymptotic expansion for forward-backward SDEs with jumps

This work provides a semi-analytic approximation method for decoupled forward-backward SDEs (FBSDEs) with jumps. In particular, we construct an asymptotic expansion method for FBSDEs driven by the random Poisson measures with σ-finite compensators as well as the standard Brownian motions around the small-variance limit of the forward SDE. We provide a semi-analytic solution technique as well as its error estimate for which we only need to solve essentially a system of linear ODEs. In the case of a finite jump measure with a bounded intensity, the method can also handle state-dependent and hence non-Poissonian jumps, which are quite relevant for many practical applications.     

Φ-entropy inequality and application for SDEs with jumps

By using the Φ-entropy inequality derived in [16] and [2] for Poisson measures, the same type of inequality is established for a class of stochastic differential equations driven by purely jump Lévy processes. This inequality implies the exponential convergence in Φ-entropy of the associated Markov semigroup. The semigroup Φ-entropy inequality for SDEs driven by Poisson point processes is also considered.     

A stepsize control algorithm for SDEs with small noise based on stochastic Runge–Kutta Maruyama methods

A variable stepsize control algorithm for solution of stochastic differential equations (SDEs) with a small noise parameter ?? is presented. In order to determine the optimal stepsize for each stage of the algorithm, an estimate of the global error is introduced based on the local error of the Stochastic Runge?CKutta Maruyama (SRKM) methods. Based on the relation of the stepsize and the small noise parameter, the local mean-square stochastic convergence order can be different from stage to stage. Using this relation, a strategy for producing and controlling the stepsize in the numerical integration of SDEs is proposed. Numerical experiments on several standard SDEs with small noise are presented to illustrate the effectiveness of this approach.     

Absolute Continuity and Convergence in Variation for Distributions of Functionals of Poisson Point Measure

General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the functionals on the probability space generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form \mathbbR⁺×\mathbbU{\mathbb{R}}^{+}\times{\mathbb{U}}, and its intensity measure to equal dt Π(du). We introduce the family of time stretching transformations of the configurations of the point measure. Sufficient conditions for absolute continuity and convergence in variation are given in terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDEs driven by Poisson point measures, including SDEs with non-constant jump rate.     

Stochastic differential equations with diffusion and jumps modeling currency markets

An efficient currency market with zero transaction costs is considered. The dynamics of the exchange rate in this market is described by stochastic differential equations (SDEs) with diffusion and jumps; the latter are assumed to be described by a Lévy process. Adjusting theoretical arbitrage-free option prices computed within these models to market option prices requires properly choosing the coefficients in the SDEs. For this purpose, an expression for local volatility in a diffusion model is found and a relation between local and implied volatilities is determined. For a market model with diffusion and jumps, expressions for the local volatility and the local rate function are given. Moreover, in Merton’s model, where the jump component is a compound Poisson process with normal jumps, a relation between the local and the implied volatilities is determined.     

DISCRETIZATION OF JUMP STOCHASTIC DIFFERENTIAL EQUATIONS IN TERMS OF MULTIPLE STOCHASTIC INTEGRALS

1.IntroductionFOrthestrongdiscretizationofSDEs,anynumericalmethodwhichonlydependsonthevaluesofBrownianpathsorPoissonpathsatthepartitionnodescannotachieveanorderhigherthan0.5ingeneral[')'1'].Thereforetheevaluationofmultiplestochasticintegralsontheintervalsbetweennodesisamajorobstaclethatmustbeovercome.Someattemptshavebeenmadepreviouslyindifferentapproachestoapproximatemul-tiplestochasticintegrals.[2]suggestsanapproximationintermsofFourierGaussiancoefficientsoftheBrownianbridgeprocess.Asthel…     

Numerical methods for a class of jump–diffusion systems with random magnitudes

Stochastic differential delay equations with Poisson driven jumps of random magnitude are popular as models in mathematical finance. In this paper, we shall deal with convergence of the semi-implicit Euler method for nonlinear stochastic differential delay equations with random jump magnitudes and show that the approximate solutions strongly converge to the exact solutions with the order 1 − 1/q (q > 1). This result is more general than what they deal with the jump of deterministic magnitude.     

Law of the iterated logarithm for solutions of stochastic differential equations

We prove the law of the iterated logarithm for solutions of Stochastic Differential Equations (SDEs) driven by continuous semiraartingales, under suitable conditions. This extends a result of Kulinich for classical diffusions to solutions of SDEs which are not necessarily Markov     

Interacting Particle System based estimation of reach probability of General Stochastic Hybrid Systems

For diffusions, a well-developed approach in rare event estimation is to introduce a suitable factorization of the reach probability and then to estimate these factors through simulation of an Interacting Particle System (IPS). This paper studies IPS based reach probability estimation for General Stochastic Hybrid Systems (GSHS). The continuous-time executions of a GSHS evolve in a hybrid state space under influence of combinations of diffusions, spontaneous jumps and forced jumps. In applying IPS to a GSHS, simulation of the GSHS execution plays a central role. From literature, two basic approaches in simulating GSHS execution are known. One approach is direct simulation of a GSHS execution. An alternative is to first transform the spontaneous jumps of a GSHS to forced transitions, and then to simulate executions of this transformed version. This paper will show that the latter transformation yields an extra Markov state component that should be treated as being unobservable for the IPS process. To formally make this state component unobservable for IPS, this paper also develops an enriched GSHS transformation prior to transforming spontaneous jumps to forced jumps. The expected improvements in IPS reach probability estimation are also illustrated through simulation results for a simple GSHS example.     

Integration by parts formulas concerning maxima of some SDEs with applications to study on density functions

In this article, we prove integration by parts (IBP) formulas concerning maxima of solutions to some stochastic differential equations (SDEs). We will deal with three types of maxima. First, we consider discrete time maximum, and then continuous time maximum in the case of one-dimensional SDEs. Finally, we deal with the maximum of the components of a solution to multi-dimensional SDEs. Applications to study their probability density functions by means of the IBP formulas are also discussed.     

A unifying formulation of the Fokker–Planck–Kolmogorov equation for general stochastic hybrid systems

A general formulation of the Fokker–Planck–Kolmogorov (FPK) equation for stochastic hybrid systems is presented, within the framework of Generalized Stochastic Hybrid Systems (GSHSs). The FPK equation describes the time evolution of the probability law of the hybrid state. Our derivation is based on the concept of mean jump intensity, which is related to both the usual stochastic intensity (in the case of spontaneous jumps) and the notion of probability current (in the case of forced jumps). This work unifies all previously known instances of the FPK equation for stochastic hybrid systems, and provides GSHS practitioners with a tool to derive the correct evolution equation for the probability law of the state in any given example.     

分数Brown运动时变随机种群收获系统数值解的均方散逸性

讨论了一类带分数Brown运动时变随机种群收获系统数值解的均方散逸性.在一定条件下,利用It公式和Bellman-Gronwall-Type引理,研究了方程(1)具有均方散逸性.分别利用带补偿的倒向Euler方法和分步倒向Euler方法讨论数值解的均方散逸性存在的充分条件,并通过数值算例对所给出的结论进行了验证.     

Weak regenerative structure of an open Jackson queueing network

A general wide-sense regenerative structure for an open queueing network is proposed. Some examples and simulation applications are considered. The Poisson case is discussed in more detail. Supported by the Nordic Council of Ministers and INTAS-93-0893. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part II.     

Removing logarithms in Poisson approximation to the partial sum process of a Markov chain

He and Xia (1997, Stochastic Processes Appl. 68, pp. 101–111) gave some error bounds for a Wasserstein distance between the distributions of the partial sum process of a Markov chain and a Poisson point process on the positive half-line. However, all these bounds increase logarithmically with the mean of the Poisson point process. In this paper, using the coupling method and a general deep result for estimating the errors of Poisson process approximation in Brown and Xia (2001, Ann. Probab. 29, pp. 1373–1403), we give a new error bound for the above Wasserstein distance. In contrast to the previous results of He and Xia (1997), our new error bound has no logarithm anymore and it is bounded and asymptotically remains constant as the mean increases.     

Time inhomogeneous Stochastic Differential Equations involving the local time of the unknown process,and associated parabolic operators

In this paper we study time inhomogeneous versions of one-dimensional Stochastic Differential Equations (SDE) involving the Local Time of the unknown process on curves. After proving existence and uniqueness for these SDEs under mild assumptions, we explore their link with Parabolic Differential Equations (PDE) with transmission conditions. We study the regularity of solutions of such PDEs and ensure the validity of a Feynman–Kac representation formula. These results are then used to characterize the solutions of these SDEs as time inhomogeneous Markov Feller processes.     

Stochastic piecewise affine control with application to pitch control of helicopter

In this paper, at first the stability condition which gives an upper stochastic bound for a class of Stochastic Hybrid Systems (SHS) with deterministic jumps is derived. Here, additive noise signals are considered that do not vanish at equilibrium points. The presented theorem gives an upper bound for the second stochastic moment or variance of the system trajectories. Then, the linear case of SHS is investigated to show the application of the theorem. For the linear case of such stochastic hybrid systems, the stability criterion is obtained in terms of Linear Matrix Inequality (LMI) and an upper bound on state covariance is obtained for them. Then utilizing the stability theorem, an output feedback controller design procedure is proposed which requires the Bilinear Matrix Inequalities (BMI) to be solved. Next, the pitch dynamics of a helicopter is approximated with a set of linear stochastic systems, and the proposed controller is designed for the approximated model and implemented on the main nonlinear system to demonstrate the effectiveness of the proposed theorem and the control design method.