Showing papers on "Stochastic process published in 2004"

Lévy Processes and Stochastic Calculus

Abstract: Levy processes form a wide and rich class of random process, and have many applications ranging from physics to finance Stochastic calculus is the mathematics of systems interacting with random noise Here, the author ties these two subjects together, beginning with an introduction to the general theory of Levy processes, then leading on to develop the stochastic calculus for Levy processes in a direct and accessible way This fully revised edition now features a number of new topics These include: regular variation and subexponential distributions; necessary and sufficient conditions for Levy processes to have finite moments; characterisation of Levy processes with finite variation; Kunita's estimates for moments of Levy type stochastic integrals; new proofs of Ito representation and martingale representation theorems for general Levy processes; multiple Wiener-Levy integrals and chaos decomposition; an introduction to Malliavin calculus; an introduction to stability theory for Levy-driven SDEs

Random walks on complex networks.

Abstract: We investigate random walks on complex networks and derive an exact expression for the mean firstpassage time (MFPT) between two nodes. We introduce for each node the random walk centrality C, which is the ratio between its coordination number and a characteristic relaxation time, and show that it determines essentially the MFPT. The centrality of a node determines the relative speed by which a node can receive and spread information over the network in a random process. Numerical simulations of an ensemble of random walkers moving on paradigmatic network models confirm this analytical prediction.

Stochastic properties of the random waypoint mobility model

Abstract: The random waypoint model is a commonly used mobility model for simulations of wireless communication networks. By giving a formal description of this model in terms of a discrete-time stochastic process, we investigate some of its fundamental stochastic properties with respect to: (a) the transition length and time of a mobile node between two waypoints, (b) the spatial distribution of nodes, (c) the direction angle at the beginning of a movement transition, and (d) the cell change rate if the model is used in a cellular-structured system area. The results of this paper are of practical value for performance analysis of mobile networks and give a deeper understanding of the behavior of this mobility model. Such understanding is necessary to avoid misinterpretation of simulation results. The movement duration and the cell change rate enable us to make a statement about the "degree of mobility" of a certain simulation scenario. Knowledge of the spatial node distribution is essential for all investigations in which the relative location of the mobile nodes is important. Finally, the direction distribution explains in an analytical manner the effect that nodes tend to move back to the middle of the system area.

Random Walks on Complex Networks

Abstract: We investigate random walks on complex networks and derive an exact expression for the mean firstpassage time (MFPT) between two nodes. We introduce for each node the random walk centrality C, which is the ratio between its coordination number and a characteristic relaxation time, and show that it determines essentially the MFPT. The centrality of a node determines the relative speed by which a node can receive and spread information over the network in a random process. Numerical simulations of an ensemble of random walkers moving on paradigmatic network models confirm this analytical prediction.

Agreement over random networks

Abstract: We consider the agreement problem over random information networks. In a random network, the existence of an information channel between a pair of elements at each time instance is probabilistic and independent of other channels; hence, the topology of the network varies over time. In such a framework, we address the asymptotic agreement for the networked elements via notions from stochastic stability. Furthermore, we delineate on the rate of convergence as it relates to the algebraic connectivity of random graphs.

Random walks in peer-to-peer networks

Abstract: We quantify the effectiveness of random walks for searching and construction of unstructured peer-to-peer (P2P) networks. We have identified two cases where the use of random walks for searching achieves better results than flooding: a) when the overlay topology is clustered, and h) when a client re-issues the same query while its horizon does not change much. For construction, we argue that an expander can he maintained dynamically with constant operations per addition. The key technical ingredient of our approach is a deep result of stochastic processes indicating that samples taken from consecutive steps of a random walk can achieve statistical properties similar to independent sampling (if the second eigenvalue of the transition matrix is hounded away from 1, which translates to good expansion of the network; such connectivity is desired, and believed to hold, in every reasonable network and network model). This property has been previously used in complexity theory for construction of pseudorandom number generators. We reveal another facet of this theory and translate savings in random bits to savings in processing overhead.

A stochastic control strategy for hybrid electric vehicles

Abstract: The supervisory control strategy of a hybrid vehicle coordinates the operation of vehicle sub-systems to achieve performance targets such as maximizing fuel economy and reducing exhaust emissions. This high-level control problem is commonly referred as the power management problem. In the past, many supervisory control strategies were developed on the basis of a few pre-defined driving cycles, using intuition and heuristics. The resulting control strategy was often inherently cycle-beating and lacked a guaranteed level of optimality. In this study, the power management problem is tackled from a stochastic viewpoint. An infinite-horizon stochastic dynamic optimization problem is formulated. The power demand from the driver is modeled as a random Markov process. The optimal control strategy is then obtained by using stochastic dynamic programming (SDP). The obtained control law is in the form of a stationary full-state feedback and can be directly implemented. Simulation results over standard driving cycles and random driving cycles are presented to demonstrate the effectiveness of the proposed stochastic approach. It was found that the obtained SDP control algorithm outperforms a sub-optimal rule-based control strategy trained from deterministic DP results.

Convergence of proportional-fair sharing algorithms under general conditions

Abstract: We are concerned with the allocation of the base station transmitter time in time-varying mobile communications with many users who are transmitting data. Time is divided into small scheduling intervals, and the channel rates for the various users are available at the start of the intervals. Since the rates vary randomly, in selecting the current user there is a conflict between full use (by selecting the user with the highest current rate) and fairness (which entails consideration for users with poor throughput to date). The proportional fair scheduler of the Qualcomm High Data Rate system and related algorithms are designed to deal with such conflicts. The aim here is to put such algorithms on a sure mathematical footing and analyze their behavior. The available analysis, while obtaining interesting information, does not address the actual convergence for arbitrarily many users under general conditions. Such algorithms are of the stochastic approximation type and results of stochastic approximation are used to analyze the long-term properties. It is shown that the limiting behavior of the sample paths of the throughputs converges to the solution of an intuitively reasonable ordinary differential equation, which is akin to a mean flow. We show that the ordinary differential equation (ODE) has a unique equilibrium and that it is characterized as optimizing a concave utility function, which shows that PFS is not ad-hoc, but actually corresponds to a reasonable maximization problem. These results may be used to analyze the performance of PFS. The results depend on the fact that the mean ODE has a special form that arises in problems with certain types of competitive behavior. There is a large set of such algorithms, each one corresponding to a concave utility function. This set allows a choice of tradeoffs between the current rate and throughout. Extensions to multiple antenna and frequency systems are given. Finally, the infinite backlog assumption is dropped and the data is allowed to arrive at random. This complicates the analysis, but the same results hold.

Time-dependent Hurst exponent in financial time series

Abstract: We calculate the Hurst exponent H ( t ) of several time series by dynamical implementation of a recently proposed scaling technique: the detrending moving average (DMA). In order to assess the accuracy of the technique, we calculate the exponent H ( t ) for artificial series, simulating monofractal Brownian paths, with assigned Hurst exponents H. We next calculate the exponent H ( t ) for the return of high-frequency (tick-by-tick sampled every minute) series of the German market. We find a much more pronounced time-variability in the local scaling exponent of financial series compared to the artificial ones. The DMA algorithm allows the calculation of the exponent H ( t ) , without any a priori assumption on the stochastic process and on the probability distribution function of the random variables, as happens, for example, in the case of the Kitagawa grid and the extended Kalmann filtering methods. The present technique examines the local scaling exponent H ( t ) around a given instant of time. This is a significant advance with respect to the standard wavelet transform or to the higher-order power spectrum technique, which instead operate on the global properties of the series by Legendre or Fourier transform of qth-order moments.

Lévy Processes—From Probability to Finance and Quantum Groups

Abstract: 1320 NOTICES OF THE AMS VOLUME 51, NUMBER 11 T he theory of stochastic processes was one of the most important mathematical developments of the twentieth century. Intuitively, it aims to model the interaction of “chance” with “time”. The tools with which this is made precise were provided by the great Russian mathematician A. N. Kolmogorov in the 1930s. He realized that probability can be rigorously founded on measure theory, and then a stochastic process is a family of random variables (X(t), t ≥ 0) defined on a probability space (Ω,F , P ) and taking values in a measurable space (E,E) . Here Ω is a set (the sample space of possible outcomes), F is a σ-algebra of subsets of Ω (the events), and P is a positive measure of total mass 1 on (Ω,F ) (the probability). E is sometimes called the state space. Each X(t) is a (F ,E) measurable mapping from Ω to E and should be thought of as a random observation made on E made at time t . For many developments, both theoretical and applied, E is Euclidean space Rd (often with d = 1); however, there is also considerable interest in the case where E is an infinite dimensional Hilbert or Banach space, or a finite-dimensional Lie group or manifold. In all of these cases E can be taken to be the Borel σalgebra generated by the open sets. To model probabilities arising within quantum theory, the scheme described above is insufficiently general and must be embedded into a suitable noncommutative structure. Stochastic processes are not only mathematically rich objects. They also have an extensive range of applications in, e.g., physics, engineering, ecology, and economics—indeed, it is difficult to conceive of a quantitative discipline in which they do not feature. There is a limited amount that can be said about the general concept, and much of both theory and applications focusses on the properties of specific classes of process that possess additional structure. Many of these, such as random walks and Markov chains, will be well known to readers. Others, such as semimartingales and measure-valued diffusions, are more esoteric. In this article, I will give an introduction to a class of stochastic processes called Levy processes, in honor of the great French probabilist Paul Levy, who first studied them in the 1930s. Their basic structure was understood during the “heroic age” of probability in the 1930s and 1940s and much of this was due to Paul Levy himself, the Russian mathematician A. N. Khintchine, and to K. Ito in Japan. During the past ten years, there has been a great revival of interest in these processes, due to new theoretical developments and also a wealth of novel applications—particularly to option pricing in mathematical finance. As well as a vast number of research papers, a number of books on the subject have been published ([3], [11], [1], [2], [12]) and there have been annual international conferences devoted to these processes since 1998. Before we begin the main part of the article, it is worth David Applebaum is professor of probability and statistics at the University of Sheffield. His email address is D.Applebaum@sheffield.ac.uk. He is the author of Levy Processes and Stochastic Calculus, Cambridge University Press, 2004, on which part of this article is based.

Ergodicity of the 2D Navier-Stokes Equations with Degenerate Stochastic Forcing

Abstract: The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in $Ł^2_0(\TT^2)$. Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the \textit{asymptotic strong Feller} property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds under a H{o}rmander-type condition. This requires some interesting nonadapted stochastic analysis.

Binomial leap methods for simulating stochastic chemical kinetics.

Abstract: This paper discusses efficient simulation methods for stochastic chemical kinetics. Based on the τ-leap and midpoint τ-leap methods of Gillespie [D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001)], binomial random variables are used in these leap methods rather than Poisson random variables. The motivation for this approach is to improve the efficiency of the Poisson leap methods by using larger stepsizes. Unlike Poisson random variables whose range of sample values is from zero to infinity, binomial random variables have a finite range of sample values. This probabilistic property has been used to restrict possible reaction numbers and to avoid negative molecular numbers in stochastic simulations when larger stepsize is used. In this approach a binomial random variable is defined for a single reaction channel in order to keep the reaction number of this channel below the numbers of molecules that undergo this reaction channel. A sampling technique is also designed for the total reaction number of a reactant species that undergoes two or more reaction channels. Samples for the total reaction number are not greater than the molecular number of this species. In addition, probability properties of the binomial random variables provide stepsize conditions for restricting reaction numbers in a chosen time interval. These stepsize conditions are important properties of robust leap control strategies. Numerical results indicate that the proposed binomial leap methods can be applied to a wide range of chemical reaction systems with very good accuracy and significant improvement on efficiency over existing approaches.

From Diffusion to Anomalous Diffusion: A Century after Einstein's Brownian Motion

Abstract: Einstein's explanation of Brownian motion provided one of the cornerstones which underlie the modern approaches to stochastic processes. His approach is based on a random walk picture and is valid for Markovian processes lacking long-term memory. The coarse-grained behavior of such processes is described by the diffusion equation. However, many natural processes do not possess the Markovian property and exhibit to anomalous diffusion. We consider here the case of subdiffusive processes, which are semi-Markovian and correspond to continuous-time random walks in which the waiting time for a step is given by a probability distribution with a diverging mean value. Such a process can be considered as a process subordinated to normal diffusion under operational time which depends on this pathological waiting-time distribution. We derive two different but equivalent forms of kinetic equations, which reduce to know fractional diffusion or Fokker-Planck equations for waiting-time distributions following a power-law. For waiting time distributions which are not pure power laws one or the other form of the kinetic equation is advantageous, depending on whether the process slows down or accelerates in the course of time.

Cubature on Wiener space

Abstract: It is well known that there is a mathematical equivalence between ‘solving’ parabolic partial differential equations (PDEs) and ‘the integration’ of certain functionals on Wiener space. Monte Carlo simulation of stochastic differential equations (SDEs) is a naive approach based on this underlying principle. In finite dimensions, it is well known that cubature can be a very effective approach to integration. We discuss the appropriate extension of this idea to Wiener space. In the process we develop high–order numerical schemes valid for high–dimensional SDEs and semi–elliptic PDEs.

An introduction to statistical signal processing

Abstract: This volume describes the essential tools and techniques of statistical signal processing. At every stage, theoretical ideas are linked to specific applications in communications and signal processing. The book begins with an overview of basic probability, random objects, expectation, and second-order moment theory, followed by a wide variety of examples of the most popular random process models and their basic uses and properties. Specific applications to the analysis of random signals and systems for communicating, estimating, detecting, modulating, and other processing of signals are interspersed throughout the text.

Synchronization of trajectories in canonical molecular-dynamics simulations: observation, explanation, and exploitation.

Abstract: For two methods commonly used to achieve canonical-ensemble sampling in a molecular-dynamics simulation, the Langevin thermostat and the Andersen [H. C. Andersen, J. Chem. Phys. 72, 2384 (1980)] thermostat, we observe, as have others, synchronization of initially independent trajectories in the same potential basin when the same random number sequence is employed. For the first time, we derive the time dependence of this synchronization for a harmonic well and show that the rate of synchronization is proportional to the thermostat coupling strength at weak coupling and inversely proportional at strong coupling with a peak in between. Explanations for the synchronization and the coupling dependence are given for both thermostats. Observation of the effect for a realistic 97-atom system indicates that this phenomenon is quite general. We discuss some of the implications of this effect and propose that it can be exploited to develop new simulation techniques. We give three examples: efficient thermalization (a concept which was also noted by Fahy and Hamann [S. Fahy and D. R. Hamann, Phys. Rev. Lett. 69, 761 (1992)]), time-parallelization of a trajectory in an infrequent-event system, and detecting transitions in an infrequent-event system.

Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation.

Abstract: A detailed study is presented for a large class of uncoupled continuous-time random walks (CTRWs). The master equation is solved for the Mittag-Leffler survival probability. The properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed. Finally, some common objections found in the literature are thoroughly reviewed.

Ruin probabilities and overshoots for general Lévy insurance risk processes

Abstract: We formulate the insurance risk process in a general Levy process setting, and give general theorems for the ruin probability and the asymptotic distribution of the overshoot of the process above a high level, when the process drifts to −∞ a.s. and the positive tail of the Levy measure, or of the ladder height measure, is subexponential or, more generally, convolution equivalent. Results of Asmussen and Kluppelberg [Stochastic Process. Appl. 64 (1996) 103–125] and Bertoin and Doney [Adv. in Appl. Probab. 28 (1996) 207–226] for ruin probabilities and the overshoot in random walk and compound Poisson models are shown to have analogues in the general setup. The identities we derive open the way to further investigation of general renewal-type properties of Levy processes.

Verification and planning for stochastic processes with asynchronous events

Abstract: Asynchronous stochastic systems are abundant in the real world. Examples include queuing systems, telephone exchanges, and computer networks. Yet, little attention has been given to such systems in the model checking and planning literature, at least not without making limiting and often unrealistic assumptions regarding the dynamics of the systems. The most common assumption is that of history-independence: the Markov assumption. In this thesis, we consider the problems of verification and planning for stochastic processes with asynchronous events, without relying on the Markov assumption. We establish the foundation for statistical probabilistic model checking, an approach to probabilistic model checking based on hypothesis testing and simulation. We demonstrate that this approach is competitive with state-of-the-art numerical solution methods for probabilistic model checking. While the verification result can be guaranteed only with some probability of error, we can set this error bound arbitrarily low (at the cost of efficiency). Our contribution in planning consists of a formalism, the generalized semi-Markov decision process (GSMDP), for planning with asynchronous stochastic events. We consider both goal directed and decision theoretic planning. In the former case, we rely on statistical model checking to verify plans, and use the simulation traces to guide plan repair. In the latter case, we present the use of phase-type distributions to approximate a GSMDP with a continuous-time MDP, which can then be solved using existing techniques. We demonstrate that the introduction of phases permits us to take history into account when making action choices, and this can result in policies of higher quality than we would get if we ignored history dependence.

Stochastic models in population biology and their deterministic analogs.

Abstract: We introduce a class of stochastic population models based on “patch dynamics.” The size of the patch may be varied, and this allows one to quantify the departures of these stochastic models from various mean-field theories, which are generally valid as the patch size becomes very large. These models may be used to formulate a broad range of biological processes in both spatial and nonspatial contexts. Here, we concentrate on two-species competition. We present both a mathematical analysis of the patch model, in which we derive the precise form of the competition mean-field equations (and their first-order corrections in the nonspatial case), and simulation results. These mean-field equations differ, in some important ways, from those which are normally written down on phenomenological grounds. Our general conclusion is that mean-field theory is more robust for spatial models than for a single isolated patch. This is due to the dilution of stochastic effects in a spatial setting resulting from repeated rescue events mediated by interpatch diffusion. However, discrete effects due to modest patch sizes lead to striking deviations from mean-field theory even in a spatial setting.

Numerical methods for strong solutions of stochastic differential equations: an overview

Abstract: This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Probabilistic analysis of directed polymers in a random environment: a review

Abstract: Directed polymers in random environment can be thought of as a model of statistical mechanics in which paths of stochastic processes interact with a quenched disorder (impurities), depending on both time and space. We review here main results which have been obtained during the last fifteen years, with proofs to most of the results. The material covers the diffusive behavior of the polymers in weak disorder phase studied by J. Imbrie, T. Spencer, E. Bolthausen, R. Song and X. Y. Zhou [11, 3, 25], and localization of the paths in strong disordered phase recently obtained by P. Carmona, Y. Hu, and the authors of the present article [4, 5].

Quadratic Hedging and Mean-Variance Portfolio Selection with Random Parameters in an Incomplete Market

Abstract: This paper concerns the problems ofquadratic hedging andpricing, andmean-variance portfolio selection in anincomplete market setting with continuous trading, multiple assets, and Brownian information. In particular, we assume throughout that the parameters describing the market model may be random processes. We approach these problems from the perspective oflinear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs); that is, we focus on the so-calledstochastic Riccati equation (SRE) associated with the problem. Excepting certain special cases, solvability of the SRE remains an open question. Our primary theoretical contribution is a proof of existence and uniqueness of solutions of the SRE associated with the quadratic hedging and mean-variance problems. In addition, we derive closed-form expressions for the optimal portfolios and efficient frontier in terms of the solution of the SRE. A generalization of theMutual Fund Theorem is also obtained.

Decoupling quantum dissipation interaction via stochastic fields

Abstract: Based on the Hubbard–Stratonovich transformation, the dissipative interaction between the system of interest and the heat bath is decoupled and the separated system and bath thus evolve in common classical random fields. This manipulation allows us to establish a novel theoretical methodology by which the reduced density matrix is formulated as an ensemble average of its random realizations in the auxiliary white noise fields. Within the stochastic description, the interaction between the system and the bath is reflected in the mutually induced mean fields. The relationship between the bath-induced field and the influence functional in the path integral framework is revealed. As a demonstration of this approach, we derive the exact master equations for two model systems.

A stochastic integral equation method for modeling the rough surface effect on interconnect capacitance

Abstract: In This work we describe a stochastic integral equation method for computing the mean value and the variance of capacitance of interconnects with random surface roughness. An ensemble average Green's function is combined with a matrix Neumann expansion to compute nominal capacitance and its variance. This method avoids the time-consuming Monte Carlo simulations and the discretization of rough surfaces. Numerical experiments show that the results of the new method agree very well with Monte Carlo simulation results.

Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems

Abstract: As an extension of the theory of Dyson’s Brownian motion models for the standard Gaussian random-matrix ensembles, we report a systematic study of Hermitian matrix-valued processes and their eigenvalue processes associated with the chiral and nonstandard random-matrix ensembles. In addition to the noncolliding Brownian motions, we introduce a one-parameter family of temporally homogeneous noncolliding systems of the Bessel processes and a two-parameter family of temporally inhomogeneous noncolliding systems of Yor’s generalized meanders and show that all of the ten classes of eigenvalue statistics in the Altland–Zirnbauer classification are realized as particle distributions in the special cases of these diffusion particle systems. As a corollary of each equivalence in distribution of a temporally inhomogeneous eigenvalue process and a noncolliding diffusion process, a stochastic-calculus proof of a version of the Harish–Chandra (Itzykson–Zuber) formula of integral over unitary group is established.