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This paper revisits the definition of linear time-invariant (LTI) stochastic process within a behavioral systems framework. Building on [Willems, 2013], we derive a canonical representation of an LTI stochastic process and a physically grounded notion of interconnection between independent stochastic processes. We use this framework to analyze the invariance properties enjoyed by distances between spectral densities of LTI processes.

We present a study of residence time statistics for $N$ blinking quantum dots. With numerical simulations and exact calculations we show sharp transitions for a critical number of dots. In contrast to expectation the fluctuations in the limit of $N \to \infty$ are non-trivial. Besides quantum dots our work describes residence time statistics in several other many particle systems for example $N$ Brownian particles. Our work provides a natural framework to detect non-ergodic kinetics from measurements of many blinking chromophores, without the need to reach the single molecule limit.

Body 13; #013: Obtaining Homochirality Through Stochastic Processes 1. Obtaining Homochirality Through Stochastic Processes

Body 13; #013: Obtaining Homochirality Through Stochastic Processes 1. Obtaining Homochirality Through Stochastic Processes

The following topics are the principal ones investigated under this grant: (1) Hypothesis testing for Poisson-like processes. (2) Hilbert space methods in time series analysis. (3) Limit distributions of branching processes. (4) Hausdorff dimension in stochastic processes. (5) Sufficient statistics for stochastic processes. (6) Absolute continuity and orthogonality of stochastic processes. (7) Subordination of stochastic processes. (8) Slowly varying functions in probability. (9) Mixture problems and the Glivenko- Cantelli Theorem. (10) Random power series.

The investigation was designed to prepare a monograph on certain mathematical aspects of the inference theory of stochastic processes, the principal components of which are substantially completed. These include substantive treatments of the foundations of inference theory, i.e., the projective limits of probability spaces, of conditional probability distributions and expectations, which occupy a central position in the analysis of essentially all the problems of inference, some new or simplified proofs of the standard theory of martingales together with a demonstration of the equivalence of the martingale convergence and the Andersen-Jessen theory, of stochastic difference and differential equations in both the physical and social sciences, of Gaussian processes, and of hypothesis testing, parametric estimation, and prediction, as the latter three topics relate to inference theory.

Stochastic Processes and Their Applications 1989: Volume 31 , Issue Index. Digitized from IA1653419-02 . Previous issue: sim_stochastic-processes-and-their-applications_1988-12_30_2 . Next issue: sim_stochastic-processes-and-their-applications_1989-03_31_1 .

Stochastic Processes and Their Applications 1990: Volume 36 , Issue Index. Digitized from IA1653419-02 . Previous issue: sim_stochastic-processes-and-their-applications_1990-08_35_2 . Next issue: sim_stochastic-processes-and-their-applications_1990-10_36_1 .

Stochastic Processes and Their Applications 1997: Volume 68 , Issue Index. Digitized from IA1653419-05 . Previous issue: sim_stochastic-processes-and-their-applications_1997-05-16_67_2 . Next issue: sim_stochastic-processes-and-their-applications_1997-05-30_68_1 .

Stochastic Processes and Their Applications 1995: Volume 58 , Issue Index. Digitized from IA1652405-03 . Previous issue: sim_stochastic-processes-and-their-applications_1995-06_57_2_0 . Next issue: sim_stochastic-processes-and-their-applications_1995-07_58_1_0 .

Stochastic Processes and Their Applications 1995: Volume 58 , Issue Index. Digitized from IA1652405-03 . Previous issue: sim_stochastic-processes-and-their-applications_1995-06_57_2_0 . Next issue: sim_stochastic-processes-and-their-applications_1995-07_58_1_0 .

Stochastic Processes and Their Applications 1995: Volume 58 , Issue Index. Digitized from IA1652405-03 . Previous issue: sim_stochastic-processes-and-their-applications_1995-06_57_2_0 . Next issue: sim_stochastic-processes-and-their-applications_1995-07_58_1_0 .

Stochastic Processes and Their Applications 1995: Volume 58 , Issue Index. Digitized from IA1652405-03 . Previous issue: sim_stochastic-processes-and-their-applications_1995-06_57_2_0 . Next issue: sim_stochastic-processes-and-their-applications_1995-07_58_1_0 .

We introduce a new distance, a Lipschitz-Prokhorov distance $d_{LP}$, on the set $\mathcal {PM}$ of isomorphism classes of pairs $(X, P)$ where $X$ is a compact metric space and $P$ is the law of a continuous stochastic process on $X$. We show that $(\mathcal {PM}, d_{LP})$ is a complete metric space. For Markov processes on Riemannian manifolds, we study relative compactness and convergence.

This text is a survey of the general theory of stochastic processes, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which were developed by the French probability school and which are usually written in French. The material presented in the last three chapters is less standard and takes into account some recent developments.

We present the formalism of sequential and asynchronous processes defined in terms of random or quantum grammars and argue that these processes have relevance in genomics. To make the article accessible to the non-mathematicians, we keep the mathematical exposition as elementary as possible, focusing on some general ideas behind the formalism and stating the implications of the known mathematical results. We close with a set of open challenging problems.

An analytical study of the return time distribution of extreme events for stochastic processes with power-law correlation has been carried on. The calculation is based on an epsilon-expansion in the correlation exponent: C(t)=|t|^{-1+epsilon}. The fixed point of the theory is associated with stretched exponential scaling of the distribution; analytical expressions, valid in the pre-asymptotic regime, have been provided. Also the permanence time distribution appears to be characterized by stretched exponential scaling. The conditions for application of the theory to non-Gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed.

An analytical study of the return time distribution of extreme events for stochastic processes with power-law correlation has been carried on. The calculation is based on an epsilon-expansion in the correlation exponent: C(t)=|t|^{-1+epsilon}. The fixed point of the theory is associated with stretched exponential scaling of the distribution; analytical expressions, valid in the pre-asymptotic regime, have been provided. Also the permanence time distribution appears to be characterized by stretched exponential scaling. The conditions for application of the theory to non-Gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed.

Given a Gaussian stationary increment processes with spectral density, we show that a Wick-Ito integral with respect to this process can be naturally obtained using Hida's white noise space theory. We use the Bochner-Minlos theorem to associate a probability space to the process, and define the counterpart of the S-transform in this space. We then use this transform to define the stochastic integral and prove an associated Ito formula.

In this paper, we develop new optional stopping theorems for scenarios where the stopping rules are defined by bounded continuity regions. Moreover, we establish a wide variety of inequalities on the supremums and infimums of functions of stochastic processes over the whole range of time indexes.

In this paper, we adopt a Bayesian point of view for predicting real continuous-time processes. We give two equivalent definitions of a Bayesian predictor and study some properties: admissibility, prediction sufficiency, non-unbiasedness, comparison with efficient predictors. Prediction of Poisson process and prediction of Ornstein-Uhlenbeck process in the continuous and sampled situations are considered. Various simulations illustrate comparison with non-Bayesian predictors.

This paper consists of two independent parts. In the first one, we contribute to the study of the class $(\Sigma)$. For instance, we provide a new way to characterize stochastic processes of this class. We also present some new properties and solve the Bachelier equation. In the second part, we study the class of stochastic processes $\Sigma(H)$. This class was introduced in \cite{f} where from tools of the theory of martingales with respect to a signed measure of \cite{chav}, the authors provide a general framework and methods for dealing with processes of this class. In this work, after developing some new properties, we embed a non-atomic measure $\nu$ in $X$, a process of the class $\Sigma(H)$. More precisely, we find a stopping time $T

Stochastic integration with respect to Gaussian processes, such as fractional Brownian motion (fBm) or multifractional Brownian motion (mBm), has raised strong interest in recent years, motivated in particular by applications in finance, Internet traffic modeling and biomedicine. The aim of this work to define and develop, using White Noise Theory, an anticipative stochastic calculus with respect to a large class of Gaussian processes, denoted G, that contains, among many other processes, Volterra processes (and thus fBm) and also mBm. This stochastic calculus includes a definition of a stochastic integral, It\^o formulas (both for tempered distributions and for functions with sub-exponential growth), a Tanaka Formula as well as a definition, and a short study, of (both weighted and non weighted) local times of elements of G . In that view, a white noise derivative of any Gaussian process G of G is defined and used to integrate, with respect to G, a large class of stochastic processes, using Wick products. A comparison of our integral wrt elements of G to the ones provided by Malliavin calculus in [AMN01] and by It\^o stochastic calculus is also made. Moreover, one shows that the stochastic calculus with respect to Gaussian processes provided in this work generalizes the stochastic calculus originally proposed for fBm in [EVdH03, BS{\O}W04, Ben03a] and for mBm in [LLV14, Leb13, LLVH14]. Likewise, it generalizes results given in [NT06] and some results given in [AMN01]. In addition, it offers alternative conditions to the ones required in [AMN01] when one deals with stochastic calculus with respect to Gaussian processes.

Gaussian stochastic diffusion processes are used to derive cosmic mass functions. To get analytic relations previous studies exploited the sharp $k$-space filter assumption yielding zero drift terms in the corresponding Fokker-Planck (Kolmogorov's forward) equation and thus simplifying analytic treatments significantly (excursion set formalism). In the present paper methods are described to derive for given diffusion processes and Gaussian random fields the corresponding mass and filter functions by solving the Kolmogorov's forward and backward equations including nonzero drift terms. This formalism can also be used in cases with non-sharp $k$-space filters and for diffusion processes exhibiting correlations between different mass scales.

There is a whole range of emergent phenomena in non-equilibrium behaviors can be well described by a set of stochastic differential equations. Inspired by an insight gained during our study of robustness and stability in phage lambda genetic switch in modern biology, we found that there exists a classification of generic nonequilibrium processes: In the continuous description in terms of stochastic differential equations, there exists four dynamical elements: the potential function $\phi$, the friction matrix $ S$, the anti-symmetric matrix $ T $, and the noise. The generic feature of absence of detailed balance is then precisely represented by $T$. For dynamical near a fixed point, whether or not it is stable or not, the stochastic dynamics is linear. A rather complete analysis has been carried out (Kwon, Ao, Thouless, cond-mat/0506280; PNAS, {\bf 102} (2005) 13029), referred to as SDS I. One important and persistent question is the existence of a potential function with nonlinear force and with multiplicative noise, with both nice local dynamical and global steady state properties. Here we demonstrate that a dynamical structure built into stochastic differential equation allows us to construct such a global optimization potential function. First, we provide the construction. One of most important ingredient is the generalized Einstein relation. We then present an approximation scheme: The gradient expansion which turns every order into linear matrix equations. The consistent of such methodology with other known stochastic treatments will be discussed in next paper, SDS III; and the explicitly connection to statistical mechanics and thermodynamics will be discussed in a forthcoming paper, SDS IV.

This is a set of four lectures devoted to simple ideas about turbulent transport, a ubiquitous non-equilibrium phenomenon. In the course similar to that given by the author in 2006 in Warwick [45], we discuss lessons which have been learned from naive models of turbulent mixing that employ simple random velocity ensembles and study related stochastic processes. In the first lecture, after a brief reminder of the turbulence phenomenology, we describe the intimate relation between the passive advection of particles and fields by hydrodynamical flows. The second lecture is devoted to some useful tools of the multiplicative ergodic theory for random dynamical systems. In the third lecture, we apply these tools to the example of particle flows in the Kraichnan ensemble of smooth velocities that mimics turbulence at intermediate Reynolds numbers. In the fourth lecture, we extends the discussion of particle flows to the case of non-smooth Kraichnan velocities that model fully developed turbulence. We stress the unconventional aspects of particle flows that appear in this regime and lead to phase transitions in the presence of compressibility. The intermittency of scalar fields advected by fully turbulent velocities and the scenario linking it to hidden statistical conservation laws of multi-particle flows are briefly explained.

This is a set of four lectures devoted to simple ideas about turbulent transport, a ubiquitous non-equilibrium phenomenon. In the course similar to that given by the author in 2006 in Warwick [45], we discuss lessons which have been learned from naive models of turbulent mixing that employ simple random velocity ensembles and study related stochastic processes. In the first lecture, after a brief reminder of the turbulence phenomenology, we describe the intimate relation between the passive advection of particles and fields by hydrodynamical flows. The second lecture is devoted to some useful tools of the multiplicative ergodic theory for random dynamical systems. In the third lecture, we apply these tools to the example of particle flows in the Kraichnan ensemble of smooth velocities that mimics turbulence at intermediate Reynolds numbers. In the fourth lecture, we extends the discussion of particle flows to the case of non-smooth Kraichnan velocities that model fully developed turbulence. We stress the unconventional aspects of particle flows that appear in this regime and lead to phase transitions in the presence of compressibility. The intermittency of scalar fields advected by fully turbulent velocities and the scenario linking it to hidden statistical conservation laws of multi-particle flows are briefly explained.

Markov decision processes (MDPs) and simple stochastic games (SSGs) provide a rich mathematical framework to study many important problems related to probabilistic systems. MDPs and SSGs with finite-horizon objectives, where the goal is to maximize the probability to reach a target state in a given finite time, is a classical and well-studied problem. In this work we consider the strategy complexity of finite-horizon MDPs and SSGs. We show that for all $\epsilon>0$, the natural class of counter-based strategies require at most $\log \log (\frac{1}{\epsilon}) + n+1$ memory states, and memory of size $\Omega(\log \log (\frac{1}{\epsilon}) + n)$ is required. Thus our bounds are asymptotically optimal. We then study the periodic property of optimal strategies, and show a sub-exponential lower bound on the period for optimal strategies.

In this paper we present a new model for modeling the diffusion and relative dispersion of particles in homogeneous isotropic turbulence. We use an Heisenberg-like Hamiltonian to incorporate spatial correlations between fluid particles, which are modeled by stochastic processes correlated in time. We are able to reproduce the ballistic regime in the mean squared displacement of single particles and the transition to a normal diffusion regime for long times. For the dispersion of particle pairs we find a $t^{2}$-dependence of the mean squared separation at short times and a $t$-dependence for long ones. For intermediate times indications for a Richardson $t^{3}$ law are observed in certain situations. Finally the influence of inertia of real particles on the dispersion is investigated.

The purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes. The theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts. Emphasis is placed on stationary cases but other important classes (e.g. Mark of sequences) are included. Significant ideas and methods are described rather than details, and in particular the nature and role of important underlying point processes (such as exceedances and upcrossings) are emphasized. Applications are given to particular classes of process (e.g. normal, moving average) and connections with related theory (such as convergence of sums) are indicated.

Some new classes of discrete time non-stationary processes, related to the harmonizable and V-bounded classes, are introduced. A few characterizations are obtained which, in turn, unify the V-bounded theory. Our main results depend on a special form of Grothendieck inequality.

We formulate and prove a new criterion for stability of e-processes. It says that any e-process which is averagely bounded and concentrating is asymptotically stable. In the second part, we show how this general result applies to some shell models (the Goy and the Sabra model). Indeed, we manage to prove that the processes corresponding to these models satisfy the e-process property. They are also averagely bounded and concentrating. Consequently, their stability follows.

We investigate a functional limit theorem (homogenization) for Reflected Stochastic Differential Equations on a half-plane with stationary coefficients when it is necessary to analyze both the effective Brownian motion and the effective local time. We prove that the limiting process is a reflected non-standard Brownian motion. Beyond the result, this problem is known as a prototype of non-translation invariant problem making the usual method of the "environment as seen from the particle" inefficient.

The paper studies stochastic integration with respect to Gaussian processes and fields. It is more convenient to work with a field than a process: by definition, a field is a collection of stochastic integrals for a class of deterministic integrands. The problem is then to extend the definition to random integrands. An orthogonal decomposition of chaos space of the random field leads to two such extensions, corresponding to the \Ito-Skorokhod and the Stratononovich integrals, and provides an efficient tool to study these integrals, both analytically and numerically. For a Gaussian process, a natural definition of the integral follows from a canonical correspondence between random processes and a special class of random fields.

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, the author considers the problems of verification and planning for stochastic processes with asynchronous events, without relying on the Markov assumption. He establishes the foundation for statistical probabilistic model checking, an approach to probabilistic model checking based on hypothesis testing and simulation. He demonstrates 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, he can set this error bound arbitrarily low (at the cost of efficiency). His contribution in planning consists of a formalism, the generalized semi-Markov decision process (GSMDP), for planning with asynchronous stochastic events. He considers both goal-directed and decision theoretic planning. In the former case, he relies on statistical model checking to verify plans, and uses the simulation traces to guide plan repair. In the latter case, he presents the use of phase-type distributions to approximate a GSMDP with a continuous-time MDP, which can then be solved using existing techniques. He demonstrates that the introduction of phases permits him to take history into account when making action choices, and this can result in policies of higher quality than he would get if he ignored history dependence.

In this paper we generalize the martingale of Kella and Whitt to the setting of L\'{e}vy-type processes and show that the (local) martingales obtained are in fact square integrable martingales which upon dividing by the time index converge to zero a.s. and in $L^2$. The reflected L\'{e}vy-type process is considered as an example.

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.

These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.

We extend the work of Tanase-Nicola and Kurchan on the structure of diffusion processes and the associated supersymmetry algebra by examining the responses of a simple statistical system to external disturbances of various kinds. We consider both the stochastic differential equations (SDEs) for the process and the associated diffusion equation. The influence of the disturbances can be understood by augmenting the original SDE with an equation for {\it slave variables}. The evolution of the slave variables describes the behaviour of line elements carried along in the stochastic flow. These line elements together with the associated surface and volume elements constructed from them provide the basis of the supersymmetry properties of the theory. For ease of visualisation, and in order to emphasise a helpful electromagnetic analogy, we work in three dimensions. The results are all generalisable to higher dimensions and can be specialised to one and two dimensions. The electromagnetic analogy is a useful starting point for calculating asymptotic results at low temperature that can be compared with direct numerical evaluations. We also examine the problems that arise in a direct numerical simulation of the stochastic equation together with the slave equations. We pay special attention to the dependence of the slave variable statistics on temperature. We identify in specific models the critical temperature below which the slave variable distribution ceases to have a variance and consider the effect on estimates of susceptibilities.

By introducing a new stochastic integral, we investigate the energetics of classical stochastic systems driven by non-Gaussian white noises. In particular, we introduce a decomposition of the total-energy difference into the work and the heat for each trajectory, and derive a formula to calculate the heat from experimental data on the dynamics. We apply our formulation and results to a Langevin system driven by a Poisson noise.

This paper studies a novel approach for approximating the behavior of compartmental spreading processes. In contrast to prior work, the methods developed describe a dynamics which bound the exact moment dynamics, without explicitly requiring a priori knowledge of non-negative (or non-positive) covariance between pairs of system variables. Moreover, we provide systems which provide both upper- and lower- bounds on the process moments. We then show that when system variables are shown to be non-negatively (or non-positively) correlated for all time in the system's evolution, we may leverage the knowledge to create better approximating systems. We then apply the technique to several previously studied compartmental spreading processes, and compare the bounding systems' performance to the standard approximations studied in prior literature.

We propose a mathematical framework for natural selection in finite populations. Traditionally, many of the selection-based processes used to describe cultural and genetic evolution (such as imitation and birth-death models) have been studied on a case-by-case basis. Over time, these models have grown in sophistication to include population structure, differing phenotypes, and various forms of interaction asymmetry, among other features. Furthermore, many processes inspired by natural selection, such as evolutionary algorithms in computer science, possess characteristics that should fall within the realm of a "selection process," but so far there is no overarching theory encompassing these evolutionary processes. The framework of $\textit{stochastic selection processes}$ we present here provides such a theory and consists of three main components: a $\textit{population state space}$, an $\textit{aggregate payoff function}$, and an $\textit{update rule}$. A population state space is a generalization of the notion of population structure, and it can include non-spatial information such as strategy-mutation rates and phenotypes. An aggregate payoff function allows one to generically talk about the fitness of traits without explicitly specifying a method of payoff accounting or even the nature of the interactions that determine payoff/fitness. An update rule is a fitness-based function that updates a population based on its current state, and it includes as special cases the classical update mechanisms (Moran, Wright-Fisher, etc.) as well as more complicated mechanisms involving chromosomal crossover, mutation, and even complex cultural syntheses of strategies of neighboring individuals. Our framework covers models with variable population size as well as with arbitrary, measurable trait spaces.

We discuss one-dimensional stochastic processes defined through the Temperley-Lieb algebra related to the Q=1 Potts model. For various boundary conditions, we formulate a conjecture relating the probability distribution which describes the stationary state, to the enumeration of a symmetry class of alternating sign matrices, objects that have received much attention in combinatorics.

During the project period, Ph.D. student Amites Dasgupta has been investigating "Fractional Brownian Motion: its Properties and Applications to Stochastic Integration".

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We study the problem of extracting a prescribed number of random bits by reading the smallest possible number of symbols from non-ideal stochastic processes. The related interval algorithm proposed by Han and Hoshi has asymptotically optimal performance; however, it assumes that the distribution of the input stochastic process is known. The motivation for our work is the fact that, in practice, sources of randomness have inherent correlations and are affected by measurement's noise. Namely, it is hard to obtain an accurate estimation of the distribution. This challenge was addressed by the concepts of seeded and seedless extractors that can handle general random sources with unknown distributions. However, known seeded and seedless extractors provide extraction efficiencies that are substantially smaller than Shannon's entropy limit. Our main contribution is the design of extractors that have a variable input-length and a fixed output length, are efficient in the consumption of symbols from the source, are capable of generating random bits from general stochastic processes and approach the information theoretic upper bound on efficiency.

Many applications require stochastic processes specified on two- or higher-dimensional domains; spatial or spatial-temporal modelling, for example. In these applications it is attractive, for conceptual simplicity and computational tractability, to propose a covariance function that is separable; e.g., the product of a covariance function in space and one in time. This paper presents a representation theorem for such a proposal, and shows that all processes with continuous separable covariance functions are second-order identical to the product of second-order uncorrelated processes. It discusses the implications of separable or nearly separable prior covariances for the statistical emulation of complicated functions such as computer codes, and critically reexamines the conventional wisdom concerning emulator structure, and size of design.

Many applications require stochastic processes specified on two- or higher-dimensional domains; spatial or spatial-temporal modelling, for example. In these applications it is attractive, for conceptual simplicity and computational tractability, to propose a covariance function that is separable; e.g., the product of a covariance function in space and one in time. This paper presents a representation theorem for such a proposal, and shows that all processes with continuous separable covariance functions are second-order identical to the product of second-order uncorrelated processes. It discusses the implications of separable or nearly separable prior covariances for the statistical emulation of complicated functions such as computer codes, and critically reexamines the conventional wisdom concerning emulator structure, and size of design.