Regular Variation and Financial Time Series Models
|
|
- Elaine Brook Ray
- 5 years ago
- Views:
Transcription
1 Regular Variaion and Financial Time Series Models Richard A. Davis Colorado Sae Universiy Thomas Mikosch Universiy of Copenhagen Bojan Basrak Eurandom
2 Ouline Characerisics of some financial ime series IBM reurns Muliplicaive models for log-reurns (GARCH, SV) Regular variaion univariae case mulivariae case new characerizaion: is RV c is RV? Applicaions of regular variaion Sochasic recurrence equaions (GARCH) Poin process convergence Wrap-up Exremes and exremal index Limi behavior of sample correlaions
3 Characerisics of some financial ime series Define = ln (P ) - ln (P - ) (log reurns) heavy ailed P( > x) ~ C x α, 0 < α < 4. uncorrelaed ˆ ρ ( h) near 0 for all lags h > 0 (MGD sequence) and have slowly decaying auocorrelaions ρˆ ( ) and ˆ h ρ ( h) converge o 0 slowly as h increases. process exhibis volailiy clusering. 3
4 Log reurns for IBM /3/6-/3/00 (blue=96-98) 00*log(reurns) ime 4
5 Sample ACF IBM (a) 96-98, (b) (a) ACF of IBM (s half) (b) ACF of IBM (nd half) ACF ACF Lag Lag 5
6 Sample ACF of abs values for IBM (a) 96-98, (b) (a) ACF, Abs Values of IBM (s half) (b) ACF, Abs Values of IBM (nd half) ACF ACF Lag Lag 6
7 Hill s plo of ail index for IBM (96-98, ) Hill Hill m m 7
8 Muliplicaive models for log(reurns) Basic model = ln (P ) - ln (P - ) (log reurns) = σ Z, where {Z } is IID wih mean 0, variance (if exiss). (e.g. N(0,) or a -disribuion wih ν df.) {σ } is he volailiy process σ and Z are independen. Properies: E = 0, Cov(, +h ) = 0, h>0 (uncorrelaed if Var( ) < ) condiional heeroscedasic (condiion on σ ). 8
9 Muliplicaive models for log(reurns)-con = σ Z (observaion eqn in sae-space formulaion) Two classes of models for volailiy: (i) GARCH(p,q) process (General AuoRegressive Condiional Heeroscedasic-observaion-driven specificaion) σ = α 0 + α - + L+ α p -p +β σ - + L+β q σ -q. Special case: ARCH(): = ( α = α 0 Z = A +α B - +α )Z 0 Z (sochasic recurrence eqn) h ρ ( h) = α, if α < /3. 9
10 Muliplicaive models for log(reurns)-con GARCH(,): = σ Z, σ = α 0 + α - + α - +β σ -. Then Y = (, )' follows he SRE given by σ - σ - αz = Z - - +β α σ α 0 0 Quesions: Exisence of a unique saionary soluion o he SRE? Regular variaion of he join disribuions? 0
11 Muliplicaive models for log(reurns)-con = σ Z (observaion eqn in sae-space formulaion) (ii) sochasic volailiy process (parameer-driven specificaion) logσ = ψ jε j, ψ j j= j= <,{ ε }~ IIDN(0, σ ) ρ 4 ( h) = Cor( σ, σ h)/ EZ + Quesion: Join disribuions of process regularly varying if disr of Z is regularly varying?
12 Regular variaion univariae case Def: The random variable is regularly varying wih index α if P( > x)/p( >) x α and P(> )/P( >) p, or, equivalenly, if P(> x)/p( >) px α and P(< x)/p( >) qx α, where 0 p and p+q=. Equivalence: is RV(α) if and only if P( ) /P( >) v µ( ) ( v vague convergence of measures on R\{0}). In his case, µ(dx) = (pα x α I(x>0) + qα (-x) -α I(x<0)) dx Noe: µ(a) = -α µ(a) for every and A bounded away from 0.
13 Regular variaion univariae case (con) Anoher formulaion (polar coordinaes): Define he ± valued rv θ, P(θ = ) = p, P(θ = ) = p = q. Then is RV(α) if and only if or P( x, / S) P( > ) > α x P( θ S) P( x,/ ) P( > ) > α v x P( θ ) ( v vague convergence of measures on S 0 = {-,}). 3
14 Regular variaion mulivariae case Mulivariae regular variaion of =(,..., m ): There exiss a random vecor θ S m- such ha Equivalence: P( > x, / )/P( >) v x α P( θ ) ( v vague convergence on S m-, uni sphere in R m ). P( θ ) is called he specral measure α is he index of. P( ) v µ ( ) P ( > ) µ is a measure on R m which saisfies for x > 0 and A bounded away from 0, µ(xb) = x α µ(xa). 4
15 Regular variaion mulivariae case (con) Examples:. If > 0 and > 0 are iid RV(α), hen = (, ) is mulivariae regularly varying wih index α and specral disribuion P( θ =(0,) ) = P( θ =(,0) ) =.5 (mass on axes). Inerpreaion: Unlikely ha and are very large a he same ime. Figure: plo of (, ) for realizaion of 0,000. x_ x_ 5
16 . If = > 0, hen = (, ) is mulivariae regularly varying wih index α and specral disribuion P( θ = (/, / ) ) =. 3. AR(): = Z, {Z }~IID symmeric sable (.8) { ±(,.9)/sqr(.8), W.P Disr of θ: ±(0,), W.P..00 Figure: plo of (, + ) for realizaion of 0,000. x_{+} x_ 6
17 Applicaions of mulivariae regular variaion Domain of aracion for sums of iid random vecors (Rvaceva, 96). Tha is, when does he parial sum a n = converge for some consans a n? n Specral measure of mulivariae sable vecors. Domain of aracion for componenwise maxima of iid random vecors (Resnick, 987). Limi behavior of a n n = Weak convergence of poin processes wih iid poins. Soluion o sochasic recurrence equaions, Y = A Y - + B Weak convergence of sample auocovariances. 7
18 Operaions on regularly varying vecors producs Producs (Breiman 965). Suppose, Y > 0 are independen wih ~RV(α) and EY α+ε < for some ε > 0. Then Y ~ RV(α) wih P(Y > x) ~ EY α P( > x). Mulivariae version. Suppose he random vecor is regularly varying and A is a marix independen of wih Then 0 < E A α+ε <. A is regularly varying wih index α. 8
19 Applicaions of mulivariae regular variaion (con) Linear combinaions: ~RV(α) all linear combinaions of are regularly varying i.e., here exis α and slowly varying fcn L(.), s.. P(c T > )/( -α L()) w(c), exiss for all real-valued c, where w(c) = α w(c). Use vague convergence wih A c ={y: c T y > }, i.e., T P( A c ) P ( c > ) = µ (A ) : w( ), c = c α L ( ) P( > ) A c where -α L() = P( > ). 9
20 Converse? Applicaions of mulivariae regular variaion (con) ~RV(α) all linear combinaions of are regularly varying? There exis α and slowly varying fcn L(.), s.. (LC) P(c T > )/( -α L()) w(c), exiss for all real-valued c. Theorem (Basrak, Davis, Mikosch, `0). Le be a random vecor.. If saisfies (LC) wih α non-ineger, hen is RV(α).. If > 0 saisfies (LC) for non-negaive c and α is non-ineger, hen is RV(α). 3. If > 0 saisfies (LC) wih α an odd ineger, hen is RV(α). 0
21 Applicaions of mulivariae regular variaion (con) Idea of argumen: Define he measures m ( )= P( )/ ( -α L()) By assumpion we know ha for fixed c, m (A c ) µ(a c ). {m } is igh: For B bded away from 0, sup m (B) <. Do subsequenial limis of {m } coincide? If m ' v µ and m '' v µ, hen µ (A c ) = µ (A c ) for all c 0. A c Problem: Need µ = µ bu only have equaliy on A c, no a π-sysem. In general, equaliy need no hold (see Ex in Meerschaer & Scheffler (00)).
22 Applicaions of heorem. Kesen (973). Under general condiions, (LC) holds wih L()= for sochasic recurrence equaions of he form Y = A Y - + B, (A, B ) ~ IID, A d d random marices, B random d-vecors. I follows ha he disribuions of Y, and in fac all of he finie dim l disrs of Y are regularly varying (if α is non-even).. GARCH processes. Since squares of a GARCH process can be embedded in a SRE, he finie dimensional disribuions of a GARCH are regularly varying.
23 Examples Example of ARCH(): =(α 0 +α -) / Z, {Z }~IID. α found by solving E α Z α/ =. α α Disr of θ: P(θ ) = E{ (B,Z) α I(arg((B,Z)) )}/ E (B,Z) α where P(B = ) = P(B = -) =.5 3
24 Examples (con) Example of ARCH(): α 0 =, α =, α=, =(α 0 +α -) / Z, {Z }~IID Figures: plos of (, + ) and esimaed disribuion of θ for realizaion of 0,000. x_{+} x_ hea 4
25 Applicaions of heorem (con) Example: SV model = σ Z Suppose Z ~ RV(α) and logσ = j= ψ ε j j, j= ψ j <,{ ε }~ IIDN(0, σ ). Then Z n =(Z,,Z n ) is regulary varying wih index α and so is n = (,, n ) = diag(σ,, σ n ) Z n wih specral disribuion concenraed on (±,0), (0, ±). Figure: plo of (, + ) for realizaion of 0,000. x_ x_ 5
26 6 Poin process applicaion Theorem Le { } be an iid sequence of random vecors saisfying of he 3 condiions in he heorem. Then if and only if for every c 0 where {a n } saisfies np( > a n ), and N is a Poisson process wih inensiy measure µ. {P i } are Poisson ps corresponding o he radial par, i.e., has inensiy measure α x α (dx). {θ i } are iid wih he specral disribuion given by he RV, : : ' / ', = = ε = ε = j P d n a n i i n N N θ c c c c, : : / = = ε = ε = j P d n a n i i n N N θ
27 Poin process convergence Theorem (Davis & Hsing `95, Davis & Mikosch `97). Le { } be a saionary sequence of random m-vecors. Suppose (i) finie dimensional disribuions are joinly regularly varying (le (θ k,..., θ k ) be he vecor in S (k+)m- in he definiion). (ii) mixing condiion A (a n ) or srong mixing. (iii) Then limlimsupp( > an y 0 k n k r exiss. If γ > 0, hen n > a n y) = 0. k ( k) α ( k) ( k) α γ = lime( θ0 θ j ) + / E θ0 k j= N n : = n = ε a d N : = / n Pi ij i= j= ε Q, (exremal index) 7
28 Poin process convergence(con) (P i ) are poins of a Poisson process on (0, ) wih inensiy funcion ν(dy)=γαy α dy. j= εq, i, are iid poin process wih disribuion Q, and Q is he weak limi of ij k k k ( k) α ( k) ( k) α ( k) 0 θ j ) + I ( ε ( k )/ E( θ θ θ j + j= 0 ) j= k lim E( θ ) Remarks:. GARCH and SV processes saisfy he condiions of he heorem.. Limi disribuion for sample exremes and sample ACF follows from his heorem. 8
29 Exremes for GARCH and SV processes Seup = σ Z, {Z } ~ IID (0,) is RV (α) Choose {b n } s.. np( > b n ) Then P n ( b x) exp{ x α n }. Then, wih M n = max{,..., n }, (i) GARCH: P( b γ is exremal index ( 0 < γ < ). (ii) SV model: P( b M x) exp{ γx α n n M x) exp{ x }, α n n }, exremal index γ = no clusering. 9
30 (i) GARCH: P( b (ii) SV model: P( b Exremes for GARCH and SV processes (con) M x) exp{ γx α n n M x) exp{ x α n n } } Remarks abou exremal index. (i) (ii) γ < implies clusering of exceedances Numerical example. Suppose c is a hreshold such ha Then, if γ =.5, (iii) /γ is he mean cluser size of exceedances. (iv) Use γ o discriminae beween GARCH and SV models. (v) P n ( bn P( b n M n c) ~.95 c) ~ (.95) =.975 Even for he ligh-ailed SV model (i.e., {Z } ~IID N(0,), he exremal index is (see Breid and Davis `98 ).5 30
31 Exremes for GARCH and SV processes (con) ** * * *** *** ime 3
32 Summary of resuls for ACF of GARCH(p,q) and SV models GARCH(p,q) α (0,): d ( ρˆ ( h)) h=, K, m ( Vh / V0 ) h=, K, m, α (,4): α (4, ): / α d ( n ρ ( h) ) γ (0)( V ). ˆ h=, K, m h h=, K, m / d ( n ρ ( h) ) (0)( G ). ˆ γ h=, K, m h h=, K, m Remark: Similar resuls hold for he sample ACF based on and. 3
33 Summary of resuls for ACF of GARCH(p,q) and SV models (con) SV Model α (0,): / α d h+ α h ( n / ln n) ρˆ ( h). σ σ σ α S S 0 α (, ): / d ( n ρ ( h) ) (0)( G ). ˆ γ h=, K, m h h=, K, m 33
34 Sample ACF for GARCH and SV Models (000 reps) (a) GARCH(,) Model, n= (b) SV Model, n=
35 Sample ACF for Squares of GARCH (000 reps) (a) GARCH(,) Model, n= b) GARCH(,) Model, n=
36 Sample ACF for Squares of SV (000 reps) (c) SV Model, n= (d) SV Model, n=
37 .40 Amazon reurns May 6, 997 o June 6, 004. Series Residual ACF: Abs values Residual ACF: Squares
38 Wrap-up Regular variaion is a flexible ool for modeling boh dependence and ail heaviness. Useful for esablishing poin process convergence of heavy-ailed ime series. Exremal index γ < for GARCH and γ =for SV. Unresolved issues relaed o RV (LC) α = n? here is an example for which, > 0, and (c, ) and (c, ) have he same limis for all c > 0. α = n and > 0 (no rue in general). 38
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 informationSample 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 informationNonlinear Time Series Modeling
Nonlinear Time Series Modeling Part II: Time Series Models in Finance Richard A. Davis Colorado State University (http://www.stat.colostate.edu/~rdavis/lectures) MaPhySto Workshop Copenhagen September
More information14 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 informationInstitute 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 informationChapter 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 informationRecent Developments in the Unit Root Problem for Moving Averages
Recen Developmens in he Uni Roo Problem for Moving Averages Richard A. Davis Colorado Sae Universiy Mei-Ching Chen Chaoyang Insiue of echnology homas Miosch Universiy of Groningen Non-inverible MA() Model
More informationOBJECTIVES 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 informationElements 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 informationFinancial 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 informationHomework 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 informationStationary 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 informationEMS SCM joint meeting. On stochastic partial differential equations of parabolic type
EMS SCM join meeing Barcelona, May 28-30, 2015 On sochasic parial differenial equaions of parabolic ype Isván Gyöngy School of Mahemaics and Maxwell Insiue Edinburgh Universiy 1 I. Filering problem II.
More informationI. 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 informationStructural Break Detection in Time Series Models
Srucural Break Deecion in Time Series Models Richard A. Davis Thomas Lee Gabriel Rodriguez-Yam Colorado Sae Universiy (hp://www.sa.colosae.edu/~rdavis/lecures) This research suppored in par by an IBM faculy
More informationLecture 20: Riccati Equations and Least Squares Feedback Control
34-5 LINEAR SYSTEMS Lecure : Riccai Equaions and Leas Squares Feedback Conrol 5.6.4 Sae Feedback via Riccai Equaions A recursive approach in generaing he marix-valued funcion W ( ) equaion for i for he
More informationDYNAMIC ECONOMETRIC MODELS vol NICHOLAS COPERNICUS UNIVERSITY - TORUŃ Józef Stawicki and Joanna Górka Nicholas Copernicus University
DYNAMIC ECONOMETRIC MODELS vol.. - NICHOLAS COPERNICUS UNIVERSITY - TORUŃ 996 Józef Sawicki and Joanna Górka Nicholas Copernicus Universiy ARMA represenaion for a sum of auoregressive processes In he ime
More informationVectorautoregressive 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 information12: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME. Σ j =
1: AUTOREGRESSIVE AND MOVING AVERAGE PROCESSES IN DISCRETE TIME Moving Averages Recall ha a whie noise process is a series { } = having variance σ. The whie noise process has specral densiy f (λ) = of
More informationUnit 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 informationDiebold, 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 information1 Solutions to selected problems
1 Soluions o seleced problems 1. Le A B R n. Show ha in A in B bu in general bd A bd B. Soluion. Le x in A. Then here is ɛ > 0 such ha B ɛ (x) A B. This shows x in B. If A = [0, 1] and B = [0, 2], hen
More informationAn Introduction to Malliavin calculus and its applications
An Inroducion o Malliavin calculus and is applicaions Lecure 5: Smoohness of he densiy and Hörmander s heorem David Nualar Deparmen of Mahemaics Kansas Universiy Universiy of Wyoming Summer School 214
More informationØkonomisk 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 informationR 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 informationnon -negative cone Population dynamics motivates the study of linear models whose coefficient matrices are non-negative or positive.
LECTURE 3 Linear/Nonnegaive Marix Models x ( = Px ( A= m m marix, x= m vecor Linear sysems of difference equaions arise in several difference conexs: Linear approximaions (linearizaion Perurbaion analysis
More informationLicenciatura 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 informationDEPARTMENT 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 informationLecture 4: Processes with independent increments
Lecure 4: Processes wih independen incremens 1. A Wienner process 1.1 Definiion of a Wienner process 1.2 Reflecion principle 1.3 Exponenial Brownian moion 1.4 Exchange of measure (Girsanov heorem) 1.5
More informationStructural Break Detection in Time Series Models
Srucural Break Deecion in Time Series Models Richard A. Davis Thomas Lee Gabriel Rodriguez-Yam Colorado Sae Universiy (hp://www.sa.colosae.edu/~rdavis/lecures) This research suppored in par by: Naional
More informationNature 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 informationDiscrete Markov Processes. 1. Introduction
Discree Markov Processes 1. Inroducion 1. Probabiliy Spaces and Random Variables Sample space. A model for evens: is a family of subses of such ha c (1) if A, hen A, (2) if A 1, A 2,..., hen A1 A 2...,
More informationState-Space Models. Initialization, Estimation and Smoothing of the Kalman Filter
Sae-Space Models Iniializaion, Esimaion and Smoohing of he Kalman Filer Iniializaion of he Kalman Filer The Kalman filer shows how o updae pas predicors and he corresponding predicion error variances when
More informationEcon107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)
I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression
More informationMultivariate Generalized Ornstein-Uhlenbeck Processes
Mulivariae Generalized Ornsein-Uhlenbeck Processes Ania Behme and Alexander Lindner Absrac De Haan and Karandikar [12] inroduced generalized Ornsein Uhlenbeck processes as one-dimensional processes (V
More informationTransform Techniques. Moment Generating Function
Transform Techniques A convenien way of finding he momens of a random variable is he momen generaing funcion (MGF). Oher ransform echniques are characerisic funcion, z-ransform, and Laplace ransform. Momen
More information5. 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 informationSchool and Workshop on Market Microstructure: Design, Efficiency and Statistical Regularities March 2011
2229-12 School and Workshop on Marke Microsrucure: Design, Efficiency and Saisical Regulariies 21-25 March 2011 Some mahemaical properies of order book models Frederic ABERGEL Ecole Cenrale Paris Grande
More informationΓ(h)=0 h 0. Γ(h)=cov(X 0,X 0-h ). A stationary process is called white noise if its autocovariance
A family, Z,of random vecors : Ω R k defined on a probabiliy space Ω, A,P) is called a saionary process if he mean vecors E E =E M = M k E and he auocovariance marices are independen of. k cov, -h )=E
More informationf(s)dw Solution 1. Approximate f by piece-wise constant left-continuous non-random functions f n such that (f(s) f n (s)) 2 ds 0.
Advanced Financial Models Example shee 3 - Michaelmas 217 Michael Tehranchi Problem 1. Le f : [, R be a coninuous (non-random funcion and W a Brownian moion, and le σ 2 = f(s 2 ds and assume σ 2
More informationProperties of Autocorrelated Processes Economics 30331
Properies of Auocorrelaed Processes Economics 3033 Bill Evans Fall 05 Suppose we have ime series daa series labeled as where =,,3, T (he final period) Some examples are he dail closing price of he S&500,
More informationOptimality Conditions for Unconstrained Problems
62 CHAPTER 6 Opimaliy Condiions for Unconsrained Problems 1 Unconsrained Opimizaion 11 Exisence Consider he problem of minimizing he funcion f : R n R where f is coninuous on all of R n : P min f(x) x
More informationGeneralized Least Squares
Generalized Leas Squares Augus 006 1 Modified Model Original assumpions: 1 Specificaion: y = Xβ + ε (1) Eε =0 3 EX 0 ε =0 4 Eεε 0 = σ I In his secion, we consider relaxing assumpion (4) Insead, assume
More informationTesting the Random Walk Model. i.i.d. ( ) r
he random walk heory saes: esing he Random Walk Model µ ε () np = + np + Momen Condiions where where ε ~ i.i.d he idea here is o es direcly he resricions imposed by momen condiions. lnp lnp µ ( lnp lnp
More informationWe just finished the Erdős-Stone Theorem, and ex(n, F ) (1 1/(χ(F ) 1)) ( n
Lecure 3 - Kövari-Sós-Turán Theorem Jacques Versraëe jacques@ucsd.edu We jus finished he Erdős-Sone Theorem, and ex(n, F ) ( /(χ(f ) )) ( n 2). So we have asympoics when χ(f ) 3 bu no when χ(f ) = 2 i.e.
More informationAsymptotic instability of nonlinear differential equations
Elecronic Journal of Differenial Equaions, Vol. 1997(1997), No. 16, pp. 1 7. ISSN: 172-6691. URL: hp://ejde.mah.sw.edu or hp://ejde.mah.un.edu fp (login: fp) 147.26.13.11 or 129.12.3.113 Asympoic insabiliy
More informationOn a Fractional Stochastic Landau-Ginzburg Equation
Applied Mahemaical Sciences, Vol. 4, 1, no. 7, 317-35 On a Fracional Sochasic Landau-Ginzburg Equaion Nguyen Tien Dung Deparmen of Mahemaics, FPT Universiy 15B Pham Hung Sree, Hanoi, Vienam dungn@fp.edu.vn
More informationEmpirical 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 informationBOUNDED VARIATION SOLUTIONS TO STURM-LIOUVILLE PROBLEMS
Elecronic Journal of Differenial Equaions, Vol. 18 (18, No. 8, pp. 1 13. ISSN: 17-6691. URL: hp://ejde.mah.xsae.edu or hp://ejde.mah.un.edu BOUNDED VARIATION SOLUTIONS TO STURM-LIOUVILLE PROBLEMS JACEK
More informationDISCRETE GRONWALL LEMMA AND APPLICATIONS
DISCRETE GRONWALL LEMMA AND APPLICATIONS JOHN M. HOLTE MAA NORTH CENTRAL SECTION MEETING AT UND 24 OCTOBER 29 Gronwall s lemma saes an inequaliy ha is useful in he heory of differenial equaions. Here is
More informationY 0.4Y 0.45Y Y to a proper ARMA specification.
HG Jan 04 ECON 50 Exercises II - 0 Feb 04 (wih answers Exercise. Read secion 8 in lecure noes 3 (LN3 on he common facor problem in ARMA-processes. Consider he following process Y 0.4Y 0.45Y 0.5 ( where
More informationStochastic Modelling of Electricity and Related Markets: Chapter 3
Sochasic Modelling of Elecriciy and Relaed Markes: Chaper 3 Fred Espen Benh, Jūraė Šalyė Benh and Seen Koekebakker Presener: Tony Ware Universiy of Calgary Ocober 14, 2009 The Schwarz model S() = S(0)
More informationExcel-Based Solution Method For The Optimal Policy Of The Hadley And Whittin s Exact Model With Arma Demand
Excel-Based Soluion Mehod For The Opimal Policy Of The Hadley And Whiin s Exac Model Wih Arma Demand Kal Nami School of Business and Economics Winson Salem Sae Universiy Winson Salem, NC 27110 Phone: (336)750-2338
More information- The whole joint distribution is independent of the date at which it is measured and depends only on the lag.
Saionary Processes Sricly saionary - The whole join disribuion is indeenden of he dae a which i is measured and deends only on he lag. - E y ) is a finie consan. ( - V y ) is a finie consan. ( ( y, y s
More informationADDITIONAL PROBLEMS (a) Find the Fourier transform of the half-cosine pulse shown in Fig. 2.40(a). Additional Problems 91
ddiional Problems 9 n inverse relaionship exiss beween he ime-domain and freuency-domain descripions of a signal. Whenever an operaion is performed on he waveform of a signal in he ime domain, a corresponding
More informationExercises: Similarity Transformation
Exercises: Similariy Transformaion Problem. Diagonalize he following marix: A [ 2 4 Soluion. Marix A has wo eigenvalues λ 3 and λ 2 2. Since (i) A is a 2 2 marix and (ii) i has 2 disinc eigenvalues, we
More informationHamilton- J acobi Equation: Explicit Formulas In this lecture we try to apply the method of characteristics to the Hamilton-Jacobi equation: u t
M ah 5 2 7 Fall 2 0 0 9 L ecure 1 0 O c. 7, 2 0 0 9 Hamilon- J acobi Equaion: Explici Formulas In his lecure we ry o apply he mehod of characerisics o he Hamilon-Jacobi equaion: u + H D u, x = 0 in R n
More informationChapter 1 Fundamental Concepts
Chaper 1 Fundamenal Conceps 1 Signals A signal is a paern of variaion of a physical quaniy, ofen as a funcion of ime (bu also space, disance, posiion, ec). These quaniies are usually he independen variables
More informationChapter 3 Boundary Value Problem
Chaper 3 Boundary Value Problem A boundary value problem (BVP) is a problem, ypically an ODE or a PDE, which has values assigned on he physical boundary of he domain in which he problem is specified. Le
More informationTests of Nonlinear Resonse Theory. We compare the results of direct NEMD simulation against Kawasaki and TTCF for 2- particle colour conductivity.
ess of Nonlinear Resonse heory We compare he resuls of direc NEMD simulaion agains Kawasaki and CF for 2- paricle colour conduciviy. 0.0100 0.0080 0.0060 Direc CF, BK and RK J x 0.0040 0.0020 DIR KAW CF
More informationAnn. Funct. Anal. 2 (2011), no. 2, A nnals of F unctional A nalysis ISSN: (electronic) URL:
Ann. Func. Anal. 2 2011, no. 2, 34 41 A nnals of F uncional A nalysis ISSN: 2008-8752 elecronic URL: www.emis.de/journals/afa/ CLASSIFICAION OF POSIIVE SOLUIONS OF NONLINEAR SYSEMS OF VOLERRA INEGRAL EQUAIONS
More informationInnova Junior College H2 Mathematics JC2 Preliminary Examinations Paper 2 Solutions 0 (*)
Soluion 3 x 4x3 x 3 x 0 4x3 x 4x3 x 4x3 x 4x3 x x 3x 3 4x3 x Innova Junior College H Mahemaics JC Preliminary Examinaions Paper Soluions 3x 3 4x 3x 0 4x 3 4x 3 0 (*) 0 0 + + + - 3 3 4 3 3 3 3 Hence x or
More informationGMM - Generalized Method of Moments
GMM - Generalized Mehod of Momens Conens GMM esimaion, shor inroducion 2 GMM inuiion: Maching momens 2 3 General overview of GMM esimaion. 3 3. Weighing marix...........................................
More information. Now define y j = log x j, and solve the iteration.
Problem 1: (Disribued Resource Allocaion (ALOHA!)) (Adaped from M& U, Problem 5.11) In his problem, we sudy a simple disribued proocol for allocaing agens o shared resources, wherein agens conend for resources
More informationPredator - Prey Model Trajectories and the nonlinear conservation law
Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories
More informationVehicle 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 informationEcon 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 informationReferences are appeared in the last slide. Last update: (1393/08/19)
SYSEM IDEIFICAIO Ali Karimpour Associae Professor Ferdowsi Universi of Mashhad References are appeared in he las slide. Las updae: 0..204 393/08/9 Lecure 5 lecure 5 Parameer Esimaion Mehods opics o be
More informationStructural Break Detection for a Class of Nonlinear Time Series Models
Srucural Break Deecion for a Class of Nonlinear Time Series Models Richard A. Davis Thomas Lee Gabriel Rodriguez-Yam Colorado Sae Universiy (hp://www.sa.colosae.edu/~rdavis/lecures) This research suppored
More informationHomogenization of random Hamilton Jacobi Bellman Equations
Probabiliy, Geomery and Inegrable Sysems MSRI Publicaions Volume 55, 28 Homogenizaion of random Hamilon Jacobi Bellman Equaions S. R. SRINIVASA VARADHAN ABSTRACT. We consider nonlinear parabolic equaions
More information6. 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 informationL p -L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity
ANNALES POLONICI MATHEMATICI LIV.2 99) L p -L q -Time decay esimae for soluion of he Cauchy problem for hyperbolic parial differenial equaions of linear hermoelasiciy by Jerzy Gawinecki Warszawa) Absrac.
More informationEXERCISES FOR SECTION 1.5
1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler
More informationLecture #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 informationHomework 4 (Stats 620, Winter 2017) Due Tuesday Feb 14, in class Questions are derived from problems in Stochastic Processes by S. Ross.
Homework 4 (Sas 62, Winer 217) Due Tuesday Feb 14, in class Quesions are derived from problems in Sochasic Processes by S. Ross. 1. Le A() and Y () denoe respecively he age and excess a. Find: (a) P{Y
More informationPOSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE
Urainian Mahemaical Journal, Vol. 55, No. 2, 2003 POSITIVE AND MONOTONE SYSTEMS IN A PARTIALLY ORDERED SPACE A. G. Mazo UDC 517.983.27 We invesigae properies of posiive and monoone differenial sysems wih
More informationt is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...
Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger
More informationBasic notions of probability theory (Part 2)
Basic noions of probabiliy heory (Par 2) Conens o Basic Definiions o Boolean Logic o Definiions of probabiliy o Probabiliy laws o Random variables o Probabiliy Disribuions Random variables Random variables
More informationMath 10B: Mock Mid II. April 13, 2016
Name: Soluions Mah 10B: Mock Mid II April 13, 016 1. ( poins) Sae, wih jusificaion, wheher he following saemens are rue or false. (a) If a 3 3 marix A saisfies A 3 A = 0, hen i canno be inverible. True.
More informationHow 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 informationFinancial Econometrics Introduction to Realized Variance
Financial Economerics Inroducion o Realized Variance Eric Zivo May 16, 2011 Ouline Inroducion Realized Variance Defined Quadraic Variaion and Realized Variance Asympoic Disribuion Theory for Realized Variance
More informationConvergence of the Neumann series in higher norms
Convergence of he Neumann series in higher norms Charles L. Epsein Deparmen of Mahemaics, Universiy of Pennsylvania Version 1.0 Augus 1, 003 Absrac Naural condiions on an operaor A are given so ha he Neumann
More informationPart 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 informationDistribution of Estimates
Disribuion of Esimaes From Economerics (40) Linear Regression Model Assume (y,x ) is iid and E(x e )0 Esimaion Consisency y α + βx + he esimaes approach he rue values as he sample size increases Esimaion
More informationIntroduction 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 informationStatistics 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 informationModeling and Forecasting Volatility Autoregressive Conditional Heteroskedasticity Models. Economic Forecasting Anthony Tay Slide 1
Modeling and Forecasing Volailiy Auoregressive Condiional Heeroskedasiciy Models Anhony Tay Slide 1 smpl @all line(m) sii dl_sii S TII D L _ S TII 4,000. 3,000.1.0,000 -.1 1,000 -. 0 86 88 90 9 94 96 98
More informationHamilton- J acobi Equation: Weak S olution We continue the study of the Hamilton-Jacobi equation:
M ah 5 7 Fall 9 L ecure O c. 4, 9 ) Hamilon- J acobi Equaion: Weak S oluion We coninue he sudy of he Hamilon-Jacobi equaion: We have shown ha u + H D u) = R n, ) ; u = g R n { = }. ). In general we canno
More informationThe motions of the celt on a horizontal plane with viscous friction
The h Join Inernaional Conference on Mulibody Sysem Dynamics June 8, 18, Lisboa, Porugal The moions of he cel on a horizonal plane wih viscous fricion Maria A. Munisyna 1 1 Moscow Insiue of Physics and
More informationDYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń Piotr Fiszeder Nicolaus Copernicus University in Toruń
DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 006 Pior Fiszeder Nicolaus Copernicus Universiy in Toruń Consequences of Congruence for GARCH Modelling. Inroducion In 98 Granger formulaed
More informationMATH 4330/5330, Fourier Analysis Section 6, Proof of Fourier s Theorem for Pointwise Convergence
MATH 433/533, Fourier Analysis Secion 6, Proof of Fourier s Theorem for Poinwise Convergence Firs, some commens abou inegraing periodic funcions. If g is a periodic funcion, g(x + ) g(x) for all real x,
More information2 int T. is the Fourier transform of f(t) which is the inverse Fourier transform of f. i t e
PHYS67 Class 3 ourier Transforms In he limi T, he ourier series becomes an inegral ( nt f in T ce f n f f e d, has been replaced by ) where i f e d is he ourier ransform of f() which is he inverse ourier
More informationIntroduction 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 informationRepresentation of Stochastic Process by Means of Stochastic Integrals
Inernaional Journal of Mahemaics Research. ISSN 0976-5840 Volume 5, Number 4 (2013), pp. 385-397 Inernaional Research Publicaion House hp://www.irphouse.com Represenaion of Sochasic Process by Means of
More informationHeavy Tails of Discounted Aggregate Claims in the Continuous-time Renewal Model
Heavy Tails of Discouned Aggregae Claims in he Coninuous-ime Renewal Model Qihe Tang Deparmen of Saisics and Acuarial Science The Universiy of Iowa 24 Schae er Hall, Iowa Ciy, IA 52242, USA E-mail: qang@sa.uiowa.edu
More informationCH Sean Han QF, NTHU, Taiwan BFS2010. (Joint work with T.-Y. Chen and W.-H. Liu)
CH Sean Han QF, NTHU, Taiwan BFS2010 (Join work wih T.-Y. Chen and W.-H. Liu) Risk Managemen in Pracice: Value a Risk (VaR) / Condiional Value a Risk (CVaR) Volailiy Esimaion: Correced Fourier Transform
More informationEndpoint Strichartz estimates
Endpoin Sricharz esimaes Markus Keel and Terence Tao (Amer. J. Mah. 10 (1998) 955 980) Presener : Nobu Kishimoo (Kyoo Universiy) 013 Paricipaing School in Analysis of PDE 013/8/6 30, Jeju 1 Absrac of he
More information10. State Space Methods
. Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he
More informationFiltering Turbulent Signals Using Gaussian and non-gaussian Filters with Model Error
Filering Turbulen Signals Using Gaussian and non-gaussian Filers wih Model Error June 3, 3 Nan Chen Cener for Amosphere Ocean Science (CAOS) Couran Insiue of Sciences New York Universiy / I. Ouline Use
More informationInternational Journal of Scientific & Engineering Research, Volume 4, Issue 10, October ISSN
Inernaional Journal of Scienific & Engineering Research, Volume 4, Issue 10, Ocober-2013 900 FUZZY MEAN RESIDUAL LIFE ORDERING OF FUZZY RANDOM VARIABLES J. EARNEST LAZARUS PIRIYAKUMAR 1, A. YAMUNA 2 1.
More information