School and Workshop on Market Microstructure: Design, Efficiency and Statistical Regularities March 2011

Size: px
Start display at page:

Download "School and Workshop on Market Microstructure: Design, Efficiency and Statistical Regularities March 2011"

Transcription

1 School and Workshop on Marke Microsrucure: Design, Efficiency and Saisical Regulariies March 2011 Some mahemaical properies of order book models Frederic ABERGEL Ecole Cenrale Paris Grande Voie des Vignes Chaenay Malabry FRANCE

2 Some empirical and mahemaical properies of limi order books Frédéric Abergel Chair of Quaniaive Finance École Cenrale Paris hp://fiquan.mas.ecp.fr

3 Limi order books Join works (some in progress) wih I. Muni Toke, A. Jedidi References I. Muni Toke, Marke making behaviour and is impac on he Bid-Ask spread, in Econophysics of Order-driven Markes, Abergel, F.; Chakrabari, B.K.; Chakrabori, A.; Mira, M. (Eds.), Springer, 2011 F. Abergel, A. Jedidi, A mahemaical approach o order book modelling, hp://papers.ssrn.com/sol3/papers.cfm?absrac_id= F. Abergel, A. Chakrabori, I. Muni Toke, M. Pariarca, Econophysics I: empirical facs and Econophysics II: agen-based models, o appear in Quaniaive Finance

4 Limi order books Summary Empirical properies of he order book Saionary saisical properies Dynamical saisical properies Mahemaical models Mahemaical framework Price dynamics

5 Limi order book

6 Saisical properies I A hos of empirical sudies going back o ~20 years, addressing he wo following quesions: When will he nex even ake place? Where will he nex even ake place? Under independence assumpions: Zero-inelligence models

7 Saisical properies I Such uncondiional saisics do no fully reflec he dynamics of a limi order book. Many ineresing phenomena are no described his way Volailiy clusering Leverage Auocorrelaion of he order flow In real markes, agens observe he sae of he marke and adap o i An example (Muni-Toke): empirical evidence of marke making and marke aking

8 Saisical properies II Marke making Following a marke order New limi orders arrive more rapidly han uncondiional limi orders No significan correlaion beween he respecive signs of he marke and limi orders

9 Saisical properies II Marke making

10 Saisical properies II Marke aking Following a limi order New marke orders do no arrive more rapidly excep when he limi order fell wihin he spread

11 Saisical properies II Marke aking

12 Saisical properies II Several recen sudies accouning for dependencies (Large 2007, Muni Toke 2010, Eisler 2010) Condiional iner-even duraion Lead and lag relaionship Condiional price and volume disribuions lead o models involving Sae-dependen inensiies and placemen Muually excied processes

13 Mahemaical framework The limi order book: a vecor valued poin process Main quesions o be addressed Saionariy Price and spread dynamics Scaling and long ime asympoics

14 Mahemaical framework Back o he simples example: zero-inelligence model wih limi orders, marke orders and cancellaions (Farmer, Smih, Guillemo, Krishnamurhy, 2003) dl dm dc λ i ± i ± L λ ± ± M λ Δ P τ 1 τ a i, τ λ i ± i + i C C b i τ

15 Mahemaical framework Two ses of variables Coupled dynamics a,..., a ; b,..., b 1 N 1 i i ( ) ( ) a a iδp b b iδp A = a B = b i i i i k= 1 k= 1 N i i Two basic ypes of evens Jump: a change in he quaniies Shif : renumbering afer a change of one of he bes quoes

16 Saionariy In his simple model, here exiss a Lyapunov funcion (he oal available volume), hanks o he exponenial damping effec of cancellaions Therefore, here exiss a saionary disribuion wih exponenial convergence This resul can be generalized o sae-dependen inensiies

17 Exension o Hawkes processes Hawkes processes: a poin process wih sochasic inensiy The inensiy is excied by he previous jumps (auoregressive process) N j j ( ) P = 0 + jp s dns p = 1 λ λ ϕ Typical choice: exponenial kernels N j j β jp ( s) P = + 0 jpe dns p= 1 λ λ α Becomes a Markov process in 1D (or higher wih equal decay raes)

18 Exension o Hawkes processes Clusering of orders easily described Leverage modelled hanks o asymeric kernels Saionariy condiions relaed o he values of he Hawkes parameers α 1 ( ( j )) jp ( j E λ = Id λ ) 0 β jp Saionariy condiions are found saisfied in empirical sudies (Muni Toke, Hewle, Large )

19 Hawkes processes Spread disribuion A consequence of beer modelling: spread disribuion

20 Price dynamics Price dynamics depend on Evens affecing he bes limis The firs gap process A useful represenaion for he bes Ask and Bid prices: 1 1 ia + 1 dp P (( A ( τ ) A ( 0) )( dm dc )) ( ( 0) ) B i dl 1 ( 0) + A + i+ =Δ + i< B 1 1 ib 1 dp P (( B ( τ ) B ( 0) )( dm dc )) ( ( 0) ) A i dl 1 ( 0) + B i = Δ + + i< A

21 Price dynamics The expressions above provide a naural inerpreaion of he price changes: hey are due o New limi orders ha fall wihin he spread, for which one can safely assume some independence assumpions Evens ha modify he bes quoes (eiher cancellaions or marke orders), for which he price changes depends on he firs gaps 1 A τ A 1 0 and 1 B τ B 1 0 ( ( ) ( )) ( ) ( ) ( ) The price process has he following represenaion i X dn A Bachelier marke has a similar represenaion wih i.i.d. marks The marks may be assumed o be idenically disribued (under saionariy), bu no independen. dp = i i The long ime dynamics is sensiive o he dependence srucure of hese processes

22 Price dynamics A mahemaical resul The cenered price process in a zero-inelligence model wih proporional cancellaion rae scales o a brownian moion in he long ime limi No a surprise from he physicis s poin of view A firs general resul relaing order book models and classical price models The spurrious randomness of he volailiy due o he memory of he order book vanishes exponenially fas in his simple case Exensions o sae dependen inensiies, Hawkes processes

23 Price dynamics The case of local (endogenous) or sochasic (exogenous) inensiies allows one o mimick some classical local and sochasic volailiy models The leaner he order book, he closer he dynamics is o sandard diffusion models Long memory may appear in he case of slow cancellaion raes, slow decay kernel

24 Order book modelling Conclusion Empirical sudies of he order book A large body of empirical resuls Condiional quaniies conain a lo of relevan informaion The behaviour of marke paricipans a he bes limis ends o conrol he dynamics of price and spread Mahemaical modelling A general framework suiable for many exensions An approach bridging he gap beween order book dynamics and price process

5. Stochastic processes (1)

5. Stochastic processes (1) Lec05.pp S-38.45 - Inroducion o Teleraffic Theory Spring 2005 Conens Basic conceps Poisson process 2 Sochasic processes () Consider some quaniy in a eleraffic (or any) sysem I ypically evolves in ime randomly

More information

Exponential Weighted Moving Average (EWMA) Chart Under The Assumption of Moderateness And Its 3 Control Limits

Exponential Weighted Moving Average (EWMA) Chart Under The Assumption of Moderateness And Its 3 Control Limits DOI: 0.545/mjis.07.5009 Exponenial Weighed Moving Average (EWMA) Char Under The Assumpion of Moderaeness And Is 3 Conrol Limis KALPESH S TAILOR Assisan Professor, Deparmen of Saisics, M. K. Bhavnagar Universiy,

More information

Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.

Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time. Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike

More information

Licenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A

Licenciatura de ADE y Licenciatura conjunta Derecho y ADE. Hoja de ejercicios 2 PARTE A Licenciaura de ADE y Licenciaura conjuna Derecho y ADE Hoja de ejercicios PARTE A 1. Consider he following models Δy = 0.8 + ε (1 + 0.8L) Δ 1 y = ε where ε and ε are independen whie noise processes. In

More information

Vehicle Arrival Models : Headway

Vehicle Arrival Models : Headway Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where

More information

Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate.

Introduction D P. r = constant discount rate, g = Gordon Model (1962): constant dividend growth rate. Inroducion Gordon Model (1962): D P = r g r = consan discoun rae, g = consan dividend growh rae. If raional expecaions of fuure discoun raes and dividend growh vary over ime, so should he D/P raio. Since

More information

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles

Diebold, Chapter 7. Francis X. Diebold, Elements of Forecasting, 4th Edition (Mason, Ohio: Cengage Learning, 2006). Chapter 7. Characterizing Cycles Diebold, Chaper 7 Francis X. Diebold, Elemens of Forecasing, 4h Ediion (Mason, Ohio: Cengage Learning, 006). Chaper 7. Characerizing Cycles Afer compleing his reading you should be able o: Define covariance

More information

in Engineering Prof. Dr. Michael Havbro Faber ETH Zurich, Switzerland Swiss Federal Institute of Technology

in Engineering Prof. Dr. Michael Havbro Faber ETH Zurich, Switzerland Swiss Federal Institute of Technology Risk and Saey in Engineering Pro. Dr. Michael Havbro Faber ETH Zurich, Swizerland Conens o Today's Lecure Inroducion o ime varian reliabiliy analysis The Poisson process The ormal process Assessmen o he

More information

Stationary Time Series

Stationary Time Series 3-Jul-3 Time Series Analysis Assoc. Prof. Dr. Sevap Kesel July 03 Saionary Time Series Sricly saionary process: If he oin dis. of is he same as he oin dis. of ( X,... X n) ( X h,... X nh) Weakly Saionary

More information

Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé

Bias in Conditional and Unconditional Fixed Effects Logit Estimation: a Correction * Tom Coupé Bias in Condiional and Uncondiional Fixed Effecs Logi Esimaion: a Correcion * Tom Coupé Economics Educaion and Research Consorium, Naional Universiy of Kyiv Mohyla Academy Address: Vul Voloska 10, 04070

More information

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,

More information

1. Consider a pure-exchange economy with stochastic endowments. The state of the economy

1. Consider a pure-exchange economy with stochastic endowments. The state of the economy Answer 4 of he following 5 quesions. 1. Consider a pure-exchange economy wih sochasic endowmens. The sae of he economy in period, 0,1,..., is he hisory of evens s ( s0, s1,..., s ). The iniial sae is given.

More information

Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution

Physics 127b: Statistical Mechanics. Fokker-Planck Equation. Time Evolution Physics 7b: Saisical Mechanics Fokker-Planck Equaion The Langevin equaion approach o he evoluion of he velociy disribuion for he Brownian paricle migh leave you uncomforable. A more formal reamen of his

More information

Elements of Stochastic Processes Lecture II Hamid R. Rabiee

Elements of Stochastic Processes Lecture II Hamid R. Rabiee Sochasic Processes Elemens of Sochasic Processes Lecure II Hamid R. Rabiee Overview Reading Assignmen Chaper 9 of exbook Furher Resources MIT Open Course Ware S. Karlin and H. M. Taylor, A Firs Course

More information

Understanding the asymptotic behaviour of empirical Bayes methods

Understanding the asymptotic behaviour of empirical Bayes methods Undersanding he asympoic behaviour of empirical Bayes mehods Boond Szabo, Aad van der Vaar and Harry van Zanen EURANDOM, 11.10.2011. Conens 2/20 Moivaion Nonparameric Bayesian saisics Signal in Whie noise

More information

Chapter 5. Heterocedastic Models. Introduction to time series (2008) 1

Chapter 5. Heterocedastic Models. Introduction to time series (2008) 1 Chaper 5 Heerocedasic Models Inroducion o ime series (2008) 1 Chaper 5. Conens. 5.1. The ARCH model. 5.2. The GARCH model. 5.3. The exponenial GARCH model. 5.4. The CHARMA model. 5.5. Random coefficien

More information

Econ Autocorrelation. Sanjaya DeSilva

Econ Autocorrelation. Sanjaya DeSilva Econ 39 - Auocorrelaion Sanjaya DeSilva Ocober 3, 008 1 Definiion Auocorrelaion (or serial correlaion) occurs when he error erm of one observaion is correlaed wih he error erm of any oher observaion. This

More information

DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND

DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND DEPARTMENT OF ECONOMICS AND FINANCE COLLEGE OF BUSINESS AND ECONOMICS UNIVERSITY OF CANTERBURY CHRISTCHURCH, NEW ZEALAND Asymmery and Leverage in Condiional Volailiy Models Michael McAleer WORKING PAPER

More information

Mathematical Theory and Modeling ISSN (Paper) ISSN (Online) Vol 3, No.3, 2013

Mathematical Theory and Modeling ISSN (Paper) ISSN (Online) Vol 3, No.3, 2013 Mahemaical Theory and Modeling ISSN -580 (Paper) ISSN 5-05 (Online) Vol, No., 0 www.iise.org The ffec of Inverse Transformaion on he Uni Mean and Consan Variance Assumpions of a Muliplicaive rror Model

More information

Richard A. Davis Colorado State University Bojan Basrak Eurandom Thomas Mikosch University of Groningen

Richard A. Davis Colorado State University Bojan Basrak Eurandom Thomas Mikosch University of Groningen Mulivariae Regular Variaion wih Applicaion o Financial Time Series Models Richard A. Davis Colorado Sae Universiy Bojan Basrak Eurandom Thomas Mikosch Universiy of Groningen Ouline + Characerisics of some

More information

A Dynamic Model of Economic Fluctuations

A Dynamic Model of Economic Fluctuations CHAPTER 15 A Dynamic Model of Economic Flucuaions Modified for ECON 2204 by Bob Murphy 2016 Worh Publishers, all righs reserved IN THIS CHAPTER, OU WILL LEARN: how o incorporae dynamics ino he AD-AS model

More information

EXCHANGE RATE ECONOMICS LECTURE 3 ASYMMETRIC INFORMATION AND EXCHANGE RATES. A. Portfolio Shifts Model and the Role of Order Flow

EXCHANGE RATE ECONOMICS LECTURE 3 ASYMMETRIC INFORMATION AND EXCHANGE RATES. A. Portfolio Shifts Model and the Role of Order Flow EXCHANGE RATE ECONOMICS LECTURE 3 ASYMMETRIC INFORMATION AND EXCHANGE RATES A. Porfolio Shifs Model and he Role of Order Flow Porfolio shifs by public cause exchange rae change no common knowledge when

More information

Financial Econometrics Jeffrey R. Russell Midterm Winter 2009 SOLUTIONS

Financial Econometrics Jeffrey R. Russell Midterm Winter 2009 SOLUTIONS Name SOLUTIONS Financial Economerics Jeffrey R. Russell Miderm Winer 009 SOLUTIONS You have 80 minues o complee he exam. Use can use a calculaor and noes. Try o fi all your work in he space provided. If

More information

Final Spring 2007

Final Spring 2007 .615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o

More information

Unit Root Time Series. Univariate random walk

Unit Root Time Series. Univariate random walk Uni Roo ime Series Univariae random walk Consider he regression y y where ~ iid N 0, he leas squares esimae of is: ˆ yy y y yy Now wha if = If y y hen le y 0 =0 so ha y j j If ~ iid N 0, hen y ~ N 0, he

More information

14 Autoregressive Moving Average Models

14 Autoregressive Moving Average Models 14 Auoregressive Moving Average Models In his chaper an imporan parameric family of saionary ime series is inroduced, he family of he auoregressive moving average, or ARMA, processes. For a large class

More information

ECON 482 / WH Hong Time Series Data Analysis 1. The Nature of Time Series Data. Example of time series data (inflation and unemployment rates)

ECON 482 / WH Hong Time Series Data Analysis 1. The Nature of Time Series Data. Example of time series data (inflation and unemployment rates) ECON 48 / WH Hong Time Series Daa Analysis. The Naure of Time Series Daa Example of ime series daa (inflaion and unemploymen raes) ECON 48 / WH Hong Time Series Daa Analysis The naure of ime series daa

More information

ACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H.

ACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H. ACE 564 Spring 2006 Lecure 7 Exensions of The Muliple Regression Model: Dumm Independen Variables b Professor Sco H. Irwin Readings: Griffihs, Hill and Judge. "Dumm Variables and Varing Coefficien Models

More information

Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds

Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Kriging Models Predicing Arazine Concenraions in Surface Waer Draining Agriculural Waersheds Paul L. Mosquin, Jeremy Aldworh, Wenlin Chen Supplemenal Maerial Number

More information

6. Stochastic calculus with jump processes

6. Stochastic calculus with jump processes A) Trading sraegies (1/3) Marke wih d asses S = (S 1,, S d ) A rading sraegy can be modelled wih a vecor φ describing he quaniies invesed in each asse a each insan : φ = (φ 1,, φ d ) The value a of a porfolio

More information

Dynamic Econometric Models: Y t = + 0 X t + 1 X t X t k X t-k + e t. A. Autoregressive Model:

Dynamic Econometric Models: Y t = + 0 X t + 1 X t X t k X t-k + e t. A. Autoregressive Model: Dynamic Economeric Models: A. Auoregressive Model: Y = + 0 X 1 Y -1 + 2 Y -2 + k Y -k + e (Wih lagged dependen variable(s) on he RHS) B. Disribued-lag Model: Y = + 0 X + 1 X -1 + 2 X -2 + + k X -k + e

More information

OBJECTIVES OF TIME SERIES ANALYSIS

OBJECTIVES OF TIME SERIES ANALYSIS OBJECTIVES OF TIME SERIES ANALYSIS Undersanding he dynamic or imedependen srucure of he observaions of a single series (univariae analysis) Forecasing of fuure observaions Asceraining he leading, lagging

More information

Stochastic Model for Cancer Cell Growth through Single Forward Mutation

Stochastic Model for Cancer Cell Growth through Single Forward Mutation Journal of Modern Applied Saisical Mehods Volume 16 Issue 1 Aricle 31 5-1-2017 Sochasic Model for Cancer Cell Growh hrough Single Forward Muaion Jayabharahiraj Jayabalan Pondicherry Universiy, jayabharahi8@gmail.com

More information

Appendix to Creating Work Breaks From Available Idleness

Appendix to Creating Work Breaks From Available Idleness Appendix o Creaing Work Breaks From Available Idleness Xu Sun and Ward Whi Deparmen of Indusrial Engineering and Operaions Research, Columbia Universiy, New York, NY, 127; {xs2235,ww24}@columbia.edu Sepember

More information

Sample Autocorrelations for Financial Time Series Models. Richard A. Davis Colorado State University Thomas Mikosch University of Copenhagen

Sample Autocorrelations for Financial Time Series Models. Richard A. Davis Colorado State University Thomas Mikosch University of Copenhagen Sample Auocorrelaions for Financial Time Series Models Richard A. Davis Colorado Sae Universiy Thomas Mikosch Universiy of Copenhagen Ouline Characerisics of some financial ime series IBM reurns NZ-USA

More information

Reliability of Technical Systems

Reliability of Technical Systems eliabiliy of Technical Sysems Main Topics Inroducion, Key erms, framing he problem eliabiliy parameers: Failure ae, Failure Probabiliy, Availabiliy, ec. Some imporan reliabiliy disribuions Componen reliabiliy

More information

Lecture Notes 2. The Hilbert Space Approach to Time Series

Lecture Notes 2. The Hilbert Space Approach to Time Series Time Series Seven N. Durlauf Universiy of Wisconsin. Basic ideas Lecure Noes. The Hilber Space Approach o Time Series The Hilber space framework provides a very powerful language for discussing he relaionship

More information

ACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H.

ACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H. ACE 56 Fall 005 Lecure 5: he Simple Linear Regression Model: Sampling Properies of he Leas Squares Esimaors by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Inference in he Simple

More information

Introduction to Probability and Statistics Slides 4 Chapter 4

Introduction to Probability and Statistics Slides 4 Chapter 4 Inroducion o Probabiliy and Saisics Slides 4 Chaper 4 Ammar M. Sarhan, asarhan@mahsa.dal.ca Deparmen of Mahemaics and Saisics, Dalhousie Universiy Fall Semeser 8 Dr. Ammar Sarhan Chaper 4 Coninuous Random

More information

Regular Variation and Financial Time Series Models

Regular Variation and Financial Time Series Models Regular Variaion and Financial Time Series Models Richard A. Davis Colorado Sae Universiy www.sa.colosae.edu/~rdavis Thomas Mikosch Universiy of Copenhagen Bojan Basrak Eurandom Ouline Characerisics of

More information

Macroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3

Macroeconomic Theory Ph.D. Qualifying Examination Fall 2005 ANSWER EACH PART IN A SEPARATE BLUE BOOK. PART ONE: ANSWER IN BOOK 1 WEIGHT 1/3 Macroeconomic Theory Ph.D. Qualifying Examinaion Fall 2005 Comprehensive Examinaion UCLA Dep. of Economics You have 4 hours o complee he exam. There are hree pars o he exam. Answer all pars. Each par has

More information

R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t

R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,

More information

Internet Traffic Modeling for Efficient Network Research Management Prof. Zhili Sun, UniS Zhiyong Liu, CATR

Internet Traffic Modeling for Efficient Network Research Management Prof. Zhili Sun, UniS Zhiyong Liu, CATR Inerne Traffic Modeling for Efficien Nework Research Managemen Prof. Zhili Sun, UniS Zhiyong Liu, CATR UK-China Science Bridge Workshop 13-14 December 2011, London Ouline Inroducion Background Classical

More information

How to Deal with Structural Breaks in Practical Cointegration Analysis

How to Deal with Structural Breaks in Practical Cointegration Analysis How o Deal wih Srucural Breaks in Pracical Coinegraion Analysis Roselyne Joyeux * School of Economic and Financial Sudies Macquarie Universiy December 00 ABSTRACT In his noe we consider he reamen of srucural

More information

An introduction to the theory of SDDP algorithm

An introduction to the theory of SDDP algorithm An inroducion o he heory of SDDP algorihm V. Leclère (ENPC) Augus 1, 2014 V. Leclère Inroducion o SDDP Augus 1, 2014 1 / 21 Inroducion Large scale sochasic problem are hard o solve. Two ways of aacking

More information

Part III: Chap. 2.5,2.6 & 12

Part III: Chap. 2.5,2.6 & 12 Survival Analysis Mah 434 Fall 2011 Par III: Chap. 2.5,2.6 & 12 Jimin Ding Mah Dep. www.mah.wusl.edu/ jmding/mah434/index.hml Jimin Ding, Ocober 4, 2011 Survival Analysis, Fall 2011 - p. 1/14 Jimin Ding,

More information

( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is

( ) ( ) if t = t. It must satisfy the identity. So, bulkiness of the unit impulse (hyper)function is equal to 1. The defining characteristic is UNIT IMPULSE RESPONSE, UNIT STEP RESPONSE, STABILITY. Uni impulse funcion (Dirac dela funcion, dela funcion) rigorously defined is no sricly a funcion, bu disribuion (or measure), precise reamen requires

More information

Chapter 12: Velocity, acceleration, and forces

Chapter 12: Velocity, acceleration, and forces To Feel a Force Chaper Spring, Chaper : A. Saes of moion For moion on or near he surface of he earh, i is naural o measure moion wih respec o objecs fixed o he earh. The 4 hr. roaion of he earh has a measurable

More information

1 Answers to Final Exam, ECN 200E, Spring

1 Answers to Final Exam, ECN 200E, Spring 1 Answers o Final Exam, ECN 200E, Spring 2004 1. A good answer would include he following elemens: The equiy premium puzzle demonsraed ha wih sandard (i.e ime separable and consan relaive risk aversion)

More information

Homework 10 (Stats 620, Winter 2017) Due Tuesday April 18, in class Questions are derived from problems in Stochastic Processes by S. Ross.

Homework 10 (Stats 620, Winter 2017) Due Tuesday April 18, in class Questions are derived from problems in Stochastic Processes by S. Ross. Homework (Sas 6, Winer 7 Due Tuesday April 8, in class Quesions are derived from problems in Sochasic Processes by S. Ross.. A sochasic process {X(, } is said o be saionary if X(,..., X( n has he same

More information

Exponentially Weighted Moving Average (EWMA) Chart Based on Six Delta Initiatives

Exponentially Weighted Moving Average (EWMA) Chart Based on Six Delta Initiatives hps://doi.org/0.545/mjis.08.600 Exponenially Weighed Moving Average (EWMA) Char Based on Six Dela Iniiaives KALPESH S. TAILOR Deparmen of Saisics, M. K. Bhavnagar Universiy, Bhavnagar-36400 E-mail: kalpesh_lr@yahoo.co.in

More information

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models

A Specification Test for Linear Dynamic Stochastic General Equilibrium Models Journal of Saisical and Economeric Mehods, vol.1, no.2, 2012, 65-70 ISSN: 2241-0384 (prin), 2241-0376 (online) Scienpress Ld, 2012 A Specificaion Tes for Linear Dynamic Sochasic General Equilibrium Models

More information

Stochastic Modelling in Finance - Solutions to sheet 8

Stochastic Modelling in Finance - Solutions to sheet 8 Sochasic Modelling in Finance - Soluions o shee 8 8.1 The price of a defaulable asse can be modeled as ds S = µ d + σ dw dn where µ, σ are consans, (W ) is a sandard Brownian moion and (N ) is a one jump

More information

Comparing Means: t-tests for One Sample & Two Related Samples

Comparing Means: t-tests for One Sample & Two Related Samples Comparing Means: -Tess for One Sample & Two Relaed Samples Using he z-tes: Assumpions -Tess for One Sample & Two Relaed Samples The z-es (of a sample mean agains a populaion mean) is based on he assumpion

More information

Sensors, Signals and Noise

Sensors, Signals and Noise Sensors, Signals and Noise COURSE OUTLINE Inroducion Signals and Noise: 1) Descripion Filering Sensors and associaed elecronics rv 2017/02/08 1 Noise Descripion Noise Waveforms and Samples Saisics of Noise

More information

Failure of the work-hamiltonian connection for free energy calculations. Abstract

Failure of the work-hamiltonian connection for free energy calculations. Abstract Failure of he work-hamilonian connecion for free energy calculaions Jose M. G. Vilar 1 and J. Miguel Rubi 1 Compuaional Biology Program, Memorial Sloan-Keering Cancer Cener, 175 York Avenue, New York,

More information

Inventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions

Inventory Analysis and Management. Multi-Period Stochastic Models: Optimality of (s, S) Policy for K-Convex Objective Functions Muli-Period Sochasic Models: Opimali of (s, S) Polic for -Convex Objecive Funcions Consider a seing similar o he N-sage newsvendor problem excep ha now here is a fixed re-ordering cos (> 0) for each (re-)order.

More information

Zürich. ETH Master Course: L Autonomous Mobile Robots Localization II

Zürich. ETH Master Course: L Autonomous Mobile Robots Localization II Roland Siegwar Margaria Chli Paul Furgale Marco Huer Marin Rufli Davide Scaramuzza ETH Maser Course: 151-0854-00L Auonomous Mobile Robos Localizaion II ACT and SEE For all do, (predicion updae / ACT),

More information

The Brock-Mirman Stochastic Growth Model

The Brock-Mirman Stochastic Growth Model c December 3, 208, Chrisopher D. Carroll BrockMirman The Brock-Mirman Sochasic Growh Model Brock and Mirman (972) provided he firs opimizing growh model wih unpredicable (sochasic) shocks. The social planner

More information

T L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB

T L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal

More information

Regression with Time Series Data

Regression with Time Series Data Regression wih Time Series Daa y = β 0 + β 1 x 1 +...+ β k x k + u Serial Correlaion and Heeroskedasiciy Time Series - Serial Correlaion and Heeroskedasiciy 1 Serially Correlaed Errors: Consequences Wih

More information

The consumption-based determinants of the term structure of discount rates: Corrigendum. Christian Gollier 1 Toulouse School of Economics March 2012

The consumption-based determinants of the term structure of discount rates: Corrigendum. Christian Gollier 1 Toulouse School of Economics March 2012 The consumpion-based deerminans of he erm srucure of discoun raes: Corrigendum Chrisian Gollier Toulouse School of Economics March 0 In Gollier (007), I examine he effec of serially correlaed growh raes

More information

Inventory Control of Perishable Items in a Two-Echelon Supply Chain

Inventory Control of Perishable Items in a Two-Echelon Supply Chain Journal of Indusrial Engineering, Universiy of ehran, Special Issue,, PP. 69-77 69 Invenory Conrol of Perishable Iems in a wo-echelon Supply Chain Fariborz Jolai *, Elmira Gheisariha and Farnaz Nojavan

More information

Two Coupled Oscillators / Normal Modes

Two Coupled Oscillators / Normal Modes Lecure 3 Phys 3750 Two Coupled Oscillaors / Normal Modes Overview and Moivaion: Today we ake a small, bu significan, sep owards wave moion. We will no ye observe waves, bu his sep is imporan in is own

More information

Lecture #31, 32: The Ornstein-Uhlenbeck Process as a Model of Volatility

Lecture #31, 32: The Ornstein-Uhlenbeck Process as a Model of Volatility Saisics 441 (Fall 214) November 19, 21, 214 Prof Michael Kozdron Lecure #31, 32: The Ornsein-Uhlenbeck Process as a Model of Volailiy The Ornsein-Uhlenbeck process is a di usion process ha was inroduced

More information

On Multicomponent System Reliability with Microshocks - Microdamages Type of Components Interaction

On Multicomponent System Reliability with Microshocks - Microdamages Type of Components Interaction On Mulicomponen Sysem Reliabiliy wih Microshocks - Microdamages Type of Componens Ineracion Jerzy K. Filus, and Lidia Z. Filus Absrac Consider a wo componen parallel sysem. The defined new sochasic dependences

More information

Robust estimation based on the first- and third-moment restrictions of the power transformation model

Robust estimation based on the first- and third-moment restrictions of the power transformation model h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,

More information

Cash Flow Valuation Mode Lin Discrete Time

Cash Flow Valuation Mode Lin Discrete Time IOSR Journal of Mahemaics (IOSR-JM) e-issn: 2278-5728,p-ISSN: 2319-765X, 6, Issue 6 (May. - Jun. 2013), PP 35-41 Cash Flow Valuaion Mode Lin Discree Time Olayiwola. M. A. and Oni, N. O. Deparmen of Mahemaics

More information

Simulating models with heterogeneous agents

Simulating models with heterogeneous agents Simulaing models wih heerogeneous agens Wouer J. Den Haan London School of Economics c by Wouer J. Den Haan Individual agen Subjec o employmen shocks (ε i, {0, 1}) Incomplee markes only way o save is hrough

More information

Institute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler

Institute for Mathematical Methods in Economics. University of Technology Vienna. Singapore, May Manfred Deistler MULTIVARIATE TIME SERIES ANALYSIS AND FORECASTING Manfred Deisler E O S Economerics and Sysems Theory Insiue for Mahemaical Mehods in Economics Universiy of Technology Vienna Singapore, May 2004 Inroducion

More information

Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi

Navneet Saini, Mayank Goyal, Vishal Bansal (2013); Term Project AML310; Indian Institute of Technology Delhi Creep in Viscoelasic Subsances Numerical mehods o calculae he coefficiens of he Prony equaion using creep es daa and Herediary Inegrals Mehod Navnee Saini, Mayank Goyal, Vishal Bansal (23); Term Projec

More information

Optimal Investment under Dynamic Risk Constraints and Partial Information

Optimal Investment under Dynamic Risk Constraints and Partial Information Opimal Invesmen under Dynamic Risk Consrains and Parial Informaion Wolfgang Puschögl Johann Radon Insiue for Compuaional and Applied Mahemaics (RICAM) Ausrian Academy of Sciences www.ricam.oeaw.ac.a 2

More information

Nonstationarity-Integrated Models. Time Series Analysis Dr. Sevtap Kestel 1

Nonstationarity-Integrated Models. Time Series Analysis Dr. Sevtap Kestel 1 Nonsaionariy-Inegraed Models Time Series Analysis Dr. Sevap Kesel 1 Diagnosic Checking Residual Analysis: Whie noise. P-P or Q-Q plos of he residuals follow a normal disribuion, he series is called a Gaussian

More information

AMartingaleApproachforFractionalBrownian Motions and Related Path Dependent PDEs

AMartingaleApproachforFractionalBrownian Motions and Related Path Dependent PDEs AMaringaleApproachforFracionalBrownian Moions and Relaed Pah Dependen PDEs Jianfeng ZHANG Universiy of Souhern California Join work wih Frederi VIENS Mahemaical Finance, Probabiliy, and PDE Conference

More information

Chapter 7 Response of First-order RL and RC Circuits

Chapter 7 Response of First-order RL and RC Circuits Chaper 7 Response of Firs-order RL and RC Circuis 7.- The Naural Response of RL and RC Circuis 7.3 The Sep Response of RL and RC Circuis 7.4 A General Soluion for Sep and Naural Responses 7.5 Sequenial

More information

Statistics versus mean-field limit for Hawkes process. with Sylvain Delattre (P7)

Statistics versus mean-field limit for Hawkes process. with Sylvain Delattre (P7) Saisics versus mean-field limi for Hawkes process wih Sylvain Delare (P7) The model We have individuals. Z i, := number of acions of he i-h individual unil ime. Z i, jumps (is increased by 1) a rae λ i,

More information

Estimation of Poses with Particle Filters

Estimation of Poses with Particle Filters Esimaion of Poses wih Paricle Filers Dr.-Ing. Bernd Ludwig Chair for Arificial Inelligence Deparmen of Compuer Science Friedrich-Alexander-Universiä Erlangen-Nürnberg 12/05/2008 Dr.-Ing. Bernd Ludwig (FAU

More information

Empirical Process Theory

Empirical Process Theory Empirical Process heory 4.384 ime Series Analysis, Fall 27 Reciaion by Paul Schrimpf Supplemenary o lecures given by Anna Mikusheva Ocober 7, 28 Reciaion 7 Empirical Process heory Le x be a real-valued

More information

Let us start with a two dimensional case. We consider a vector ( x,

Let us start with a two dimensional case. We consider a vector ( x, Roaion marices We consider now roaion marices in wo and hree dimensions. We sar wih wo dimensions since wo dimensions are easier han hree o undersand, and one dimension is a lile oo simple. However, our

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION DOI: 0.038/NCLIMATE893 Temporal resoluion and DICE * Supplemenal Informaion Alex L. Maren and Sephen C. Newbold Naional Cener for Environmenal Economics, US Environmenal Proecion

More information

ACE 562 Fall Lecture 8: The Simple Linear Regression Model: R 2, Reporting the Results and Prediction. by Professor Scott H.

ACE 562 Fall Lecture 8: The Simple Linear Regression Model: R 2, Reporting the Results and Prediction. by Professor Scott H. ACE 56 Fall 5 Lecure 8: The Simple Linear Regression Model: R, Reporing he Resuls and Predicion by Professor Sco H. Irwin Required Readings: Griffihs, Hill and Judge. "Explaining Variaion in he Dependen

More information

1. Diagnostic (Misspeci cation) Tests: Testing the Assumptions

1. Diagnostic (Misspeci cation) Tests: Testing the Assumptions Business School, Brunel Universiy MSc. EC5501/5509 Modelling Financial Decisions and Markes/Inroducion o Quaniaive Mehods Prof. Menelaos Karanasos (Room SS269, el. 01895265284) Lecure Noes 6 1. Diagnosic

More information

13.3 Term structure models

13.3 Term structure models 13.3 Term srucure models 13.3.1 Expecaions hypohesis model - Simples "model" a) shor rae b) expecaions o ge oher prices Resul: y () = 1 h +1 δ = φ( δ)+ε +1 f () = E (y +1) (1) =δ + φ( δ) f (3) = E (y +)

More information

Vectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1

Vectorautoregressive Model and Cointegration Analysis. Time Series Analysis Dr. Sevtap Kestel 1 Vecorauoregressive Model and Coinegraion Analysis Par V Time Series Analysis Dr. Sevap Kesel 1 Vecorauoregression Vecor auoregression (VAR) is an economeric model used o capure he evoluion and he inerdependencies

More information

Application of Speed Transform to the diagnosis of a roller bearing in variable speed

Application of Speed Transform to the diagnosis of a roller bearing in variable speed Applicaion of Speed Transform o he diagnosis of a roller bearing in variable speed Julien Roussel 1, Michel Hariopoulos 1, Edgard Sekko 1, Cécile Capdessus 1 and Jérôme Anoni 1 PRISME laboraory 1 rue de

More information

Computer Simulates the Effect of Internal Restriction on Residuals in Linear Regression Model with First-order Autoregressive Procedures

Computer Simulates the Effect of Internal Restriction on Residuals in Linear Regression Model with First-order Autoregressive Procedures MPRA Munich Personal RePEc Archive Compuer Simulaes he Effec of Inernal Resricion on Residuals in Linear Regression Model wih Firs-order Auoregressive Procedures Mei-Yu Lee Deparmen of Applied Finance,

More information

Lecture 2 April 04, 2018

Lecture 2 April 04, 2018 Sas 300C: Theory of Saisics Spring 208 Lecure 2 April 04, 208 Prof. Emmanuel Candes Scribe: Paulo Orensein; edied by Sephen Baes, XY Han Ouline Agenda: Global esing. Needle in a Haysack Problem 2. Threshold

More information

I. Return Calculations (20 pts, 4 points each)

I. Return Calculations (20 pts, 4 points each) Universiy of Washingon Spring 015 Deparmen of Economics Eric Zivo Econ 44 Miderm Exam Soluions This is a closed book and closed noe exam. However, you are allowed one page of noes (8.5 by 11 or A4 double-sided)

More information

A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients

A Note on the Equivalence of Fractional Relaxation Equations to Differential Equations with Varying Coefficients mahemaics Aricle A Noe on he Equivalence of Fracional Relaxaion Equaions o Differenial Equaions wih Varying Coefficiens Francesco Mainardi Deparmen of Physics and Asronomy, Universiy of Bologna, and he

More information

The expectation value of the field operator.

The expectation value of the field operator. The expecaion value of he field operaor. Dan Solomon Universiy of Illinois Chicago, IL dsolom@uic.edu June, 04 Absrac. Much of he mahemaical developmen of quanum field heory has been in suppor of deermining

More information

2. Nonlinear Conservation Law Equations

2. Nonlinear Conservation Law Equations . Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear

More information

Økonomisk Kandidateksamen 2005(II) Econometrics 2. Solution

Økonomisk Kandidateksamen 2005(II) Econometrics 2. Solution Økonomisk Kandidaeksamen 2005(II) Economerics 2 Soluion his is he proposed soluion for he exam in Economerics 2. For compleeness he soluion gives formal answers o mos of he quesions alhough his is no always

More information

Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims

Problem Set 5. Graduate Macro II, Spring 2017 The University of Notre Dame Professor Sims Problem Se 5 Graduae Macro II, Spring 2017 The Universiy of Nore Dame Professor Sims Insrucions: You may consul wih oher members of he class, bu please make sure o urn in your own work. Where applicable,

More information

A unit root test based on smooth transitions and nonlinear adjustment

A unit root test based on smooth transitions and nonlinear adjustment MPRA Munich Personal RePEc Archive A uni roo es based on smooh ransiions and nonlinear adjusmen Aycan Hepsag Isanbul Universiy 5 Ocober 2017 Online a hps://mpra.ub.uni-muenchen.de/81788/ MPRA Paper No.

More information

Quarterly ice cream sales are high each summer, and the series tends to repeat itself each year, so that the seasonal period is 4.

Quarterly ice cream sales are high each summer, and the series tends to repeat itself each year, so that the seasonal period is 4. Seasonal models Many business and economic ime series conain a seasonal componen ha repeas iself afer a regular period of ime. The smalles ime period for his repeiion is called he seasonal period, and

More information

Some Ramsey results for the n-cube

Some Ramsey results for the n-cube Some Ramsey resuls for he n-cube Ron Graham Universiy of California, San Diego Jozsef Solymosi Universiy of Briish Columbia, Vancouver, Canada Absrac In his noe we esablish a Ramsey-ype resul for cerain

More information

A Study of Inventory System with Ramp Type Demand Rate and Shortage in The Light Of Inflation I

A Study of Inventory System with Ramp Type Demand Rate and Shortage in The Light Of Inflation I Inernaional Journal of Mahemaics rends and echnology Volume 7 Number Jan 5 A Sudy of Invenory Sysem wih Ramp ype emand Rae and Shorage in he Ligh Of Inflaion I Sangeea Gupa, R.K. Srivasava, A.K. Singh

More information

HIGGS&AT&HADRON&COLLIDER

HIGGS&AT&HADRON&COLLIDER IGGS&AT&ADRON&COLLIDER iggs&proper,es&and&precision&tes& Lecure&1& Shahram&Rahalou Fisica&delle&Par,celle&Elemenari,&Anno&Accademico&014815 hp://www.roma1.infn.i/people/rahalou/paricelle/ WY&AND&WIC&BOSON?

More information

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check

More information

THE TERM STRUCTURE OF INTEREST RATES IN A MARKOV SETTING

THE TERM STRUCTURE OF INTEREST RATES IN A MARKOV SETTING THE TERM STRUCTURE OF INTEREST RATES IN A MARKOV SETTING Rober J. Ellio Haskayne School of Business Universiy of Calgary Calgary, Albera, Canada rellio@ucalgary.ca Craig A. Wilson College of Commerce Universiy

More information