DOCSLIB.ORG

An Introduction to Stochastic Processes in Continuous Time

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
An Introduction to Stochastic Processes in Continuous Time An Introduction to Stochastic Processes in Continuous Time Harry van Zanten November 8, 2004 (this version) always under construction ii Preface iv Contents 1 Stochastic processes 1 1.1 Stochastic processes 1 1.2 Finite-dimensional distributions 3 1.3 Kolmogorov's continuity criterion 4 1.4 Gaussian processes 7 1.5 Non-di®erentiability of the Brownian sample paths 10 1.6 Filtrations and stopping times 11 1.7 Exercises 17 2 Martingales 21 2.1 De¯nitions and examples 21 2.2 Discrete-time martingales 22 2.2.1 Martingale transforms 22 2.2.2 Inequalities 24 2.2.3 Doob decomposition 26 2.2.4 Convergence theorems 27 2.2.5 Optional stopping theorems 31 2.3 Continuous-time martingales 33 2.3.1 Upcrossings in continuous time 33 2.3.2 Regularization 34 2.3.3 Convergence theorems 37 2.3.4 Inequalities 38 2.3.5 Optional stopping 38 2.4 Applications to Brownian motion 40 2.4.1 Quadratic variation 40 2.4.2 Exponential inequality 42 2.4.3 The law of the iterated logarithm 43 2.4.4 Distribution of hitting times 45 2.5 Exercises 47 3 Markov processes 49 3.1 Basic de¯nitions 49 3.2 Existence of a canonical version 52 3.3 Feller processes 55 3.3.1 Feller transition functions and resolvents 55 3.3.2 Existence of a cadlag version 59 3.3.3 Existence of a good ¯ltration 61 3.4 Strong Markov property 64 vi Contents 3.4.1 Strong Markov property of a Feller process 64 3.4.2 Applications to Brownian motion 68 3.5 Generators 70 3.5.1 Generator of a Feller process 70 3.5.2 Characteristic operator 74 3.6 Killed Feller processes 76 3.6.1 Sub-Markovian processes 76 3.6.2 Feynman-Kac formula 78 3.6.3 Feynman-Kac formula and arc-sine law for the BM 79 3.7 Exercises 82 4 Special Feller processes 85 4.1 Brownian motion in Rd 85 4.2 Feller di®usions 88 4.3 Feller processes on discrete spaces 93 4.4 L¶evy processes 97 4.4.1 De¯nition, examples and ¯rst properties 98 4.4.2 Jumps of a L¶evy process 101 4.4.3 L¶evy-It^o decomposition 106 4.4.4 L¶evy-Khintchine formula 109 4.4.5 Stable processes 111 4.5 Exercises 113 A Elements of measure theory 117 A.1 De¯nition of conditional expectation 117 A.2 Basic properties of conditional expectation 118 A.3 Uniform integrability 119 A.4 Monotone class theorem 120 B Elements of functional analysis 123 B.1 Hahn-Banach theorem 123 B.2 Riesz representation theorem 124 References 125 1 Stochastic processes 1.1 Stochastic processes Loosely speaking, a stochastic process is a phenomenon that can be thought of as evolving in time in a random manner. Common examples are the location of a particle in a physical system, the price of a stock in a ¯nancial market, interest rates, etc. A basic example is the erratic movement of pollen grains suspended in water, the so-called Brownian motion. This motion was named after the English botanist R. Brown, who ¯rst observed it in 1827. The movement of the pollen grain is thought to be due to the impacts of the water molecules that surround it. These hits occur a large number of times in each small time interval, they are independent of each other, and the impact of one single hit is very small compared to the total e®ect. This suggest that the motion of the grain can be viewed as a random process with the following properties: (i) The displacement in any time interval [s; t] is independent of what hap- pened before time s. (ii) Such a displacement has a Gaussian distribution, which only depends on the length of the time interval [s; t]. (iii) The motion is continuous. The mathematical model of the Brownian motion will be the main object of investigation in this course. Figure 1.1 shows a particular realization of this stochastic process. The picture suggest that the BM has some remarkable properties, and we will see that this is indeed the case. Mathematically, a stochastic process is simply an indexed collection of random variables. The formal de¯nition is as follows. 2 Stochastic processes 0.4 0.2 0.0 -0.2 -0.4 -0.6 -0.8 -1.0 0.0 0.2 0.4 0.6 0.8 1.0 Figure 1.1: A realization of the Brownian motion De¯nition 1.1.1. Let T be a set and (E; ) a measurable space. A stochastic E process indexed by T , taking values in (E; ), is a collection X = (Xt)t T of E 2 measurable maps Xt from a probability space (­; ; P) to (E; ). The space (E; ) is called the state space of the process. F E E We think of the index t as a time parameter, and view the index set T as the set of all possible time points. In these notes we will usually have T = Z+ = 0; 1; : : : or T = R+ = [0; ). In the former case we say that time is discrete, fin the latterg we say time is1continuous. Note that a discrete-time process can always be viewed as a continuous-time process which is constant on the intervals [n 1; n) for all n N. The state space (E; ) will most often be a Euclidean space¡ Rd, endowed2with its Borel σ-algebra E(Rd). If E is the state space of a process, we call the process E-valued. B For every ¯xed t T the stochastic process X gives us an E-valued random 2 element Xt on (­; ; P). We can also ¯x ! ­ and consider the map t Xt(!) on T . These mapsFare called the trajectories2 , or sample paths of the7!process. The sample paths are functions from T to E, i.e. elements of ET . Hence, we can view the process X as a random element of the function space ET . (Quite often, the sample paths are in fact elements of some nice subset of this space.) The mathematical model of the physical Brownian motion is a stochastic process that is de¯ned as follows. De¯nition 1.1.2. The stochastic process W = (Wt)t 0 is called a (standard) Brownian motion, or Wiener process, if ¸ (i) W0 = 0 a.s., (ii) W W is independent of (W : u s) for all s t, t ¡ s u · · (iii) W W has a N(0; t s)-distribution for all s t, t ¡ s ¡ · (iv) almost all sample paths of W are continuous. 1.2 Finite-dimensional distributions 3 We abbreviate `Brownian motion' to BM in these notes. Property (i) says that a standard BM starts in 0. A process with property (ii) is called a process with independent increments. Property (iii) implies that that the distribution of the increment Wt Ws only depends on t s. This is called the stationarity of the increments. A sto¡chastic process which ¡has property (iv) is called a continuous process. Similarly, we call a stochastic process right-continuous if almost all of its sample paths are right-continuous functions. We will often use the acronym cadlag (continu `a droite, limites `a gauche) for processes with sample paths that are right-continuous have ¯nite left-hand limits at every time point. It is not clear from the de¯nition that the BM actually exists! We will have to prove that there exists a stochastic process which has all the properties required in De¯nition 1.1.2. 1.2 Finite-dimensional distributions In this section we recall Kolmogorov's theorem on the existence of stochastic processes with prescribed ¯nite-dimensional distributions. We use it to prove the existence of a process which has properties (i), (ii) and (iii) of De¯nition 1.1.2. De¯nition 1.2.1. Let X = (Xt)t T be a stochastic process. The distributions 2 of the ¯nite-dimensional vectors of the form (Xt1 ; : : : ; Xtn ) are called the ¯nite- dimensional distributions (fdd's) of the process. It is easily veri¯ed that the fdd's of a stochastic process form a consistent system of measures, in the sense of the following de¯nition. De¯nition 1.2.2. Let T be a set and (E; ) a measurable space. For all E n n t1; : : : ; tn T , let ¹t1;:::;tn be a probability measure on (E ; ). This collection of measures2 is called consistent if it has the following properties:E (i) For all t1; : : : ; tn T , every permutation ¼ of 1; : : : ; n and all A ; : : : ; A 2 f g 1 n 2 E ¹ (A A ) = ¹ (A A ): t1;:::;tn 1 £ ¢ ¢ ¢ £ n t¼(1);:::;t¼(n) ¼(1) £ ¢ ¢ ¢ £ ¼(n) (ii) For all t ; : : : ; t T and A ; : : : ; A 1 n+1 2 1 n 2 E ¹ (A A E) = ¹ (A A ): t1;:::;tn+1 1 £ ¢ ¢ ¢ £ n £ t1;:::;tn 1 £ ¢ ¢ ¢ £ n The Kolmogorov consistency theorem states that conversely, under mild regularity conditions, every consistent family of measures is in fact the family of fdd's of some stochastic process. Some assumptions are needed on the state space (E; ). We will assume that E is a Polish space (a complete, separable metric space)E and is its Borel σ-algebra, i.e. the σ-algebra generated by the open sets. Clearly, Ethe Euclidean spaces (Rn; (Rn)) ¯t into this framework. B 4 Stochastic processes Theorem 1.2.3 (Kolmogorov's consistency theorem).
Load more

Recommended publications
  • Integral Representations of Martingales for Progressive Enlargements Of
    INTEGRAL REPRESENTATIONS OF MARTINGALES FOR PROGRESSIVE ENLARGEMENTS OF FILTRATIONS ANNA AKSAMIT, MONIQUE JEANBLANC AND MAREK RUTKOWSKI Abstract. We work in the setting of the progressive enlargement G of a reference filtration F through the observation of a random time τ. We study an integral representation property for some classes of G-martingales stopped at τ. In the first part, we focus on the case where F is a Poisson filtration and we establish a predictable representation property with respect to three G-martingales. In the second part, we relax the assumption that F is a Poisson filtration and we assume that τ is an F-pseudo-stopping time. We establish integral representations with respect to some G-martingales built from F-martingales and, under additional hypotheses, we obtain a predictable representation property with respect to two G-martingales. Keywords: predictable representation property, Poisson process, random time, progressive enlargement, pseudo-stopping time Mathematics Subjects Classification (2010): 60H99 1. Introduction We are interested in the stability of the predictable representation property in a filtration en- largement setting: under the postulate that the predictable representation property holds for F and G is an enlargement of F, the goal is to find under which conditions the predictable rep- G G resentation property holds for . We focus on the progressive enlargement = (Gt)t∈R+ of a F reference filtration = (Ft)t∈R+ through the observation of the occurrence of a random time τ and we assume that the hypothesis (H′) is satisfied, i.e., any F-martingale is a G-semimartingale. It is worth noting that no general result on the existence of a G-semimartingale decomposition after τ of an F-martingale is available in the existing literature, while, for any τ, any F-martingale stopped at time τ is a G-semimartingale with an explicit decomposition depending on the Az´ema supermartingale of τ.
    [Show full text]
  • Superprocesses and Mckean-Vlasov Equations with Creation of Mass
    Sup erpro cesses and McKean-Vlasov equations with creation of mass L. Overb eck Department of Statistics, University of California, Berkeley, 367, Evans Hall Berkeley, CA 94720, y U.S.A. Abstract Weak solutions of McKean-Vlasov equations with creation of mass are given in terms of sup erpro cesses. The solutions can b e approxi- mated by a sequence of non-interacting sup erpro cesses or by the mean- eld of multityp e sup erpro cesses with mean- eld interaction. The lat- ter approximation is asso ciated with a propagation of chaos statement for weakly interacting multityp e sup erpro cesses. Running title: Sup erpro cesses and McKean-Vlasov equations . 1 Intro duction Sup erpro cesses are useful in solving nonlinear partial di erential equation of 1+ the typ e f = f , 2 0; 1], cf. [Dy]. Wenowchange the p oint of view and showhowtheyprovide sto chastic solutions of nonlinear partial di erential Supp orted byanFellowship of the Deutsche Forschungsgemeinschaft. y On leave from the Universitat Bonn, Institut fur Angewandte Mathematik, Wegelerstr. 6, 53115 Bonn, Germany. 1 equation of McKean-Vlasovtyp e, i.e. wewant to nd weak solutions of d d 2 X X @ @ @ + d x; + bx; : 1.1 = a x; t i t t t t t ij t @t @x @x @x i j i i=1 i;j =1 d Aweak solution = 2 C [0;T];MIR satis es s Z 2 t X X @ @ a f = f + f + d f + b f ds: s ij s t 0 i s s @x @x @x 0 i j i Equation 1.1 generalizes McKean-Vlasov equations of twodi erenttyp es.
    [Show full text]
  • Conditional Generalizations of Strong Laws Which Conclude the Partial Sums Converge Almost Surely1
    The Annals ofProbability 1982. Vol. 10, No.3. 828-830 CONDITIONAL GENERALIZATIONS OF STRONG LAWS WHICH CONCLUDE THE PARTIAL SUMS CONVERGE ALMOST SURELY1 By T. P. HILL Georgia Institute of Technology Suppose that for every independent sequence of random variables satis­ fying some hypothesis condition H, it follows that the partial sums converge almost surely. Then it is shown that for every arbitrarily-dependent sequence of random variables, the partial sums converge almost surely on the event where the conditional distributions (given the past) satisfy precisely the same condition H. Thus many strong laws for independent sequences may be immediately generalized into conditional results for arbitrarily-dependent sequences. 1. Introduction. Ifevery sequence ofindependent random variables having property A has property B almost surely, does every arbitrarily-dependent sequence of random variables have property B almost surely on the set where the conditional distributions have property A? Not in general, but comparisons of the conditional Borel-Cantelli Lemmas, the condi­ tional three-series theorem, and many martingale results with their independent counter­ parts suggest that the answer is affirmative in fairly general situations. The purpose ofthis note is to prove Theorem 1, which states, in part, that if "property B" is "the partial sums converge," then the answer is always affIrmative, regardless of "property A." Thus many strong laws for independent sequences (even laws yet undiscovered) may be immediately generalized into conditional results for arbitrarily-dependent sequences. 2. Main Theorem. In this note, IY = (Y1 , Y2 , ••• ) is a sequence ofrandom variables on a probability triple (n, d, !?I'), Sn = Y 1 + Y 2 + ..
    [Show full text]
  • On Hitting Times for Jump-Diffusion Processes with Past Dependent Local Characteristics
    Stochastic Processes and their Applications 47 ( 1993) 131-142 131 North-Holland On hitting times for jump-diffusion processes with past dependent local characteristics Manfred Schal Institut fur Angewandte Mathematik, Universitiit Bonn, Germany Received 30 March 1992 Revised 14 September 1992 It is well known how to apply a martingale argument to obtain the Laplace transform of the hitting time of zero (say) for certain processes starting at a positive level and being skip-free downwards. These processes have stationary and independent increments. In the present paper the method is extended to a more general class of processes the increments of which may depend both on time and past history. As a result a generalized Laplace transform is obtained which can be used to derive sharp bounds for the mean and the variance of the hitting time. The bounds also solve the control problem of how to minimize or maximize the expected time to reach zero. AMS 1980 Subject Classification: Primary 60G40; Secondary 60K30. hitting times * Laplace transform * expectation and variance * martingales * diffusions * compound Poisson process* random walks* M/G/1 queue* perturbation* control 1. Introduction Consider a process X,= S,- ct + uW, where Sis a compound Poisson process with Poisson parameter A which is disturbed by an independent standard Wiener process W This is the point of view of Dufresne and Gerber (1991). One can also look on X as a diffusion disturbed by jumps. This is the point of view of Ethier and Kurtz (1986, Section 4.10). Here u and A may be zero so that the jump term or the diffusion term may vanish.
    [Show full text]
  • PREDICTABLE REPRESENTATION PROPERTY for PROGRESSIVE ENLARGEMENTS of a POISSON FILTRATION Anna Aksamit, Monique Jeanblanc, Marek Rutkowski
    PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION Anna Aksamit, Monique Jeanblanc, Marek Rutkowski To cite this version: Anna Aksamit, Monique Jeanblanc, Marek Rutkowski. PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION. 2015. hal- 01249662 HAL Id: hal-01249662 https://hal.archives-ouvertes.fr/hal-01249662 Preprint submitted on 2 Jan 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PREDICTABLE REPRESENTATION PROPERTY FOR PROGRESSIVE ENLARGEMENTS OF A POISSON FILTRATION Anna Aksamit Mathematical Institute, University of Oxford, Oxford OX2 6GG, United Kingdom Monique Jeanblanc∗ Laboratoire de Math´ematiques et Mod´elisation d’Evry´ (LaMME), Universit´ed’Evry-Val-d’Essonne,´ UMR CNRS 8071 91025 Evry´ Cedex, France Marek Rutkowski School of Mathematics and Statistics University of Sydney Sydney, NSW 2006, Australia 10 December 2015 Abstract We study problems related to the predictable representation property for a progressive en- largement G of a reference filtration F through observation of a finite random time τ. We focus on cases where the avoidance property and/or the continuity property for F-martingales do not hold and the reference filtration is generated by a Poisson process.
    [Show full text]
  • THE MEASURABILITY of HITTING TIMES 1 Introduction
    Elect. Comm. in Probab. 15 (2010), 99–105 ELECTRONIC COMMUNICATIONS in PROBABILITY THE MEASURABILITY OF HITTING TIMES RICHARD F. BASS1 Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009 email: [email protected] Submitted January 18, 2010, accepted in final form March 8, 2010 AMS 2000 Subject classification: Primary: 60G07; Secondary: 60G05, 60G40 Keywords: Stopping time, hitting time, progressively measurable, optional, predictable, debut theorem, section theorem Abstract Under very general conditions the hitting time of a set by a stochastic process is a stopping time. We give a new simple proof of this fact. The section theorems for optional and predictable sets are easy corollaries of the proof. 1 Introduction A fundamental theorem in the foundations of stochastic processes is the one that says that, under very general conditions, the first time a stochastic process enters a set is a stopping time. The proof uses capacities, analytic sets, and Choquet’s capacibility theorem, and is considered hard. To the best of our knowledge, no more than a handful of books have an exposition that starts with the definition of capacity and proceeds to the hitting time theorem. (One that does is [1].) The purpose of this paper is to give a short and elementary proof of this theorem. The proof is simple enough that it could easily be included in a first year graduate course in probability. In Section 2 we give a proof of the debut theorem, from which the measurability theorem follows. As easy corollaries we obtain the section theorems for optional and predictable sets. This argument is given in Section 3.
    [Show full text]
  • Lecture 19 Semimartingales
    Lecture 19:Semimartingales 1 of 10 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 19 Semimartingales Continuous local martingales While tailor-made for the L2-theory of stochastic integration, martin- 2,c gales in M0 do not constitute a large enough class to be ultimately useful in stochastic analysis. It turns out that even the class of all mar- tingales is too small. When we restrict ourselves to processes with continuous paths, a naturally stable family turns out to be the class of so-called local martingales. Definition 19.1 (Continuous local martingales). A continuous adapted stochastic process fMtgt2[0,¥) is called a continuous local martingale if there exists a sequence ftngn2N of stopping times such that 1. t1 ≤ t2 ≤ . and tn ! ¥, a.s., and tn 2. fMt gt2[0,¥) is a uniformly integrable martingale for each n 2 N. In that case, the sequence ftngn2N is called the localizing sequence for (or is said to reduce) fMtgt2[0,¥). The set of all continuous local loc,c martingales M with M0 = 0 is denoted by M0 . Remark 19.2. 1. There is a nice theory of local martingales which are not neces- sarily continuous (RCLL), but, in these notes, we will focus solely on the continuous case. In particular, a “martingale” or a “local martingale” will always be assumed to be continuous. 2. While we will only consider local martingales with M0 = 0 in these notes, this is assumption is not standard, so we don’t put it into the definition of a local martingale. tn 3.
    [Show full text]
  • Deep Optimal Stopping
    Journal of Machine Learning Research 20 (2019) 1-25 Submitted 4/18; Revised 1/19; Published 4/19 Deep Optimal Stopping Sebastian Becker [email protected] Zenai AG, 8045 Zurich, Switzerland Patrick Cheridito [email protected] RiskLab, Department of Mathematics ETH Zurich, 8092 Zurich, Switzerland Arnulf Jentzen [email protected] SAM, Department of Mathematics ETH Zurich, 8092 Zurich, Switzerland Editor: Jon McAuliffe Abstract In this paper we develop a deep learning method for optimal stopping problems which directly learns the optimal stopping rule from Monte Carlo samples. As such, it is broadly applicable in situations where the underlying randomness can efficiently be simulated. We test the approach on three problems: the pricing of a Bermudan max-call option, the pricing of a callable multi barrier reverse convertible and the problem of optimally stopping a fractional Brownian motion. In all three cases it produces very accurate results in high- dimensional situations with short computing times. Keywords: optimal stopping, deep learning, Bermudan option, callable multi barrier reverse convertible, fractional Brownian motion 1. Introduction N We consider optimal stopping problems of the form supτ E g(τ; Xτ ), where X = (Xn)n=0 d is an R -valued discrete-time Markov process and the supremum is over all stopping times τ based on observations of X. Formally, this just covers situations where the stopping decision can only be made at finitely many times. But practically all relevant continuous- time stopping problems can be approximated with time-discretized versions. The Markov assumption means no loss of generality. We make it because it simplifies the presentation and many important problems already are in Markovian form.
    [Show full text]
  • Optimal Stopping Time and Its Applications to Economic Models
    OPTIMAL STOPPING TIME AND ITS APPLICATIONS TO ECONOMIC MODELS SIVAKORN SANGUANMOO Abstract. This paper gives an introduction to an optimal stopping problem by using the idea of an It^odiffusion. We then prove the existence and unique- ness theorems for optimal stopping, which will help us to explicitly solve opti- mal stopping problems. We then apply our optimal stopping theorems to the economic model of stock prices introduced by Samuelson [5] and the economic model of capital goods. Both economic models show that those related agents are risk-loving. Contents 1. Introduction 1 2. Preliminaries: It^odiffusion and boundary value problems 2 2.1. It^odiffusions and their properties 2 2.2. Harmonic function and the stochastic Dirichlet problem 4 3. Existence and uniqueness theorems for optimal stopping 5 4. Applications to an economic model: Stock prices 11 5. Applications to an economic model: Capital goods 14 Acknowledgments 18 References 19 1. Introduction Since there are several outside factors in the real world, many variables in classi- cal economic models are considered as stochastic processes. For example, the prices of stocks and oil do not just increase due to inflation over time but also fluctuate due to unpredictable situations. Optimal stopping problems are questions that ask when to stop stochastic pro- cesses in order to maximize utility. For example, when should one sell an asset in order to maximize his profit? When should one stop searching for the keyword to obtain the best result? Therefore, many recent economic models often include op- timal stopping problems. For example, by using optimal stopping, Choi and Smith [2] explored the effectiveness of the search engine, and Albrecht, Anderson, and Vroman [1] discovered how the search cost affects the search for job candidates.
    [Show full text]
  • The Ito Integral
    Notes on the Itoˆ Calculus Steven P. Lalley May 15, 2012 1 Continuous-Time Processes: Progressive Measurability 1.1 Progressive Measurability Measurability issues are a bit like plumbing – a bit ugly, and most of the time you would prefer that it remains hidden from view, but sometimes it is unavoidable that you actually have to open it up and work on it. In the theory of continuous–time stochastic processes, measurability problems are usually more subtle than in discrete time, primarily because sets of measures 0 can add up to something significant when you put together uncountably many of them. In stochastic integration there is another issue: we must be sure that any stochastic processes Xt(!) that we try to integrate are jointly measurable in (t; !), so as to ensure that the integrals are themselves random variables (that is, measurable in !). Assume that (Ω; F;P ) is a fixed probability space, and that all random variables are F−measurable. A filtration is an increasing family F := fFtgt2J of sub-σ−algebras of F indexed by t 2 J, where J is an interval, usually J = [0; 1). Recall that a stochastic process fYtgt≥0 is said to be adapted to a filtration if for each t ≥ 0 the random variable Yt is Ft−measurable, and that a filtration is admissible for a Wiener process fWtgt≥0 if for each t > 0 the post-t increment process fWt+s − Wtgs≥0 is independent of Ft. Definition 1. A stochastic process fXtgt≥0 is said to be progressively measurable if for every T ≥ 0 it is, when viewed as a function X(t; !) on the product space ([0;T ])×Ω, measurable relative to the product σ−algebra B[0;T ] × FT .
    [Show full text]
  • Interacting Particle Systems MA4H3
    Interacting particle systems MA4H3 Stefan Grosskinsky Warwick, 2009 These notes and other information about the course are available on http://www.warwick.ac.uk/˜masgav/teaching/ma4h3.html Contents Introduction 2 1 Basic theory3 1.1 Continuous time Markov chains and graphical representations..........3 1.2 Two basic IPS....................................6 1.3 Semigroups and generators.............................9 1.4 Stationary measures and reversibility........................ 13 2 The asymmetric simple exclusion process 18 2.1 Stationary measures and conserved quantities................... 18 2.2 Currents and conservation laws........................... 23 2.3 Hydrodynamics and the dynamic phase transition................. 26 2.4 Open boundaries and matrix product ansatz.................... 30 3 Zero-range processes 34 3.1 From ASEP to ZRPs................................ 34 3.2 Stationary measures................................. 36 3.3 Equivalence of ensembles and relative entropy................... 39 3.4 Phase separation and condensation......................... 43 4 The contact process 46 4.1 Mean field rate equations.............................. 46 4.2 Stochastic monotonicity and coupling....................... 48 4.3 Invariant measures and critical values....................... 51 d 4.4 Results for Λ = Z ................................. 54 1 Introduction Interacting particle systems (IPS) are models for complex phenomena involving a large number of interrelated components. Examples exist within all areas of natural and social sciences, such as traffic flow on highways, pedestrians or constituents of a cell, opinion dynamics, spread of epi- demics or fires, reaction diffusion systems, crystal surface growth, chemotaxis, financial markets... Mathematically the components are modeled as particles confined to a lattice or some discrete geometry. Their motion and interaction is governed by local rules. Often microscopic influences are not accesible in full detail and are modeled as effective noise with some postulated distribution.
    [Show full text]
  • Tight Inequalities Among Set Hitting Times in Markov Chains
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, Pages 000–000 S 0002-9939(XX)0000-0 TIGHT INEQUALITIES AMONG SET HITTING TIMES IN MARKOV CHAINS SIMON GRIFFITHS, ROSS J. KANG, ROBERTO IMBUZEIRO OLIVEIRA, AND VIRESH PATEL Abstract. Given an irreducible discrete-time Markov chain on a finite state space, we consider the largest expected hitting time T (α) of a set of stationary measure at least α for α ∈ (0, 1). We obtain tight inequalities among the values of T (α) for different choices of α. One consequence is that T (α) ≤ T (1/2)/α for all α < 1/2. As a corollary we have that, if the chain is lazy in a certain sense as well as reversible, then T (1/2) is equivalent to the chain’s mixing time, answering a question of Peres. We furthermore demonstrate that the inequalities we establish give an almost everywhere pointwise limiting characterisation of possible hitting time functions T (α) over the domain α ∈ (0, 1/2]. 1. Introduction Hitting times are a classical topic in the theory of finite Markov chains, with connections to mixing times, cover times and electrical network representations [5, 6]. In this paper, we consider a natural family of extremal problems for maximum expected hitting times. In contrast to most earlier work on hitting times that considered the maximum expected hitting times of individual states, we focus on hitting sets of states of at least a given stationary measure. Informally, we are interested in the following basic question: how much more difficult is it to hit a smaller set than a larger one? (We note that other, quite different extremal problems about hitting times have been considered, e.g.
    [Show full text]