Topic
Stochastic process
About: Stochastic process is a research topic. Over the lifetime, 31227 publications have been published within this topic receiving 898736 citations. The topic is also known as: random process & stochastic processes.
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TL;DR: In this article, Markov Chain Algorithms have been used to define a criterion for recurring and transience of Brownian Motion in several dimensions, including positive recurring and negative recurring and null recurring.
Abstract: Preface to Second Edition Preface to First Edition PRELIMINARIES Introduction Linear Differential Equations Linear Difference Equations Exercises FINITE MARKOV CHAINS Definitions and Examples Large-Time Behavior and Invariant Probability Classification of States Return Times Transient States Examples Exercises COUNTABLE MARKOV CHAINS Introduction Recurrence and Transience Positive Recurrence and Null Recurrence Branching Process Exercises CONTINUOUS-TIME MARKOV CHAINS Poisson Process Finite State Space Birth-and-Death Processes General Case Exercises OPTIMAL STOPPING Optimal Stopping of Markov Chains Optimal Stopping with Cost Optimal Stopping with Discounting Exercises MARTINGALES Conditional Expectation Definition and Examples Optional Sampling Theorem Uniform Integrability Martingale Convergence Theorem Maximal Inequalities Exercises RENEWAL PROCESSES Introduction Renewal Equation Discrete Renewal Processes M/G/1 and G/M/1 Queues Exercises REVERSIBLE MARKOV CHAINS Reversible Processes Convergence to Equilibrium Markov Chain Algorithms A Criterion for Recurrence Exercises BROWNIAN MOTION Introduction Markov Property Zero Set of Brownian Motion Brownian Motion in Several Dimensions Recurrence and Transience Fractal Nature of Brownian Motion Scaling Rules Brownian Motion with Drift Exercises STOCHASTIC INTEGRATION Integration with Respect to Random Walk Integration with Respect to Brownian Motion Ito's Formula Extensions if Ito's Formula Continuous Martingales Girsanov Transformation Feynman-Kac Formula Black-Scholes Formula Simulation Exercises Suggestions for Further Reading Index
172 citations
TL;DR: Several approaches for the evaluation of upper and lower bounds on error probability of asynchronous spread spectrum multiple access communication systems are presented, utilizing an isomorphism theorem in the theory of moment spaces.
Abstract: Several approaches for the evaluation of upper and lower bounds on error probability of asynchronous spread spectrum multiple access communication systems are presented. These bounds are obtained by utilizing an isomorphism theorem in the theory of moment spaces. From this theorem, we generate closed, compact, and convex bodies, where one of the coordinates represents error probability, while the other coordinate represents a generalized moment of the multiple access interference random variable. Derivations for the second moment, fourth moment, single exponential moment, and multiple exponential moment are given in terms of the partial cross correlations of the codes used in the system. Error bounds based on the use of these moments are obtained. By using a sufficient number of terms in the multiple exponential moment, upper and lower error bounds can be made arbitrarily tight. In that case, the error probability equals the multiple exponential moment of the multiple access interference random variable. An example using partial cross correlations based on codes generated from Gold's method is presented.
172 citations
TL;DR: It is shown that under the control law based on a memoryless observer, the closed-loop equilibrium of interest is globally stable in probability, and moreover, the solution process can be regulated to the origin almost surely.
Abstract: In this paper, we investigate the adaptive output-feedback stabilisation for a class of stochastic non-linear systems with time-varying time delays First, we give some sufficient conditions to ensure the existence and uniqueness of the solution process for stochastic non-linear systems with time delays, and introduce a new stability notion and the related criterion Then, for a class of stochastic non-linear systems with time-varying time delays, uncertain parameters in both drift and diffusion terms, and general constant virtual control coefficients, we present a systematic design procedure for a memoryless adaptive output-feedback control law by using the backstepping method It is shown that under the control law based on a memoryless observer, the closed-loop equilibrium of interest is globally stable in probability, and moreover, the solution process can be regulated to the origin almost surely
172 citations
TL;DR: Simulation results demonstrate that ME-gPC is effective in improving the accuracy of gPC for a long-term integration whereas high-order gPC cannot capture the correct asymptotic behavior.
172 citations
TL;DR: In this article, a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers is developed, where the intrinsic permeability is represented in the model via its Karhunen-Loeve expansion.
Abstract: This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.
171 citations