DOCSLIB.ORG

Transient Nearest Neighbor Random Walk and Bessel Process

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Transient Nearest Neighbor Random Walk and Bessel Process TRANSIENT NEAREST NEIGHBOR RANDOM WALK AND BESSEL PROCESS Endre Csáki1 Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, H-1364, Hungary. E-mail address: [email protected] Antónia Földes2 Department of Mathematics, College of Staten Island, CUNY, 2800 Victory Blvd., Staten Island, New York 10314, U.S.A. E-mail address: [email protected] Pál Révész1 Institut für Statistik und Wahrscheinlichkeitstheorie, Technische Universität Wien, Wiedner Hauptstrasse 8-10/107 A-1040 Vienna, Austria. E-mail address: [email protected] Abstract: We prove strong invariance principle between a transient Bessel process and a certain nearest neighbor (NN) random walk that is constructed from the former by using stopping times. We show that their local times are close enough to share the same strong limit theorems. It is also shown, that if the dierence between the distributions of two NN random walks are small, then the walks themselves can be constructed on such a way that they are close enough. Finally, some consequences concerning strong limit theorems are discussed. AMS 2000 Subject Classication: Primary 60F17; Secondary 60F15, 60J10, 60J55, 60J60. Keywords: transient random walk, Bessel process, strong invariance principle, local time, strong theorems. Running head: NN random walk and Bessel process. 1Research supported by the Hungarian National Foundation for Scientific Research, Grant No. K 61052 and K 67961. 2Research supported by a PSC CUNY Grant, No. 69020-0038. 1 1. Introduction In this paper we consider a nearest neighbor (NN) random walk, dened as follows: let X0 = 0,X1,X2,... be a Markov chain with Ei := P(Xn+1 = i + 1 | Xn = i) = 1 − P(Xn+1 = i − 1 | Xn = i) (1.1) ( 1 if i = 0 = 1/2 + pi if i = 1, 2,..., where −1/2 ≤ pi ≤ 1/2, i = 1, 2,.... In case 0 < pi ≤ 1/2 the sequence {Xi} describes the motion of a particle which starts at zero, moves over the nonnegative integers and going away from 0 with a larger probability than to the direction of 0. We will be interested in the case when pi ∼ B/4i with B > 0 as i → ∞. We want to show that in certain sense, this Markov chain is a discrete analogue of a continuous Bessel process and establish a strong invariance principle between these two processes. The properties of the discrete model, often called birth and death chain, its connections with orthogonal polynomials in particular, has been treated extensively in the literature. See e.g. the classical paper by Karlin and McGregor [13], or more recent papers by Coolen- Schrijner and Van Doorn [6] and Dette [9]. In an earlier paper [7] we investigated the local time of this Markov chain in the transient case. There is a well-known result in the literature (cf. e.g. Chung [5]) characterizing those sequences {pi} for which {Xi} is transient (resp. recurrent). Theorem A: ([5], page 74) Let Xn be a Markov chain with transition probabilities given in (1.1) with −1/2 < pi < 1/2, i = 1, 2,... Dene 1 − Ei 1/2 − pi Ui := = (1.2) Ei 1/2 + pi Then Xn is transient if and only if ∞ k X Y Ui < ∞. k=1 i=1 As a consequence, the Markov chain (Xn) with pR ∼ B/4R, R → ∞ is transient if B > 1 and recurrent if B < 1. The Bessel process of order ν, denoted by Yν(t), t ≥ 0 is a diusion process on the line with generator 1 d2 2ν + 1 d + . 2 dx2 2x dx 2 d = 2ν + 2 is the dimension of the Bessel process. If d is a positive integer, then Yν(·) is the absolute value of a d-dimensional Brownian motion. The Bessel process Yν(t) is transient if and only if ν > 0. The properties of the Bessel process were extensively studied in the literature. Cf. Borodin and Salminen [2], Revuz and Yor [20], Knight [15]. Lamperti [16] determined the limiting distribution of Xn and also proved a weak conver- gence theorem in a more general setting. His result in our case reads as follows. Theorem B: ([16]) Let Xn be a Markov chain with transition probabilities given in (1.1) with −1/2 < pi < 1/2, i = 1, 2,... If limR→∞ RpR = B/4 > −1/4, then the following weak convergence holds: X √[nt] =⇒ Y (t) n (B−1)/2 in the space D[0,1]. In particular, ! x X 1 Z 2 lim P √n < x = uBe−u /2 du. n→∞ n 2B/2−1/2Γ(B/2 + 1/2) 0 In Theorems A and B the values of pi can be negative. In the sequel however we deal only with the case when pi are non-negative, and the chain is transient, which will be assumed throughout without mentioning it. Let ∞ j X Y D(R, ∞) := 1 + UR+i, (1.3) j=1 i=1 and dene 1 + p p∗ := 2 R = 1 − q∗ (1.4) R D(R, ∞) R Now let ξ(R, ∞), R = 0, 1, 2,... be the total local time at R of the Markov chain {Xn}, i.e. ξ(R, ∞) := #{n ≥ 0 : Xn = R}. (1.5) Theorem C: ([7]) For a transient NN random walk ∗ ∗ k−1 (1.6) P(ξ(R, ∞) = k) = pR(qR) , k = 1, 2,... Moreover, η(R, t), R > 0 will denote the local time of the Bessel process, i.e. 1 Z t η(R, t) := lim I{Yν(s) ∈ (R − ε, R + ε)} ds, η(R, ∞) := lim η(R, t). ε→0 2ε 0 t→∞ 3 It is well-known that η(R, ∞) has exponential distribution (see e.g. [2]). ν P(η(R, ∞) < x) = 1 − exp − x . (1.7) R For 0 < a < b let τ := τ(a, b) = min{t ≥ 0 : Yν(t) ∈/ (a, b)}. (1.8) Then we have (cf. Borodin and Salminen [2], Section 6, 3.0.1 and 3.0.4). Theorem D: For 0 < a < x < b we have x−2ν − b−2ν P (Y (τ) = a) = 1 − P (Y (τ) = b) = , (1.9) x ν x ν a−2ν − b−2ν √ √ √ √ −ατ Sν(b 2α, x 2α) + Sν(x 2α, a 2α) Exe = √ √ , (1.10) Sν(b 2α, a 2α) where −ν Sν(u, v) = (uv) (Iν(u)Kν(v) − Kν(u)Iν(v)), (1.11) Iν and Kν being the modied Bessel functions of the rst and second kind, resp. Here and in what follows Px and Ex denote conditional probability, resp. expectation under Yν(0) = x. For simplicity we will use P0 = P, and E0 = E. Now consider Yν(t), t ≥ 0, a Bessel process of order ν, Yν(0) = 0, and let Xn, n = 0, 1, 2,... be an NN random walk with p0 = p1 = 1/2, (R − 1)−2ν − R−2ν 1 p = − ,R = 2, 3,... (1.12) R (R − 1)−2ν − (R + 1)−2ν 2 Our main results are strong invariance principles concerning Bessel process, NN random walk and their local times. Theorem 1.1. On a suitable probability space we can construct a Bessel process {Yν(t), t ≥ 0}, ν > 0 and an NN random walk {Xn, n = 0, 1, 2,...} with pR as in (1.12) such that for any ε > 0, as n → ∞ we have 1/4+ε Yν(n) − Xn = O(n ) a.s. (1.13) Our strong invariance principle for local times reads as follows. 4 Theorem 1.2. Let Yν(t) and Xn as in Theorem 1.1 and let η and ξ their respective local times. As R → ∞, we have ξ(R, ∞) − η(R, ∞) = O(R1/2 log R) a.s. (1.14) We prove the following strong invariance principle between two NN random walks. Theorem 1.3. Let (1) ∞ and (2) ∞ be two NN random walk with (1) and (2), {Xn }n=0 {Xn }n=0 pj pj resp. Assume that B C (1) (1.15) pj − ≤ 4j jγ and B C (2) (1.16) pj − ≤ 4j jγ j = 1, 2,... with B > 1, 1 < γ ≤ 2 and some non-negative constant C. Then on a suitable probability space one can construct (1) and (2) such that as {Xn } {Xn } n → ∞ (1) (2) (1) (2) 2−γ 1−γ/2 |Xn − Xn | = O((Xn + Xn ) ) = O((n log log n) ) a.s. The organization of the paper is as follows. In Section 2 we will present some well-known facts and prove some preliminary results. Sections 3-5 contain the proofs of Theorems 1.1-1.3, respectively. In Section 6 we prove strong theorems (most of them are integral tests) which easily follow from Theorems 1.1 and 1.2 and the corresponding results for Bessel process. In Section 7, using our Theorem 1.3 in both directions, we prove an integral test for the local time of the NN-walk, and a strong theorem for the speed of escape of the Bessel process. 2. Preliminaries Lemma 2.1. Let Yν(·) be a Bessel process starting from x = R and let τ be the stopping time dened by (1.8) with a = R − 1 and b = R + 1. Let pR be dened by (1.12). Then as R → ∞ 2ν + 1 1 p = + O , (2.1) R 4R R2 1 E (τ) = 1 + O , (2.2) R R V arR(τ) = O(1). (2.3) 5 Proof: For ν = 1/2, i.e. for d = 3-dimensional Bessel process, in case x = R, a = R − 1, b = R + 1 we have λτ 1 ER(e ) = √ cos( 2λ) which does not depend on R. We prove that this holds asymptotically in general, when ν > 0.
Load more

Recommended publications
  • Perturbed Bessel Processes Séminaire De Probabilités (Strasbourg), Tome 32 (1998), P
    SÉMINAIRE DE PROBABILITÉS (STRASBOURG) R.A. DONEY JONATHAN WARREN MARC YOR Perturbed Bessel processes Séminaire de probabilités (Strasbourg), tome 32 (1998), p. 237-249 <http://www.numdam.org/item?id=SPS_1998__32__237_0> © Springer-Verlag, Berlin Heidelberg New York, 1998, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec les conditions générales d’utili- sation (http://www.numdam.org/conditions). Toute utilisation commerciale ou im- pression systématique est constitutive d’une infraction pénale. Toute copie ou im- pression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Perturbed Bessel Processes R.A.DONEY, J.WARREN, and M.YOR. There has been some interest in the literature in Brownian Inotion perturbed at its maximum; that is a process (Xt ; t > 0) satisfying (0.1) , where Mf = XS and (Bt; t > 0) is Brownian motion issuing from zero. The parameter a must satisfy a 1. For example arc-sine laws and Ray-Knight theorems have been obtained for this process; see Carmona, Petit and Yor [3], Werner [16], and Doney [7]. Our initial aim was to identify a process which could be considered as the process X conditioned to stay positive. This new process behaves like the Bessel process of dimension three except when at its maximum and we call it a perturbed three-dimensional Bessel process. We establish Ray-Knight theorems for the local times of this process, up to a first passage time and up to infinity (see Theorem 2.3), and observe that these descriptions coincide with those of the local times of two processes that have been considered in Yor [18].
    [Show full text]
  • Patterns in Random Walks and Brownian Motion
    Patterns in Random Walks and Brownian Motion Jim Pitman and Wenpin Tang Abstract We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented. AMS 2010 Mathematics Subject Classification: 60C05, 60G17, 60J65. 1 Introduction and Main Results We are interested in the question of embedding some continuous-time stochastic processes .Zu;0Ä u Ä 1/ into a Brownian path .BtI t 0/, without time-change or scaling, just by a random translation of origin in spacetime. More precisely, we ask the following: Question 1 Given some distribution of a process Z with continuous paths, does there exist a random time T such that .BTCu BT I 0 Ä u Ä 1/ has the same distribution as .Zu;0Ä u Ä 1/? The question of whether external randomization is allowed to construct such a random time T, is of no importance here. In fact, we can simply ignore Brownian J. Pitman ()•W.Tang Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA 94720-3860, USA e-mail: [email protected]; [email protected] © Springer International Publishing Switzerland 2015 49 C. Donati-Martin et al.
    [Show full text]
  • Bessel Process, Schramm-Loewner Evolution, and Dyson Model
    Bessel process, Schramm-Loewner evolution, and Dyson model – Complex Analysis applied to Stochastic Processes and Statistical Mechanics – ∗ Makoto Katori † 24 March 2011 Abstract Bessel process is defined as the radial part of the Brownian motion (BM) in the D-dimensional space, and is considered as a one-parameter family of one- dimensional diffusion processes indexed by D, BES(D). First we give a brief review of BES(D), in which D is extended to be a continuous positive parameter. It is well-known that D = 2 is the critical dimension such that, when D D (resp. c ≥ c D < Dc), the process is transient (resp. recurrent). Bessel flow is a notion such that we regard BES(D) with a fixed D as a one-parameter family of initial value x> 0. There is another critical dimension Dc = 3/2 and, in the intermediate values of D, Dc <D<Dc, behavior of Bessel flow is highly nontrivial. The dimension D = 3 is special, since in addition to the aspect that BES(3) is a radial part of the three-dimensional BM, it has another aspect as a conditional BM to stay positive. Two topics in probability theory and statistical mechanics, the Schramm-Loewner evolution (SLE) and the Dyson model (i.e., Dyson’s BM model with parameter β = 2), are discussed. The SLE(D) is introduced as a ‘complexification’ of Bessel arXiv:1103.4728v1 [math.PR] 24 Mar 2011 flow on the upper-half complex-plane, which is indexed by D > 1. It is explained (D) (D) that the existence of two critical dimensions Dc and Dc for BES makes SLE have three phases; when D D the SLE(D) path is simple, when D <D<D ≥ c c c it is self-intersecting but not dense, and when 1 < D D it is space-filling.
    [Show full text]
  • A Two-Dimensional Oblique Extension of Bessel Processes Dominique Lepingle
    A two-dimensional oblique extension of Bessel processes Dominique Lepingle To cite this version: Dominique Lepingle. A two-dimensional oblique extension of Bessel processes. 2016. hal-01164920v2 HAL Id: hal-01164920 https://hal.archives-ouvertes.fr/hal-01164920v2 Preprint submitted on 22 Nov 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. A TWO-DIMENSIONAL OBLIQUE EXTENSION OF BESSEL PROCESSES Dominique L´epingle1 Abstract. We consider a Brownian motion forced to stay in the quadrant by an electro- static oblique repulsion from the sides. We tackle the question of hitting the corner or an edge, and find product-form stationary measures under a certain condition, which is reminiscent of the skew-symmetry condition for a reflected Brownian motion. 1. Introduction In the present paper we study existence and properties of a new process with values in the nonnegative quadrant S = R+ ×R+ where R+ := [0; 1). It may be seen as a two-dimensional extension of a usual Bessel process. It is a two-dimensional Brownian motion forced to stay in the quadrant by electrostatic repulsive forces, in the same way as in the one-dimensional case where a Brownian motion which is prevented from becoming negative by an electrostatic drift becomes a Bessel process.
    [Show full text]
  • Bessel Processes, Stochastic Volatility, and Timer Options
    Mathematical Finance, Vol. 26, No. 1 (January 2016), 122–148 BESSEL PROCESSES, STOCHASTIC VOLATILITY, AND TIMER OPTIONS CHENXU LI Guanghua School of Management, Peking University Motivated by analytical valuation of timer options (an important innovation in realized variance-based derivatives), we explore their novel mathematical connection with stochastic volatility and Bessel processes (with constant drift). Under the Heston (1993) stochastic volatility model, we formulate the problem through a first-passage time problem on realized variance, and generalize the standard risk-neutral valuation theory for fixed maturity options to a case involving random maturity. By time change and the general theory of Markov diffusions, we characterize the joint distribution of the first-passage time of the realized variance and the corresponding variance using Bessel processes with drift. Thus, explicit formulas for a useful joint density related to Bessel processes are derived via Laplace transform inversion. Based on these theoretical findings, we obtain a Black–Scholes–Merton-type formula for pricing timer options, and thus extend the analytical tractability of the Heston model. Several issues regarding the numerical implementation are briefly discussed. KEY WORDS: timer options, volatility derivatives, realized variance, stochastic volatility models, Bessel processes. 1. INTRODUCTION Over the past decades, volatility has become one of the central issues in financial mod- eling. Both the historic volatility derived from time series of past market prices and the implied volatility derived from the market price of a traded derivative (in particular, an option) play important roles in derivatives valuation. In addition to index or stock options, a variety of volatility (or variance) derivatives, such as variance swaps, volatility swaps, and options on VIX (the Chicago board of exchange volatility index), are now actively traded in the financial security markets.
    [Show full text]
  • Asymptotics of Query Strategies Over a Sensor Network
    Asymptotics of Query Strategies over a Sensor Network Sanjay Shakkottai∗ February 11, 2004 Abstract We consider the problem of a user querying for information over a sensor network, where the user does not have prior knowledge of the location of the information. We consider three information query strategies: (i) a Source-only search, where the source (user) tries to locate the destination by initiating query which propagates as a continuous time random walk (Brownian motion); (ii) a Source and Receiver Driven “Sticky” Search, where both the source and the destination send a query or an advertisement, and these leave a “sticky” trail to aid in locating the destination; and (iii) where the destination information is spatially cached (i.e., repeated over space), and the source tries to locate any one of the caches. After a random interval of time with average t, if the information is not located, the query times-out, and the search is unsuccessful. For a source-only search, we show that the probability that a query is unsuccessful decays as (log(t))−1. When both the source and the destination send queries or advertisements, we show that the probability that a query is unsuccessful decays as t−5/8. Further, faster polynomial decay rates can be achieved by using a finite number of queries or advertisements. Finally, when a spatially periodic cache is employed, we show that the probability that a query is unsuccessful decays no faster than t−1. Thus, we can match the decay rates of the source and the destination driven search with that of a spatial caching strategy by using an appropriate number of queries.
    [Show full text]
  • Bessel Spdes with General Dirichlet Boundary Conditions Henri Elad Altman
    Bessel SPDEs with general Dirichlet boundary conditions Henri Elad Altman To cite this version: Henri Elad Altman. Bessel SPDEs with general Dirichlet boundary conditions. 2019. hal-02264627 HAL Id: hal-02264627 https://hal.archives-ouvertes.fr/hal-02264627 Preprint submitted on 7 Aug 2019 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. BESSEL SPDES WITH GENERAL DIRICHLET BOUNDARY CONDITIONS HENRI ELAD ALTMAN Abstract. We generalise the integration by parts formulae obtained in [7] to Bessel bridges on [0, 1] with arbitrary boundary values, as well as Bessel processes with arbitrary initial conditions. This allows us to write, formally, the corresponding dynamics using renormalised local times, thus extending the Bessel SPDEs of [7] to general Dirichlet boundary conditions. We also prove a dynamical result for the case of dimension 2, by providing a weak construction of the gradient dynamics corresponding to a 2-dimensional Bessel bridge. 1. Introduction 1.1. Bessel SPDEs. The purpose of this paper is to further extend the results obtained recently in [7], and which introduced Bessel SPDEs of dimension smaller than 3. Bessel processes are a one-parameter family of nonnegative real-valued diffusions which play a central role in various fields, ranging from statistical mechanics to finance.
    [Show full text]
  • Supremum Distribution of Bessel Process of Drifting Brownian Motion
    PROBABILITY AND MATHEMATICAL STATISTICS Vol. 35, Fasc. 2 (2015), pp. 201–222 SUPREMUM DISTRIBUTION OF BESSEL PROCESS OF DRIFTING BROWNIAN MOTION BY ∗ ANDRZEJ P Y C´ (WROCŁAW), GRZEGORZ S E R A F I N (WROCŁAW), AND TOMASZ ZAK˙ (WROCŁAW) (1) (2) (3) Abstract. Let us assume that (Bt ;Bt ;Bt + µt) is a three- dimensional Brownian motion with drift µ, starting at the origin. Then ∥ (1) (2) (3) ∥ Xt = (Bt ;Bt ;Bt + µt) , its distance from the starting point, is a diffusion with many applications. We investigate the supremum of (Xt), give an infinite-series formula for its distribution function and an exact esti- mate of the density of this distribution in terms of elementary functions. 2010 AMS Mathematics Subject Classification: Primary: 60J60; Secondary: 60G70. Key words and phrases: Drifting Brownian motion, Bessel process, supremum distribution, estimates of theta function. 1. INTRODUCTION In his famous paper [10] (see also [9], p. 436) David Williams showed how one can decompose the paths of a transient one-dimensional diffusion at its maximum (or minimum). One of the best-known examples of such decomposition is that of B(t) + µt, a Brownian motion with a positive drift µ, as a Brownian motion with a negative drift B(t) − µt and a diffusion with generator ∆µ, 1 d2 d (1.1) ∆ = + µ coth(µx) : µ 2 dx2 dx Namely, for µ > 0 let B(t) − µt be a Brownian motion with constant drift −µ, Zt a diffusion with generator (1.1), started at zero, and let γ be a random variable with 1 exponential distribution with parameter 2µ .
    [Show full text]
  • Models Beyond the Dirichlet Process
    Models beyond the Dirichlet process Antonio Lijoi Igor Prünster No. 129 December 2009 www.carloalberto.org/working_papers © 2009 by Antonio Lijoi and Igor Prünster. Any opinions expressed here are those of the authors and not those of the Collegio Carlo Alberto. Models beyond the Dirichlet process Antonio Lijoi1 and Igor Prunster¨ 2 1 Dipartimento Economia Politica e Metodi Quantitatavi, Universit`adegli Studi di Pavia, via San Felice 5, 27100 Pavia, Italy and CNR{IMATI, via Bassini 15, 20133 Milano. E-mail: [email protected] 2 Dipartimento di Statistica e Matematica Applicata, Collegio Carlo Alberto and ICER, Universit`a degli Studi di Torino, Corso Unione Sovietica 218/bis, 10134 Torino, Italy. E-mail: [email protected] September 2009 Abstract. Bayesian nonparametric inference is a relatively young area of research and it has recently undergone a strong development. Most of its success can be explained by the considerable degree of flexibility it ensures in statistical modelling, if compared to parametric alternatives, and by the emergence of new and efficient simulation techniques that make nonparametric models amenable to concrete use in a number of applied sta- tistical problems. Since its introduction in 1973 by T.S. Ferguson, the Dirichlet process has emerged as a cornerstone in Bayesian nonparametrics. Nonetheless, in some cases of interest for statistical applications the Dirichlet process is not an adequate prior choice and alternative nonparametric models need to be devised. In this paper we provide a review of Bayesian nonparametric models that go beyond the Dirichlet process. 1 Introduction Bayesian nonparametric inference is a relatively young area of research and it has recently under- gone a strong development.
    [Show full text]
  • Strong Markov Property of Determinantal Processes With
    Strong Markov property of determinantal processes with extended kernels Hirofumi Osada ∗ Hideki Tanemura † June 16, 2018 Abstract Noncolliding Brownian motion (Dyson’s Brownian motion model with parameter β = 2) and noncolliding Bessel processes are determinantal processes; that is, their space-time correlation functions are represented by determinants. Under a proper scaling limit, such as the bulk, soft-edge and hard-edge scaling limits, these processes converge to determinantal processes describing systems with an infinite number of particles. The main purpose of this paper is to show the strong Markov property of these limit processes, which are determinantal processes with the extended sine kernel, extended Airy kernel and extended Bessel kernel, respectively. We also determine the quasi-regular Dirichlet forms and infinite-dimensional stochastic differential equations associated with the determinantal processes. Keywords Determinantal processes Correlation kernels Random matrix theory · · · Infinite particle systems Strong Markov property Entire function and topology · · Mathematics Subject Classification (2010) 15B52, 30C15, 47D07, 60G55, 82C22 arXiv:1412.8678v2 [math.PR] 12 Feb 2015 1 Introduction In a system of N independent one-dimensional diffusion processes, if we impose the condition that the particles never collide with one another, we obtain an interacting particle system with a long-range strong repulsive force between any pair of particles. We call such a system a noncolliding diffusion process. In 1962 Dyson [2] showed that
    [Show full text]
  • Analysis of Continuous Strict Local Martingales Via H-Transforms
    Analysis of continuous strict local martingales via h-transforms Soumik Pal∗ and Philip Protter † October 30, 2018 Abstract We study strict local martingales via h-transforms, a method which first appeared in Delbaen-Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are identified. We treat examples of diffusions with various boundary behavior, size-bias sam- pling of diffusion paths, and non-colliding diffusions. A multidimen- sional generalization to conformal strict local martingales is achieved through Kelvin transform. As curious examples of non-standard be- havior, we show by various examples that strict local martingales do not behave uniformly when the function (x K)+ is applied to them. Implications to the recent literature on financial− bubbles are discussed. 1 Introduction Local martingales which are not martingales (known as “strict” local martin- gales) arise naturally in the Doob-Meyer decomposition and in multiplicative arXiv:0711.1136v4 [math.PR] 2 Jun 2010 functional decompositions, as well as in stochastic integration theory. They are nevertheless often considered to be anomalies, processes that need to be maneuvered by localization. Hence, studies focussed purely on strict local martingales are rare. One notable exception is the article by Elworthy, Li, & Yor [9] who study their properties in depth. On the other hand, applications of strict local martingales are common. See the articles by Bentata & Yor ∗Supported in part by NSF grant DMS-0306194; Departments of Mathematics – C-547 Padelford Hall, University of Washington, Seattle 98195 USA †Supported in part by NSF grant DMS-0202958 and NSA grant MDA-904-03-1-0092; ORIE – 219 Rhodes Hall, Cornell University, Ithaca, NY 14853-3801 USA 1 [2], Biane & Yor [3], Cox & Hobson [4], Fernholz & Karatzas [12], and the very recent book-length preprint of Profeta, Roynette, and Yor [33].
    [Show full text]
  • Conformally Invariant Processes in the Plane Mathematical I Surveys I and I Ponographs I
    http://dx.doi.org/10.1090/surv/114 Conformally Invariant Processes in the Plane Mathematical I Surveys I and I Ponographs I Volume 114 1 Conformally Invariant Processes in the Plane I Gregory F. Lawler EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber, Chair Michael G. Eastwood Michael P. Loss J. T. Stafford 2000 Mathematics Subject Classification. Primary 30C35, 31A15, 60H30, 60J65, 81T40, 82B27. For additional information and updates on this book, visit www.ams.org/bookpages/surv-114 Library of Congress Cataloging-in-Publication Data Lawler, Gregory F., (Gregory Francis), 1955- Conformally invariant processes in the plane / Gregory F. Lawler. p. cm. — (Mathematical surveys and monographs ; v. 114) Includes bibliographical references and index. ISBN 0-8218-3677-3 (alk. paper) 1. Conformal mapping. 2. Potential theory (Mathematics) 3. Stochastic analysis. 4. Markov processes. I. Title. II. Mathematical surveys and monographs ; no. 114. QA646.L85 2005 515'.9—dc22 2004062341 AMS softcover ISBN 978-0-8218-4624-7 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected].
    [Show full text]