Showing papers on "Stochastic process published in 1998"

Laser beam propagation through random media

Abstract: Random Processes and Random Fields Optical Turbulence in the Atmosphere Free-Space Propagation of Gaussian-Beam Waves Classical Theory of Optical Wave Propagation Line-of-Sight Propagation - Weak Fluctuation Theory, Part 1 Line-of-Sight Propagation - Weak Fluctuation Theory, Part 2 Propagation Through Random Phase Screens Laser Satellite Communication Systems Propagation Through Complex Paraxial ABCD Optical Systems Doublepassage Problems - Laser Radar Systems Line-of-Sight Propagation - Strong Fluctuation Theory Appendices - Special Functions Integral Table Tables of Beam Statistics Mathematica Programmes.

The Variance Gamma Process and Option Pricing

Abstract: A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S&P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.

The variational formulation of the Fokker-Planck equation

Abstract: The Fokker--Planck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It ...

Differential equations driven by rough signals

Abstract: This paper aims to provide a systematic approach to the treatment of differential equations of the type dyt = Si fi(yt) dxti where the driving signal xt is a rough path. Such equations are very common and occur particularly frequently in probability where the driving signal might be a vector valued Brownian motion, semi-martingale or similar process. However, our approach is deterministic, is totally independent of probability and permits much rougher paths than the Brownian paths usually discussed. The results here are strong enough to treat the main probabilistic examples and significantly widen the class of stochastic processes which can be used to drive stochastic differential equations. (For a simple example see [10], [1]). We hope our results will have an influence on infinite dimensional analysis on path spaces, loop groups, etc. as well as in more applied situations. Variable step size algorithms for the numerical integration of stochastic differential equations [8] have been constructed as a consequence of these results.

Integration with respect to fractal functions and stochastic calculus. I

Abstract: The classical Lebesgue–Stieltjes integral ∫ b a fdg of real or complex-valued functions on a finite interval (a,b) is extended to a large class of integrands f and integrators g of unbounded variation. The key is to use composition formulas and integration-by-part rules for fractional integrals and Weyl derivatives. In the special case of Holder continuous functions f and g of summed order greater than 1 convergence of the corresponding Riemann–Stieltjes sums is proved. The results are applied to stochastic integrals where g is replaced by the Wiener process and f by adapted as well as anticipating random functions. In the anticipating case we work within Slobodeckij spaces and introduce a stochastic integral for which the classical Ito formula remains valid. Moreover, this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional Brownian motion.

Tractable inference for complex stochastic processes

Abstract: The monitoring and control of any dynamic system depends crucially on the ability to reason about its current status and its future trajectory In the case of a stochastic system, these tasks typically involve the use of a belief state--a probability distribution over the state of the process at a given point in time Unfortunately, the state spaces of complex processes are very large, making an explicit representation of a belief state intractable Even in dynamic Bayesian networks (DBNs), where the process itself can be represented compactly, the representation of the belief state is intractable We investigate the idea of maintaining a compact approximation to the true belief state, and analyze the conditions under which the errors due to the approximations taken over the lifetime of the process do not accumulate to make our answers completely irrelevant We show that the error in a belief state contracts exponentially as the process evolves Thus, even with multiple approximations, the error in our process remains bounded indefinitely We show how the additional structure of a DBN can be used to design our approximation scheme, improving its performance significantly We demonstrate the applicability of our ideas in the context of a monitoring task, showing that orders of magnitude faster inference can be achieved with only a small degradation in accuracy

Langevin Equation and Thermodynamics

Abstract: We introduce a framework of energetics into the stochastic dynamics described by Langevin equation in which fluctuation force obeys the Einstein relation. The energy con­ servation holds in the individual realization of stochastic process, while the second law and steady state thermodynamics of Oono and Paniconi [Y. Oono and M. Paniconi, this issue] are obtained as ensemble properties of the process.

Combinatorial Theory of the Free Product With Amalgamation and Operator-Valued Free Probability Theory

Abstract: Preliminaries on non-crossing partitions Operator-valued multiplicative functions on the lattice of non-crossing partitions Amalgamated free products Operator-valued free probability theory Operator-valued stochastic processes and stochastic differential equations Bibliography.

Information bounds and quick detection of parameter changes in stochastic systems

Abstract: By using information-theoretic bounds and sequential hypothesis testing theory, this paper provides a new approach to optimal detection of abrupt changes in stochastic systems. This approach not only generalizes previous work in the literature on optimal detection far beyond the relatively simple models treated but also suggests alternative performance criteria which are more tractable and more appropriate for general stochastic systems. In addition, it leads to detection rules which have manageable computational complexity for on-line implementation and yet are nearly optimal under the different performance criteria considered.

Brownian motion, obstacles, and random media

Abstract: This book is aimed at graduate students and researchers. It provides an account for the non-specialist of the circle of ideas, results and techniques, which grew out in the study of Brownian motion and random obstacles. This subject has a rich phenomenology which exhibits certain paradigms, emblematic of the theory of random media. It also brings into play diverse mathematical techniques such as stochastic processes, functional analysis, potential theory, first passage percolation. In a first part, the book presents, in a concrete manner, background material related to the Feynman-Kac formula, potential theory, and eigenvalue estimates. In a second part, it discusses recent developments including the method of enlargement of obstacles, Lyapunov coefficients, and the pinning effect. The book also includes an overview of known results and connections with other areas of random media.

Analysis of low density codes and improved designs using irregular graphs

Abstract: We introduce a new set of probabilistic analysis tools based on the analysis of And-Or trees with random inputs. These tools provide a unifying, intuitive, and powerful framework for carrying out the analysis of several previously studied random processes of interest, includingrandom loss-resilient codes, solving random k-SAT formula using the pure literal rule, and thegreedy algorithm for matchings in random graphs. In addition, these tools allow generalizations of these problems not previously analyzed to be analyzed in a straightforward manner. We illustrate our methodology on the three problems listed above

Analysis of random processes via And-Or tree evaluation

Abstract: We introduce a new set of probabilistic analysis tools based on the analysis of And-Or trees with random inputs. These tools provide a unifying, intuitive, and powerful framework for carrying out the analysis of several previously studied random processes of interest, including random loss-resilient codes, solving random k-SAT formula using the pure literal rule, and the greedy algorithm for matchings in random graphs. In addition, these tools allow generalizations of these problems not previously analyzed to be analyzed in a straightforward manner. We illustrate our methodology on the three problems listed above

Stochastic response surface methods (SRSMs) for uncertainty propagation: application to environmental and biological systems.

Abstract: Comprehensive uncertainty analyses of complex models of environmental and biological systems are essential but often not feasible due to the computational resources they require. "Traditional" methods, such as standard Monte Carlo and Latin Hypercube Sampling, for propagating uncertainty and developing probability densities of model outputs, may in fact require performing a prohibitive number of model simulations. An alternative is offered, for a wide range of problems, by the computationally efficient "Stochastic Response Surface Methods (SRSMs)" for uncertainty propagation. These methods extend the classical response surface methodology to systems with stochastic inputs and outputs. This is accomplished by approximating both inputs and outputs of the uncertain system through stochastic series of "well behaved" standard random variables; the series expansions of the outputs contain unknown coefficients which are calculated by a method that uses the results of a limited number of model simulations. Two case studies are presented here involving (a) a physiologically-based pharmacokinetic (PBPK) model for perchloroethylene (PERC) for humans, and (b) an atmospheric photochemical model, the Reactive Plume Model (RPM-IV). The results obtained agree closely with those of traditional Monte Carlo and Latin Hypercube Sampling methods, while significantly reducing the required number of model simulations.

A stochastic wire-length distribution for gigascale integration (GSI). I. Derivation and validation

Abstract: Based on Rent's Rule, a well-established empirical relationship, a rigorous derivation of a complete wire-length distribution for on-chip random logic networks is performed. This distribution is compared to actual wire-length distributions for modern microprocessors, and a methodology to calculate the wire-length distribution for future gigascale integration (GSI) products is proposed.

Branching processes in Lévy processes: the exploration process

Abstract: The main idea of the present work is to associate with a general continuous branching process an exploration process that contains the desirable information about the genealogical structure. The exploration process appears as a simple local time functional of a Levy process with no negative jumps, whose Laplace exponent coincides with the branching mechanism function. This new relation between spectrally positive Levy processes and continuous branching processes provides a unified perspective on both theories. In particular, we derive the adequate formulation of the classical Ray–Knight theorem for such Levy processes. As a consequence of this theorem, we show that the path continuity of the exploration process is equivalent to the almost sure extinction of the branching process.

A Markovian approach for modeling packet traffic with long-range dependence

Abstract: We present a simple Markovian framework for modeling packet traffic with variability over several time scales. We present a fitting procedure for matching second-order properties of counts to that of a second-order self-similar process. Our models essentially consist of superpositions of two-state Markov modulated Poisson processes (MMPPs). We illustrate that a superposition of four two-state MMPPs suffices to model second-order self-similar behavior over several time scales. Our modeling approach allows us to fit to additional descriptors while maintaining the second-order behavior of the counting process. We use this to match interarrival time correlations.

Stochastic power control for cellular radio systems

Abstract: For wireless communication systems, iterative power control algorithms have been proposed to minimize the transmitter power while maintaining reliable communication between mobiles and base stations. To derive deterministic convergence results, these algorithms require perfect measurements of one or more of the following parameters: (1) the mobile's signal-to-interference ratio (SIR) at the receiver; (2) the interference experienced by the mobile; and (3) the bit-error rate. However, these quantities are often difficult to measure and deterministic convergence results neglect the effect of stochastic measurements. We develop distributed iterative power control algorithms that use readily available measurements. Two classes of power control algorithms are proposed. Since the measurements are random, the proposed algorithms evolve stochastically and we define the convergence in terms of the mean-squared error (MSE) of the power vector from the optimal power vector that is the solution of a feasible deterministic power control problem. For the first class of power control algorithms using fixed step size sequences, we obtain finite lower and upper bounds for the MSE by appropriate selection of the step size. We also show that these bounds go to zero, implying convergence in the MSE sense, as the step size goes to zero. For the second class of power control algorithms, which are based on the stochastic approximations method and use time-varying step size sequences, we prove that the MSE goes to zero. Both classes of algorithms are distributed in the sense that each user needs only to know its own channel gain to its assigned base station and its own matched filter output at its assigned base station to update its power.

The sample autocorrelations of heavy-tailed processes with applications to ARCH

Abstract: We study the sample ACVF and ACF of a general stationary sequence under a weak mixing condition and in the case that the marginal distributions are regularly varying. This includes linear and bilinear processes with regularly varying noise and ARCH processes, their squares and absolute values. We show that the distributional limits of the sample ACF can be random, provided that the Variance of the marginal distribution is infinite and the process is nonlinear. This is in contrast to infinite variance linear processes. If the process has a finite second but infinite fourth moment, then the sample ACP is consistent with scaling rates that grow at a slower rate than the standard root n. Consequently, asymptotic confidence bands are wider than those constructed in the classical theory. We demonstrate the theory in full detail far an ARCH(1) process.

Wealth distributions in asset exchange models

Abstract: A model for the evolution of the wealth distribution in an economically interacting population is introduced, in which a specified amount of assets are exchanged between two individuals when they interact. The resulting wealth distributions are determined for a variety of exchange rules. For “random” exchange, either individual is equally likely to gain in a trade, while “greedy” exchange, the richer individual gains. When the amount of asset traded is fixed, random exchange leads to a Gaussian wealth distribution, while greedy exchange gives a Fermi-like scaled wealth distribution in the long-time limit. Multiplicative processes are also investigated, where the amount of asset exchanged is a finite fraction of the wealth of one of the traders. For random multiplicative exchange, a steady state occurs, while in greedy multiplicative exchange a continuously evolving power law wealth distribution arises.

Detection of stochastic processes

Abstract: This paper reviews two streams of development, from the 1940's to the present, in signal detection theory: the structure of the likelihood ratio for detecting signals in noise and the role of dynamic optimization in detection problems involving either very large signal sets or the joint optimization of observation time and performance. This treatment deals exclusively with basic results developed for the situation in which the observations are modeled as continuous-time stochastic processes. The mathematics and intuition behind such developments as the matched filter, the RAKE receiver, the estimator-correlator, maximum-likelihood sequence detectors, multiuser detectors, sequential probability ratio tests, and cumulative-sum quickest detectors, are described.

The stochastic wave equation in two spatial dimensions

Abstract: Keywords: stochastic wave equation ; Gaussian noise ; process solution Reference PROB-ARTICLE-1998-001View record in Web of Science Record created on 2008-12-01, modified on 2017-05-12

Modern simulation and modeling

Abstract: CONVENTIONAL SIMULATION. Systems, Models, and Simulation. Random Numbers, Variates, and Stochastic Process Generation. Output Analysis of Discrete-Event Systems via Simulation. Variance Reduction Techniques. MODERN SIMULATION. Sensitivity Analysis and Optimization of Discrete-Event Static Systems (DESS). Sensitivity Analysis and Optimization of Discrete-Event Dynamic Systems: Distributed Parameters. Sensitivity of Analysis of Discrete-Event Dynamic Systems: Structural Parameters. Response Surface Methodology via the Score Function Method. Estimating Rare-Event Probabilities and Related Optimization Issues. Index.

The true self-repelling motion

Abstract: We construct and study a continuous real-valued random process, which is of a new type: It is self-interacting (self-repelling) but only in a local sense: it only feels the self-repellance due to its occupation-time measure density in the `immediate neighbourhood' of the point it is just visiting. We focus on the most natural process with these properties that we call `true self-repelling motion'. This is the continuous counterpart to the integer-valued `true' self-avoiding walk, which had been studied among others by the first author. One of the striking properties of true self-repelling motion is that, although the couple (X t , occupation-time measure of X at time t) is a continuous Markov process, X is not driven by a stochastic differential equation and is not a semi-martingale. It turns out, for instance, that it has a finite variation of order 3/2, which contrasts with the finite quadratic variation of semi-martingales. One of the key-tools in the construction of X is a continuous system of coalescing Brownian motions similar to those that have been constructed by Arratia [A1, A2]. We derive various properties of X (existence and properties of the occupation time densities L t (x), local variation, etc.) and an identity that shows that the dynamics of X can be very loosely speaking described as follows: −dX t is equal to the gradient (in space) of L t (x), in a generalized sense, even though x↦L t (x) is not differentiable.

Stochastic calculus with respect to free Brownian motion and analysis on Wigner space

Abstract: We define stochastic integrals with respect to free Brownian motion, and show that they satisfy Burkholder-Gundy type inequalities in operator norm. We prove also a version of Ito's predictable representation theorem, as well as product form and functional form of Ito's formula. Finally we develop stochastic analysis on the free Fock space, in analogy with stochastic analysis on the Wiener space.

Perfect Simulation for the Area-Interaction Point Process

Abstract: Because so many random processes arising in stochastic geometry are quite intractable to analysis, simulation is an important part of the stochastic geometry toolkit. Typically, a Markov point process such as the area-interaction point process is simulated (approximately) as the long-run equilibrium distribution of a (usually reversible) Markov chain such as a spatial birth-and-death process. This is a useful method, but it can be very hard to be precise about the length of simulation required to ensure that the long-run approximation is good. The splendid idea of Propp and Wilson [17] suggests a way forward: they propose a coupling method which delivers exact simulation of equilibrium distributions of (finite-state-space) Markov chains. In this paper their idea is extended to deal with perfect simulation of attractive area-interaction point processes in bounded windows. A simple modification of the basic algorithm is described which provides perfect simulation of the repulsive case as well (which being nonmonotonic might have been thought out of reach). Results from simulations using a C computer program are reported; these confirm the practicality of this approach in both attractive and repulsive cases. The paper concludes by mentioning other point processes which can be simulated perfectly in this way, and by speculating on useful future directions of research. Clearly workers in stochastic geometry should now seek wherever possible to incorporate the Propp and Wilson idea in their simulation algorithms.

New tools for understanding spurious regressions

Abstract: Some new tools for analyzing spurious regressions are presented. The theory utilizes the general representation of a stochastic process in terms of an orthonormal system and provides an extension of the Weierstrass theorem to include the approximation of continuous functions and stochastic processes by Wiener processes. The theory is applied to two classic examples of spurious regressions: regression of stochastic trends on time polynomials, and regressions among independent random walks. It is shown that such regressions reproduce in part and in whole the underlying orthonormal representations.

Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

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.