A Holder condition for Brownian local time

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "A Holder condition for Brownian local time"

Transcription

1 J. Math. Kyoto Univ. 1- (196) A Holder condition for Brownian local time By H. P. MCKEAN, JR.' (Communicated by Prof. K. Ito, November, 1961) Given a standard Brownian motion on l e beginning at, H. Trotter [ 3 ] proved the (simultaneous) existence of the local times: 1. t(t, a) = u m measure (s: a < x(s) <b, s <t) b+a b a t >, ae R ' and derived the law a. p r l i m I t(t, b) t(t, a)i 1 = 1. p [ u rn. \ / lg118 I give simple proofs leading to the sharper bound b. p r /. t(t, b) t(t, a)i < \/max t(t, )1 1 ji r = N/8 1g118 b is proved assuming t exists and is continuous in space ; afterwards, I go back and prove the latter statement. H. Tanaka's (unpublished) expression for the local time as a stochastic integral : l t ( t, a) = max [x(t) a, ] max [ a, O] x ( d s ) ] 1 3. P [ s x (s )> a and the bound' 4a. E [ea(t)]< 1 for the functional The support of the ONR, U.S. Govt. is gratefully acknowledged. See, for example, E. B. Dynkin

2 196 H. P. McKean, Jr. 4b. a(t) r o f [x(s)]x(ds) 1 r o f[x(s)]ds 4e.P [ s : J - T x ( s ) ] c ls < + D o i are the basic tools for this. I want to thank H. Tanaka for communicating his integral 3 and for a helpful conversation about the sample path f of 17 below. Tanaka's (unpublished) proof of 3 is as follows. Bringing in the indicator e a b o f th e interval (a, bea<b), an application of the formula for stochastic _differential gives 5. 1 measure (s : a < x (s )< b, s < t) = - r o e a b [x(s)]cls = ēa b [x (t)]- ēab[x()] ro ea b[x(s)]x(ds) with e <a 6a. e.a ea b c17) =.( a < < b n<1., b a > b 6b. -eab() Pa b dn --- n t < a ( a) a < < b (b a)( a + 11 ) > b, and, using 7a. and lim(b a) - 1 b() = max [ a, ] Ir j, 7b. E f t ( eab i \b a _ E rft( e a t, _ e a c., ) e c o )x (ds) a I- i \b a d s ]

3 A Holder condition for Brownian local time 197 Er e a b (b x(s)y d s ] b a )._ <E [ e a b ds] < constant (b a), 3 is immediate on letting b l a in 5, assuming, as I now do, the existence of the local time t(t, a). Given positive numbers a and /3, points a < b, an d putting b a=8, an application of 4 a gives 8a. P [ eabx(ds)> [a+13 max t(t, )] N/8 1g118] R i < P N e a b x (d s)>.\/ 8 ) C id i 9 /(/) lg118 = P [ e a b x(ds) e bd s > a -V8 / g ild J o a <E[e7s1,eabx(d.3) A e b d s]e -7 N/6 1g VS < e - cosig vs = and since the same bound applies to e a b x (ds) as well, 5t 8. p [ e a b x(ds) > [a+ re max t(t, )] N/8 lg118] < 8 6 g, leading at once to Pr 9. max - b= j " <i j< " lal < d < E (k - n)ai 3 < k < " lai <d < 4d- n [ - - e ) o f t e ] e a b x(ds) / 8 g la >ad-re max t(t, ) which is the general term of a convergent sum provided d=1,,

4 198 H. P. McKean, Jr. 3, etc. is fixed, ar >1, and &> is so small that (1-8)cei3-1-&<. Tanaka's integral ( =3), the Borel-Cantelli lemma, and the fact that max [ b - a, ] is piecewise smooth can now be combined with 9 to establish 1. P - li m a = i ", b= j " - - <k= j i<"! b a=& $ lal <d it(t, b)-t(t, a)i < a + R m a x t ( t, ) = 1 N/8 1g118 le, for each choice o f d>1, a i e> 1, and < 6 < a 4-1 ar +1 But now, taking into account the fact that t(t, a) is continuous in space, it is plain sailing over the course laid out by P. Lévy [] for the proof of 11. p1 I x (, ) _ x ( s), < i i 1 1 -t-s= s 1./8 1g 1/8 o<s< t< 1 to deduce from 1 1. p ri i m b ) - t ( t, a) I-lb-al-a o V 8 /g 1/8 la <d < a + R max t(t, )] = 1 for each d > 1 and ar >1, and b follows on letting d + (use near ± ), letting a g 1, and making a+ max t as small as possible subject to c = 1. I n o w g o back and prove that t exists and is continuous. Beginning with the stochastic integrals e.x(ds) e(a)(a E RI), the trick is to prove, as I now do, that e can be modified so as to be continuous in space. Because 13. P r max eabds > n - n] a = (k 1) - " b= k - " la <d <d"1 3 [ : e - d s > n - "] _<d(1e)"e [ex p(" e _ ds)] R i _

5 A Holder condition for Brownian local time _<d(1e)net c 8 ele[ex p ( " e ds)] +- = d(1e)net c e d [1 + + n i do, do, ndb, ndb, e e -- (b - 1, 1) 1 ( - 1 ) C ( b 1-1, 1-1 ) / ( ) V 71-1 N / 7 T ( 1) V 7r(,-,,) <d(1e)nef = d(1e)net c 'e l [1 + E 1 convoluted with 11\/7r 1 times] i=i = d(1e)net \ / V 1 is the general term of a convergent sum a n d ea b d s is monotone in a and h, one finds Ç e d s 14. P [ urn b lb al-51 8 lg 1/8 i a < + and so, using the obvious bound 15a. P h t t ea b x (ds)> ce+ 1, o f t ) o ea b x(ds) 16. P lim a=i - - ", b= j " 8 1g1 IS - lb-al=s io ea b ds] = P p 5:eabx(ds)_1 5t eabds > a d < e - `4 3 with cev 81g118 and 3 IV 8 (a19> 1 ) in place of a and IS to obtain 15b. P [ e a b x (ds) > c c 81g118+ e d s b V Jo < 8cdo a = lb al, it follows as in the proof of b above that

6 H. P. McKean, Jr. But this means that the modified sample path 17. f(a)u r n e ( b ) lim eb,.a(ds) a E Rl b=k " a b = k - - a o is continuous ; in addition, 18. P h f(c)dc e coo d c)x (d sd = 1 a < b a a because P[f(a)----e(a)] - =-1 (a E R '), and since measure (s : a < a b, x(s) s< t) e b d s, and fd c are all continuous in a and b, an application of 5 gives 19. P[-1 measure (s : a < x(s) b, s < t) = a b [ X ( t ) ] - - a b ( ) f leading at once to the fact that d c, a < b i= 1,. 1 t(t, a) max [x(t) a, ] max [ a, ] f(a) exists and is continuous, as was to be proved. A second application of the above method gives the bound' 1. P [ r e o bx (ds) > [a + m a x t(t, a)] \/ 1 g 118] O ct b <(1g 118), - leading at once to. P L 1 im 1t 8 (t, ) t(t o N / 8 1 g 1 ' )1 /8 < 4 ' )] 1. Given t > and a E, the conditional local time [t(t, b):b E R, 1 P( /x(t) a)] is a diffusion, and, expressing it in terms of a standard Brownian motion (via a change of scale and a time substitution), it is immediate that the bounds b and are best possible ; this beautiful result will appear in a forthcoming paper by D. B. Ray. Massachustetts Institute o f Technology November g c =lg ( lg c ).

7 A Holder condition for Brownian local time 1 REFERENCES E. B. Dynkin : Additive functionals of a Wiener process determined by stochastic integrals. Teor. Veroyatnost. i ee Primenen. 5, (196). P. Lévy : Théorie de l'addition des variables aléatoires. Paris H. Trotter : A property o f Brownian motion paths. Ill. J. Math., (1958).

A L A BA M A L A W R E V IE W

A L A BA M A L A W R E V IE W A L A BA M A L A W R E V IE W Volume 52 Fall 2000 Number 1 B E F O R E D I S A B I L I T Y C I V I L R I G HT S : C I V I L W A R P E N S I O N S A N D TH E P O L I T I C S O F D I S A B I L I T Y I N

More information

ITÔ S ONE POINT EXTENSIONS OF MARKOV PROCESSES. Masatoshi Fukushima

ITÔ S ONE POINT EXTENSIONS OF MARKOV PROCESSES. Masatoshi Fukushima ON ITÔ S ONE POINT EXTENSIONS OF MARKOV PROCESSES Masatoshi Fukushima Symposium in Honor of Kiyosi Itô: Stocastic Analysis and Its Impact in Mathematics and Science, IMS, NUS July 10, 2008 1 1. Itô s point

More information

The concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.

The concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt. The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes

More information

Wiener Measure and Brownian Motion

Wiener Measure and Brownian Motion Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u

More information

Introduction to Random Diffusions

Introduction to Random Diffusions Introduction to Random Diffusions The main reason to study random diffusions is that this class of processes combines two key features of modern probability theory. On the one hand they are semi-martingales

More information

Some Tools From Stochastic Analysis

Some Tools From Stochastic Analysis W H I T E Some Tools From Stochastic Analysis J. Potthoff Lehrstuhl für Mathematik V Universität Mannheim email: potthoff@math.uni-mannheim.de url: http://ls5.math.uni-mannheim.de To close the file, click

More information

Kolmogorov Equations and Markov Processes

Kolmogorov Equations and Markov Processes Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define

More information

Mutual Information for Stochastic Differential Equations*

Mutual Information for Stochastic Differential Equations* INFORMATION AND CONTROL 19, 265--271 (1971) Mutual Information for Stochastic Differential Equations* TYRONE E. DUNCAN Department of Computer, Information and Control Engineering, College of Engineering,

More information

Gaussian Random Fields: Geometric Properties and Extremes

Gaussian Random Fields: Geometric Properties and Extremes Gaussian Random Fields: Geometric Properties and Extremes Yimin Xiao Michigan State University Outline Lecture 1: Gaussian random fields and their regularity Lecture 2: Hausdorff dimension results and

More information

Feller Processes and Semigroups

Feller Processes and Semigroups Stat25B: Probability Theory (Spring 23) Lecture: 27 Feller Processes and Semigroups Lecturer: Rui Dong Scribe: Rui Dong ruidong@stat.berkeley.edu For convenience, we can have a look at the list of materials

More information

Central limit theorem for a simple diffusion model of interacting particles

Central limit theorem for a simple diffusion model of interacting particles HIROSHIMA MATH. J. 11 (1981), 415-423 Central limit theorem for a simple diffusion model of interacting particles Hiroshi TANAKA and Masuyuki HITSUDA (Received January 19, 1981) 1. Introduction Given a

More information

Convergence at first and second order of some approximations of stochastic integrals

Convergence at first and second order of some approximations of stochastic integrals Convergence at first and second order of some approximations of stochastic integrals Bérard Bergery Blandine, Vallois Pierre IECN, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes B.P. 239 F-5456

More information

ON CONVERGENCE OF STOCHASTIC PROCESSES

ON CONVERGENCE OF STOCHASTIC PROCESSES ON CONVERGENCE OF STOCHASTIC PROCESSES BY JOHN LAMPERTI(') 1. Introduction. The "invariance principles" of probability theory [l ; 2 ; 5 ] are mathematically of the following form : a sequence of stochastic

More information

On a class of stochastic differential equations in a financial network model

On a class of stochastic differential equations in a financial network model 1 On a class of stochastic differential equations in a financial network model Tomoyuki Ichiba Department of Statistics & Applied Probability, Center for Financial Mathematics and Actuarial Research, University

More information

A Concise Course on Stochastic Partial Differential Equations

A Concise Course on Stochastic Partial Differential Equations A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original

More information

A Stochastic Paradox for Reflected Brownian Motion?

A Stochastic Paradox for Reflected Brownian Motion? Proceedings of the 9th International Symposium on Mathematical Theory of Networks and Systems MTNS 2 9 July, 2 udapest, Hungary A Stochastic Parado for Reflected rownian Motion? Erik I. Verriest Abstract

More information

GAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM

GAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM GAUSSIAN PROCESSES; KOLMOGOROV-CHENTSOV THEOREM STEVEN P. LALLEY 1. GAUSSIAN PROCESSES: DEFINITIONS AND EXAMPLES Definition 1.1. A standard (one-dimensional) Wiener process (also called Brownian motion)

More information

ON THE MAXIMUM OF A NORMAL STATIONARY STOCHASTIC PROCESS 1 BY HARALD CRAMER. Communicated by W. Feller, May 1, 1962

ON THE MAXIMUM OF A NORMAL STATIONARY STOCHASTIC PROCESS 1 BY HARALD CRAMER. Communicated by W. Feller, May 1, 1962 ON THE MAXIMUM OF A NORMAL STATIONARY STOCHASTIC PROCESS 1 BY HARALD CRAMER Communicated by W. Feller, May 1, 1962 1. Let x(t) with oo

More information

Table of C on t en t s Global Campus 21 in N umbe r s R e g ional Capac it y D e v e lopme nt in E-L e ar ning Structure a n d C o m p o n en ts R ea

Table of C on t en t s Global Campus 21 in N umbe r s R e g ional Capac it y D e v e lopme nt in E-L e ar ning Structure a n d C o m p o n en ts R ea G Blended L ea r ni ng P r o g r a m R eg i o na l C a p a c i t y D ev elo p m ent i n E -L ea r ni ng H R K C r o s s o r d e r u c a t i o n a n d v e l o p m e n t C o p e r a t i o n 3 0 6 0 7 0 5

More information

Convergence Concepts of Random Variables and Functions

Convergence Concepts of Random Variables and Functions Convergence Concepts of Random Variables and Functions c 2002 2007, Professor Seppo Pynnonen, Department of Mathematics and Statistics, University of Vaasa Version: January 5, 2007 Convergence Modes Convergence

More information

Weak convergence and large deviation theory

Weak convergence and large deviation theory First Prev Next Go To Go Back Full Screen Close Quit 1 Weak convergence and large deviation theory Large deviation principle Convergence in distribution The Bryc-Varadhan theorem Tightness and Prohorov

More information

Simulation of diffusion. processes with discontinuous coefficients. Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan

Simulation of diffusion. processes with discontinuous coefficients. Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan Simulation of diffusion. processes with discontinuous coefficients Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan From collaborations with Pierre Étoré and Miguel Martinez . Divergence

More information

Lifshitz tail for Schödinger Operators with random δ magnetic fields

Lifshitz tail for Schödinger Operators with random δ magnetic fields Lifshitz tail for Schödinger Operators with random δ magnetic fields Takuya Mine (Kyoto Institute of Technology, Czech technical university in Prague) Yuji Nomura (Ehime University) March 16, 2010 at Arizona

More information

GROUPS OF ELLIPTIC LINEAR FRACTIONAL TRANSFORMATIONS

GROUPS OF ELLIPTIC LINEAR FRACTIONAL TRANSFORMATIONS GROUPS OF ELLIPTIC LINEAR FRACTIONAL TRANSFORMATIONS R. C LYNDON AND J. L. ULLMAN 1. Introduction. Our aim is to give a simple and self contained statement and proof of the following theorem: Every subgroup

More information

Stochastic Integration and Stochastic Differential Equations: a gentle introduction

Stochastic Integration and Stochastic Differential Equations: a gentle introduction Stochastic Integration and Stochastic Differential Equations: a gentle introduction Oleg Makhnin New Mexico Tech Dept. of Mathematics October 26, 27 Intro: why Stochastic? Brownian Motion/ Wiener process

More information

SPECTRAL ORDER PRESERVING MATRICES AND MUIRHEAD'S THEOREM

SPECTRAL ORDER PRESERVING MATRICES AND MUIRHEAD'S THEOREM TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 200, 1974 SPECTRAL ORDER PRESERVING MATRICES AND MUIRHEAD'S THEOREM BY KONG-MING chongo.1) ABSTRACT. In this paper, a characterization is given

More information

Memoirs of My Research on Stochastic Analysis

Memoirs of My Research on Stochastic Analysis Memoirs of My Research on Stochastic Analysis Kiyosi Itô Professor Emeritus, Kyoto University, Kyoto, 606-8501 Japan It is with great honor that I learned of the 2005 Oslo Symposium on Stochastic Analysis

More information

(1) N(bi, n) < cid ;-

(1) N(bi, n) < cid ;- SOME RESULTS ON ADDITIVE NUMBER THEORY1 PAUL ERDOS Let 0

More information

Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations

Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations Scaling Limits of Waves in Convex Scalar Conservation Laws under Random Initial Perturbations Jan Wehr and Jack Xin Abstract We study waves in convex scalar conservation laws under noisy initial perturbations.

More information

4 Sums of Independent Random Variables

4 Sums of Independent Random Variables 4 Sums of Independent Random Variables Standing Assumptions: Assume throughout this section that (,F,P) is a fixed probability space and that X 1, X 2, X 3,... are independent real-valued random variables

More information

GARCH processes continuous counterparts (Part 2)

GARCH processes continuous counterparts (Part 2) GARCH processes continuous counterparts (Part 2) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/

More information

Definition: Lévy Process. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes. Theorem

Definition: Lévy Process. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes. Theorem Definition: Lévy Process Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes David Applebaum Probability and Statistics Department, University of Sheffield, UK July

More information

ON THE SHAPE OF THE GROUND STATE EIGENFUNCTION FOR STABLE PROCESSES

ON THE SHAPE OF THE GROUND STATE EIGENFUNCTION FOR STABLE PROCESSES ON THE SHAPE OF THE GROUND STATE EIGENFUNCTION FOR STABLE PROCESSES RODRIGO BAÑUELOS, TADEUSZ KULCZYCKI, AND PEDRO J. MÉNDEZ-HERNÁNDEZ Abstract. We prove that the ground state eigenfunction for symmetric

More information

REPRESENTATION THEOREM FOR HARMONIC BERGMAN AND BLOCH FUNCTIONS

REPRESENTATION THEOREM FOR HARMONIC BERGMAN AND BLOCH FUNCTIONS Tanaka, K. Osaka J. Math. 50 (2013), 947 961 REPRESENTATION THEOREM FOR HARMONIC BERGMAN AND BLOCH FUNCTIONS KIYOKI TANAKA (Received March 6, 2012) Abstract In this paper, we give the representation theorem

More information

Representations of Gaussian measures that are equivalent to Wiener measure

Representations of Gaussian measures that are equivalent to Wiener measure Representations of Gaussian measures that are equivalent to Wiener measure Patrick Cheridito Departement für Mathematik, ETHZ, 89 Zürich, Switzerland. E-mail: dito@math.ethz.ch Summary. We summarize results

More information

M $ 4 65\ K;$ 5, 65\ M $ C! 4 /2 K;$ M $ /+5\ 8$ A5 =+0,7 ;* C! 4.4/ =! K;$,7 $,+7; ;J zy U;K z< mj ]!.,,+7;

M $ 4 65\ K;$ 5, 65\ M $ C! 4 /2 K;$ M $ /+5\ 8$ A5 =+0,7 ;* C! 4.4/ =! K;$,7 $,+7; ;J zy U;K z< mj ]!.,,+7; V 3U. T, SK I 1393/08/21 :,F! 1393/10/29 ::!n> 2 1 /M + - /E+4q; Z R :'!3Qi M $,7 8$ 4,!AK 4 4/ * /;K "FA ƒf\,7 /;G2 @;J\ M $ 4 65\ K;$ 5, 65\ M $ C! 4 /2 K;$ M $ /+5\ 8$ A5 =+0,7 ;* C! 4.4/ =! K;$,7 $,+7;

More information

Brownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion

Brownian Motion. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Brownian Motion Brownian Motion An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Background We have already seen that the limiting behavior of a discrete random walk yields a derivation of

More information

Divergence Theorems in Path Space. Denis Bell University of North Florida

Divergence Theorems in Path Space. Denis Bell University of North Florida Divergence Theorems in Path Space Denis Bell University of North Florida Motivation Divergence theorem in Riemannian geometry Theorem. Let M be a closed d-dimensional Riemannian manifold. Then for any

More information

Applications of controlled paths

Applications of controlled paths Applications of controlled paths Massimiliano Gubinelli CEREMADE Université Paris Dauphine OxPDE conference. Oxford. September 1th 212 ( 1 / 16 ) Outline I will exhibith various applications of the idea

More information

An introduction to rough paths

An introduction to rough paths An introduction to rough paths Antoine LEJAY INRIA, Nancy, France From works from T. Lyons, Z. Qian, P. Friz, N. Victoir, M. Gubinelli, D. Feyel, A. de la Pradelle, A.M. Davie,... SPDE semester Isaac Newton

More information

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539 Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory

More information

Basic Definitions: Indexed Collections and Random Functions

Basic Definitions: Indexed Collections and Random Functions Chapter 1 Basic Definitions: Indexed Collections and Random Functions Section 1.1 introduces stochastic processes as indexed collections of random variables. Section 1.2 builds the necessary machinery

More information

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( ) Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical

More information

A B CDE F B FD D A C AF DC A F

A B CDE F B FD D A C AF DC A F International Journal of Arts & Sciences, CD-ROM. ISSN: 1944-6934 :: 4(20):121 131 (2011) Copyright c 2011 by InternationalJournal.org A B CDE F B FD D A C A BC D EF C CE C A D ABC DEF B B C A E E C A

More information

LARGE DEVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILED DEPENDENT RANDOM VECTORS*

LARGE DEVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILED DEPENDENT RANDOM VECTORS* LARGE EVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILE EPENENT RANOM VECTORS* Adam Jakubowski Alexander V. Nagaev Alexander Zaigraev Nicholas Copernicus University Faculty of Mathematics and Computer Science

More information

THE POISSON TRANSFORM^)

THE POISSON TRANSFORM^) THE POISSON TRANSFORM^) BY HARRY POLLARD The Poisson transform is defined by the equation (1) /(*)=- /" / MO- T J _M 1 + (X t)2 It is assumed that a is of bounded variation in each finite interval, and

More information

Random Conformal Welding

Random Conformal Welding Random Conformal Welding Antti Kupiainen joint work with K. Astala, P. Jones, E. Saksman Ascona 26.5.2010 Random Planar Curves 2d Statistical Mechanics: phase boundaries Closed curves or curves joining

More information

Executive Committee and Officers ( )

Executive Committee and Officers ( ) Gifted and Talented International V o l u m e 2 4, N u m b e r 2, D e c e m b e r, 2 0 0 9. G i f t e d a n d T a l e n t e d I n t e r n a t i o n a2 l 4 ( 2), D e c e m b e r, 2 0 0 9. 1 T h e W o r

More information

Sb1) = (4, Az,.-., An),

Sb1) = (4, Az,.-., An), PROBABILITY ' AND MATHEMATICAL STATISTICS VoL 20, lux. 2 (20W), pp. 359-372 DISCRETE PROBABILITY MEASURES ON 2 x 2 STOCHASTIC MATRICES AND A FUNCTIONAL EQUATION OW '[o, 11 - -. A. MUKHERJEA AND J. S. RATTI

More information

2. T H E , ( 7 ) 2 2 ij ij. p i s

2. T H E , ( 7 ) 2 2 ij ij. p i s M O D E L O W A N I E I N Y N I E R S K I E n r 4 7, I S S N 1 8 9 6-7 7 1 X A N A L Y S I S O F T E M P E R A T U R E D I S T R I B U T I O N I N C O M P O S I T E P L A T E S D U R I N G T H E R M A

More information

GAUSSIAN PROCESSES GROWTH RATE OF CERTAIN. successfully employed in dealing with certain Gaussian processes not possessing

GAUSSIAN PROCESSES GROWTH RATE OF CERTAIN. successfully employed in dealing with certain Gaussian processes not possessing 1. Introduction GROWTH RATE OF CERTAIN GAUSSIAN PROCESSES STEVEN OREY UNIVERSITY OF MINNESOTA We will be concerned with real, continuous Gaussian processes. In (A) of Theorem 1.1, a result on the growth

More information

MATH Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited).

MATH Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited). MATH 311-504 Topics in Applied Mathematics Lecture 2-6: Isomorphism. Linear independence (revisited). Definition. A mapping f : V 1 V 2 is one-to-one if it maps different elements from V 1 to different

More information

Large Deviations for Perturbed Reflected Diffusion Processes

Large Deviations for Perturbed Reflected Diffusion Processes Large Deviations for Perturbed Reflected Diffusion Processes Lijun Bo & Tusheng Zhang First version: 31 January 28 Research Report No. 4, 28, Probability and Statistics Group School of Mathematics, The

More information

Spatial Ergodicity of the Harris Flows

Spatial Ergodicity of the Harris Flows Communications on Stochastic Analysis Volume 11 Number 2 Article 6 6-217 Spatial Ergodicity of the Harris Flows E.V. Glinyanaya Institute of Mathematics NAS of Ukraine, glinkate@gmail.com Follow this and

More information

JUST THE MATHS UNIT NUMBER LAPLACE TRANSFORMS 3 (Differential equations) A.J.Hobson

JUST THE MATHS UNIT NUMBER LAPLACE TRANSFORMS 3 (Differential equations) A.J.Hobson JUST THE MATHS UNIT NUMBER 16.3 LAPLACE TRANSFORMS 3 (Differential equations) by A.J.Hobson 16.3.1 Examples of solving differential equations 16.3.2 The general solution of a differential equation 16.3.3

More information

ITERATING THE DIVISION ALGORITHM

ITERATING THE DIVISION ALGORITHM MICHAEL E. MAYS West Virginia University, Morgantown, WV 26506 (Submitted June 1985) INTRODUCTION The division algorithm guarantees that when an arbitrary integer b is divided by a positive integer a there

More information

The Codimension of the Zeros of a Stable Process in Random Scenery

The Codimension of the Zeros of a Stable Process in Random Scenery The Codimension of the Zeros of a Stable Process in Random Scenery Davar Khoshnevisan The University of Utah, Department of Mathematics Salt Lake City, UT 84105 0090, U.S.A. davar@math.utah.edu http://www.math.utah.edu/~davar

More information

Powered by TCPDF (www.tcpdf.org)

Powered by TCPDF (www.tcpdf.org) Powered by TCPDF (www.tcpdf.org) Bol. Soc. Mat. Mexicana (3) vol. 1,1995 ON ACYCLIC CURVES. A SURVEY OF RESULTS AND PROBLEMS BY J. J. CHARATONIK Contents. 2. Dendrites. 3. Dendroids - global properties.

More information

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( ) Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca October 22nd, 2014 E. Tanré (INRIA - Team Tosca) Mathematical

More information

Nonlinear Systems Theory

Nonlinear Systems Theory Nonlinear Systems Theory Matthew M. Peet Arizona State University Lecture 2: Nonlinear Systems Theory Overview Our next goal is to extend LMI s and optimization to nonlinear systems analysis. Today we

More information

A Present Position-Dependent Conditional Fourier-Feynman Transform and Convolution Product over Continuous Paths

A Present Position-Dependent Conditional Fourier-Feynman Transform and Convolution Product over Continuous Paths International Journal of Mathematical Analysis Vol. 9, 05, no. 48, 387-406 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.589 A Present Position-Dependent Conditional Fourier-Feynman Transform

More information

[ ]:543.4(075.8) 35.20: ,..,..,.., : /... ;. 2-. ISBN , - [ ]:543.4(075.8) 35.20:34.

[ ]:543.4(075.8) 35.20: ,..,..,.., : /... ;. 2-. ISBN , - [ ]:543.4(075.8) 35.20:34. .. - 2-2009 [661.87.+661.88]:543.4(075.8) 35.20:34.2373-60..,..,..,..,.. -60 : /... ;. 2-. : -, 2008. 134. ISBN 5-98298-299-7 -., -,,. - «,, -, -», - 550800,, 240600 «-», -. [661.87.+661.88]:543.4(075.8)

More information

PRODUCTS OF INDECOMPOSABLE, APERIODIC, STOCHASTIC MATRICES1 J. WOLFOWITZ

PRODUCTS OF INDECOMPOSABLE, APERIODIC, STOCHASTIC MATRICES1 J. WOLFOWITZ PRODUCTS OF INDECOMPOSABLE, APERIODIC, STOCHASTIC MATRICES1 J. WOLFOWITZ 1. Introduction. A finite square matrix P ={/»«} is called stochastic if pij^q for all i, j, and Hipa=l for all i. A stochastic

More information

Chapter 6: The Laplace Transform. Chih-Wei Liu

Chapter 6: The Laplace Transform. Chih-Wei Liu Chapter 6: The Laplace Transform Chih-Wei Liu Outline Introduction The Laplace Transform The Unilateral Laplace Transform Properties of the Unilateral Laplace Transform Inversion of the Unilateral Laplace

More information

Itô s formula. Samy Tindel. Purdue University. Probability Theory 2 - MA 539

Itô s formula. Samy Tindel. Purdue University. Probability Theory 2 - MA 539 Itô s formula Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Itô s formula Probability Theory

More information

Lecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University

Lecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral

More information

Week 2. The Simplex method was developed by Dantzig in the late 40-ties.

Week 2. The Simplex method was developed by Dantzig in the late 40-ties. 1 The Simplex method Week 2 The Simplex method was developed by Dantzig in the late 40-ties. 1.1 The standard form The simplex method is a general description algorithm that solves any LPproblem instance.

More information

1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989),

1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer 11(2) (1989), Real Analysis 2, Math 651, Spring 2005 April 26, 2005 1 Real Analysis 2, Math 651, Spring 2005 Krzysztof Chris Ciesielski 1/12/05: sec 3.1 and my article: How good is the Lebesgue measure?, Math. Intelligencer

More information

THEOREM TO BE SHARP A SET OF GENERALIZED NUMBERS SHOWING BEURLING S. We shall give an example of e-primes and e-integers for which the prime

THEOREM TO BE SHARP A SET OF GENERALIZED NUMBERS SHOWING BEURLING S. We shall give an example of e-primes and e-integers for which the prime A SET OF GENERALIZED NUMBERS SHOWING BEURLING S THEOREM TO BE SHARP BY HAROLD G. DIAMOND Beurling [1] proved that the prime number theorem holds for generalized (henoeforh e) numbers if N (z), the number

More information

Cores for generators of some Markov semigroups

Cores for generators of some Markov semigroups Cores for generators of some Markov semigroups Giuseppe Da Prato, Scuola Normale Superiore di Pisa, Italy and Michael Röckner Faculty of Mathematics, University of Bielefeld, Germany and Department of

More information

MATH 6605: SUMMARY LECTURE NOTES

MATH 6605: SUMMARY LECTURE NOTES MATH 6605: SUMMARY LECTURE NOTES These notes summarize the lectures on weak convergence of stochastic processes. If you see any typos, please let me know. 1. Construction of Stochastic rocesses A stochastic

More information

Lecture 17 Brownian motion as a Markov process

Lecture 17 Brownian motion as a Markov process Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is

More information

A mathematical statement that asserts that two quantities are equal is called an equation. Examples: x Mathematics Division, IMSP, UPLB

A mathematical statement that asserts that two quantities are equal is called an equation. Examples: x Mathematics Division, IMSP, UPLB EQUATIONS A mathematical statement that asserts that two quantities are equal is called an equation. Examples: 1.. 3. 1 9 1 x 3 11 x 4xy y 0 Consider the following equations: 1. 1 + 9 = 1. x + 9 = 1 Equation

More information

SMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS

SMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS SMOOTHNESS OF FUNCTIONS GENERATED BY RIESZ PRODUCTS PETER L. DUREN Riesz products are a useful apparatus for constructing singular functions with special properties. They have been an important source

More information

(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS

(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS (2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS Svetlana Janković and Miljana Jovanović Faculty of Science, Department of Mathematics, University

More information

9.2 Branching random walk and branching Brownian motions

9.2 Branching random walk and branching Brownian motions 168 CHAPTER 9. SPATIALLY STRUCTURED MODELS 9.2 Branching random walk and branching Brownian motions Branching random walks and branching diffusions have a long history. A general theory of branching Markov

More information

35H MPa Hydraulic Cylinder 3.5 MPa Hydraulic Cylinder 35H-3

35H MPa Hydraulic Cylinder 3.5 MPa Hydraulic Cylinder 35H-3 - - - - ff ff - - - - - - B B BB f f f f f f f 6 96 f f f f f f f 6 f LF LZ f 6 MM f 9 P D RR DD M6 M6 M6 M. M. M. M. M. SL. E 6 6 9 ZB Z EE RC/ RC/ RC/ RC/ RC/ ZM 6 F FP 6 K KK M. M. M. M. M M M M f f

More information

e st f (t) dt = e st tf(t) dt = L {t f(t)} s

e st f (t) dt = e st tf(t) dt = L {t f(t)} s Additional operational properties How to find the Laplace transform of a function f (t) that is multiplied by a monomial t n, the transform of a special type of integral, and the transform of a periodic

More information

Karhunen-Loève Expansions of Lévy Processes

Karhunen-Loève Expansions of Lévy Processes Karhunen-Loève Expansions of Lévy Processes Daniel Hackmann June 2016, Barcelona supported by the Austrian Science Fund (FWF), Project F5509-N26 Daniel Hackmann (JKU Linz) KLE s of Lévy Processes 0 / 40

More information

Solutions of APMO 2016

Solutions of APMO 2016 Solutions of APMO 016 Problem 1. We say that a triangle ABC is great if the following holds: for any point D on the side BC, if P and Q are the feet of the perpendiculars from D to the lines AB and AC,

More information

DYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING. University of Bath. Bath BA2 7AY, U.K. University of Cambridge

DYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING. University of Bath. Bath BA2 7AY, U.K. University of Cambridge DYNAMIC WEIGHT FUNCTIONS FOR A MOVING CRACK II. SHEAR LOADING A.B. Movchan and J.R. Willis 2 School of Mathematical Sciences University of Bath Bath BA2 7AY, U.K. 2 University of Cambridge Department of

More information

The absolute continuity relationship: a fi» fi =exp fif t (X t a t) X s ds W a j Ft () is well-known (see, e.g. Yor [4], Chapter ). It also holds with

The absolute continuity relationship: a fi» fi =exp fif t (X t a t) X s ds W a j Ft () is well-known (see, e.g. Yor [4], Chapter ). It also holds with A Clarification about Hitting Times Densities for OrnsteinUhlenbeck Processes Anja Göing-Jaeschke Λ Marc Yor y Let (U t ;t ) be an OrnsteinUhlenbeck process with parameter >, starting from a R, that is

More information

MATRIX ALGEBRA. or x = (x 1,..., x n ) R n. y 1 y 2. x 2. x m. y m. y = cos θ 1 = x 1 L x. sin θ 1 = x 2. cos θ 2 = y 1 L y.

MATRIX ALGEBRA. or x = (x 1,..., x n ) R n. y 1 y 2. x 2. x m. y m. y = cos θ 1 = x 1 L x. sin θ 1 = x 2. cos θ 2 = y 1 L y. as Basics Vectors MATRIX ALGEBRA An array of n real numbers x, x,, x n is called a vector and it is written x = x x n or x = x,, x n R n prime operation=transposing a column to a row Basic vector operations

More information

Stochastic Differential Equations

Stochastic Differential Equations CHAPTER 1 Stochastic Differential Equations Consider a stochastic process X t satisfying dx t = bt, X t,w t dt + σt, X t,w t dw t. 1.1 Question. 1 Can we obtain the existence and uniqueness theorem for

More information

Chapter 2 Event-Triggered Sampling

Chapter 2 Event-Triggered Sampling Chapter Event-Triggered Sampling In this chapter, some general ideas and basic results on event-triggered sampling are introduced. The process considered is described by a first-order stochastic differential

More information

PREDICTABLE REPRESENTATION PROPERTY OF SOME HILBERTIAN MARTINGALES. 1. Introduction.

PREDICTABLE REPRESENTATION PROPERTY OF SOME HILBERTIAN MARTINGALES. 1. Introduction. Acta Math. Univ. Comenianae Vol. LXXVII, 1(28), pp. 123 128 123 PREDICTABLE REPRESENTATION PROPERTY OF SOME HILBERTIAN MARTINGALES M. EL KADIRI Abstract. We prove as for the real case that a martingale

More information

ON THE DEGREE OF APPROXIMATION BY POSITIVE LINEAR OPERATORS USING THE B SUMMABILITY METHOD.* A.S. RANADIVE and S.P. SINGH. for n +I ::; m ::; n +p

ON THE DEGREE OF APPROXIMATION BY POSITIVE LINEAR OPERATORS USING THE B SUMMABILITY METHOD.* A.S. RANADIVE and S.P. SINGH. for n +I ::; m ::; n +p Revista Colombiana de Matematicas Vol. XXV (1991) pgs. 1-10 ON THE DEGREE OF APPROXIMATION BY POSITIVE LINEAR OPERATORS USING THE B SUMMABILITY METHOD.* by A.S. RANADIVE and S.P. SINGH ABSTRACT. The aim

More information

An Introduction to Malliavin Calculus. Denis Bell University of North Florida

An Introduction to Malliavin Calculus. Denis Bell University of North Florida An Introduction to Malliavin Calculus Denis Bell University of North Florida Motivation - the hypoellipticity problem Definition. A differential operator G is hypoelliptic if, whenever the equation Gu

More information

Lecture 1s Isomorphisms of Vector Spaces (pages )

Lecture 1s Isomorphisms of Vector Spaces (pages ) Lecture 1s Isomorphisms of Vector Spaces (pages 246-249) Definition: L is said to be one-to-one if L(u 1 ) = L(u 2 ) implies u 1 = u 2. Example: The mapping L : R 4 R 2 defined by L(a, b, c, d) = (a, d)

More information

Almost Sure Convergence of the General Jamison Weighted Sum of B-Valued Random Variables

Almost Sure Convergence of the General Jamison Weighted Sum of B-Valued Random Variables Acta Mathematica Sinica, English Series Feb., 24, Vol.2, No., pp. 8 92 Almost Sure Convergence of the General Jamison Weighted Sum of B-Valued Random Variables Chun SU Tie Jun TONG Department of Statistics

More information

ECONOMETRICS II, FALL Testing for Unit Roots.

ECONOMETRICS II, FALL Testing for Unit Roots. ECONOMETRICS II, FALL 216 Testing for Unit Roots. In the statistical literature it has long been known that unit root processes behave differently from stable processes. For example in the scalar AR(1)

More information

(2) bn = JZ( )(**-*.)at,

(2) bn = JZ( )(**-*.)at, THE RELATION BETWEEN THE SEQUENCE-TO-SEQUENCE AND THE SERIES-TO-SERIES VERSIONS OF QUASI- HAUSDORFF SUMMABILITY METHODS B. KWEE 1. Introduction. Let ih, p, ) be a regular Hausdorff method of summability,

More information

Math 113 Homework 5. Bowei Liu, Chao Li. Fall 2013

Math 113 Homework 5. Bowei Liu, Chao Li. Fall 2013 Math 113 Homework 5 Bowei Liu, Chao Li Fall 2013 This homework is due Thursday November 7th at the start of class. Remember to write clearly, and justify your solutions. Please make sure to put your name

More information

An adaptive numerical scheme for fractional differential equations with explosions

An adaptive numerical scheme for fractional differential equations with explosions An adaptive numerical scheme for fractional differential equations with explosions Johanna Garzón Departamento de Matemáticas, Universidad Nacional de Colombia Seminario de procesos estocásticos Jointly

More information

The Skorokhod reflection problem for functions with discontinuities (contractive case)

The Skorokhod reflection problem for functions with discontinuities (contractive case) The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection

More information

`G 12 */" T A5&2/, ]&>b ; A%/=W, 62 S 35&.1?& S + ( A; 2 ]/0 ; 5 ; L) ( >>S.

`G 12 */ T A5&2/, ]&>b ; A%/=W, 62 S 35&.1?& S + ( A; 2 ]/0 ; 5 ; L) ( >>S. 01(( +,-. ()*) $%&' "#! : : % $& - "#$ :, (!" -&. #0 12 + 34 2567 () *+ '!" #$%& ; 2 "1? + @)&2 A5&2 () 25& 89:2 *2 72, B97I J$K

More information

An Introduction to Malliavin calculus and its applications

An Introduction to Malliavin calculus and its applications An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart

More information

9/20/2017. Elements are Pure Substances that cannot be broken down into simpler substances by chemical change (contain Only One Type of Atom)

9/20/2017. Elements are Pure Substances that cannot be broken down into simpler substances by chemical change (contain Only One Type of Atom) CAPTER 6: TE PERIODIC TABLE Elements are Pure Substances that cannot be broken down into simpler substances by chemical change (contain Only One Type of Atom) The Periodic Table (Mendeleev) In 1872, Dmitri

More information

RINGS WITH UNIQUE ADDITION

RINGS WITH UNIQUE ADDITION RINGS WITH UNIQUE ADDITION Dedicated R. E. JOHNSON to the Memory of Tibor Szele Introduction. The ring {R; +, } is said to have unique addition if there exists no other ring {R; +', } having the same multiplicative

More information

Matrix-Matrix Multiplication

Matrix-Matrix Multiplication Chapter Matrix-Matrix Multiplication In this chapter, we discuss matrix-matrix multiplication We start by motivating its definition Next, we discuss why its implementation inherently allows high performance

More information