Lecture Notes in Mathematics (cid:49)(cid:57)(cid:50)(cid:51) Editors: J.-M.Morel,Cachan F.Takens,Groningen B.Teissier,Paris Jaya P.N. Bishwal Parameter Estimation in Stochastic Differential Equations (cid:65)(cid:66)(cid:67) Author JayaP.N.Bishwal DepartmentofMathematicsandStatistics UniversityofNorthCarolinaatCharlotte 376FretwellBldg. 9201UniversityCityBlvd. CharlotteNC28223-0001 USA e-mail:[email protected] URL:http://www.math.uncc.edu/∼jpbishwa LibraryofCongressControlNumber:(cid:50)(cid:48)(cid:48)(cid:55)(cid:57)(cid:51)(cid:51)(cid:53)(cid:48)(cid:48) MathematicsSubjectClassification((cid:50)(cid:48)(cid:48)(cid:48)(cid:41)(cid:58)60H10,60H15,60J60,62M05,62M09 ISSNprintedition:(cid:48)(cid:48)(cid:55)(cid:53)(cid:45)(cid:56)(cid:52)(cid:51)(cid:52) ISSNelectronicedition:(cid:49)(cid:54)(cid:49)(cid:55)(cid:45)(cid:57)(cid:54)(cid:57)(cid:50) ISBN(cid:57)(cid:55)(cid:56)(cid:45)(cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:52)(cid:52)(cid:52)(cid:55)(cid:45)(cid:52)SpringerBerlinHeidelbergNewYork DOI(cid:49)(cid:48)(cid:46)(cid:49)(cid:48)(cid:48)(cid:55)/(cid:57)(cid:55)(cid:56)(cid:45)(cid:51)(cid:45)(cid:53)(cid:52)(cid:48)(cid:45)(cid:55)(cid:52)(cid:52)(cid:52)(cid:56)(cid:45)(cid:49) Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting, reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember(cid:57), (cid:49)(cid:57)(cid:54)(cid:53),initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare liableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com (cid:176)c Springer-VerlagBerlinHeidelberg(cid:50)(cid:48)(cid:48)(cid:56) Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply, evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws andregulationsandthereforefreeforgeneraluse. TypesettingbytheauthorsandSPiusingaSpringerLATEXmacropackage Coverdesign:design&productionGmbH,Heidelberg Printedonacid-freepaper SPIN:(cid:49)(cid:50)(cid:49)(cid:49)(cid:49)(cid:54)(cid:54)(cid:52) (cid:52)(cid:49)/SPi (cid:53)(cid:52)(cid:51)(cid:50)(cid:49)(cid:48) To the memory of my late grand father and to my parents and brothers for their love and affection Preface I am indebted to my advisor Arup Bose from who I learned inference for diffusion processes during my graduate studies. I have benefited a lot from my discussions with Yacine Ait-Sahalia, Michael Sørensen, Yuri Kutoyants and Dan Crisan. I am grateful to all of them for their advice. Charlotte, NC January 30, 2007 Jaya P.N. Bishwal Contents Basic Notations................................................XIII 1 Parametric Stochastic Differential Equations............... 1 Part I Continuous Sampling 2 RatesofWeakConvergenceofEstimatorsinHomogeneous Diffusions.................................................. 15 2.1 Introduction ............................................ 15 2.2 Berry-Esseen Bounds for Estimators in the Ornstein- Uhlenbeck Process....................................... 16 2.3 Rates of Convergence in the Bernstein-von Mises Theorem for Ergodic Diffusions .................................... 26 2.4 Rates of Convergence of the Posterior Distributions in Ergodic Diffusions..................................... 33 2.5 Berry-Esseen Bound for the Bayes Estimator................ 41 2.6 Example: Hyperbolic Diffusion Model ...................... 47 3 Large Deviations of Estimators in Homogeneous Diffusions.................................................. 49 3.1 Introduction ............................................ 49 3.2 Model, Assumptions and Preliminaries ..................... 50 3.3 Large Deviations for the Maximum Likelihood Estimator ..... 51 3.4 Large Deviations for Bayes Estimators ..................... 57 3.5 Examples............................................... 59 4 Local Asymptotic Mixed Normality for Nonhomogeneous Diffusions.................................................. 61 4.1 Introduction ............................................ 61 4.2 Model, Assumptions and Preliminaries ..................... 63 X Contents 4.3 Asymptotics of the Maximum Likelihood Estimator.......... 67 4.4 The Bernstein-von Mises Type Theorem and Asymptotics of Bayes Estimators...................................... 68 4.5 Asymptotics of Maximum Probability Estimator ............ 74 4.6 Examples............................................... 75 5 Bayes and Sequential Estimation in Stochastic PDEs ...... 79 5.1 Long Time Asymptotics.................................. 79 5.1.1 Introduction ...................................... 79 5.1.2 Model, Assumptions and Preliminaries ............... 80 5.1.3 Bernstein-von Mises Theorem....................... 82 5.1.4 Asymptotics of Bayes Estimators.................... 84 5.2 Sequential Estimation.................................... 85 5.2.1 Sequential Maximum Likelihood Estimation .......... 85 5.2.2 Example ......................................... 87 5.3 Spectral Asymptotics .................................... 88 5.3.1 Introduction ...................................... 88 5.3.2 Model and Preliminaries ........................... 90 5.3.3 The Bernstein-von Mises Theorem................... 93 5.3.4 Bayes Estimation.................................. 95 5.3.5 Example: Stochastic Heat Equation.................. 96 6 Maximum Likelihood Estimation in Fractional Diffusions.................................................. 99 6.1 Introduction ............................................ 99 6.2 Fractional Stochastic Calculus ............................100 6.3 Maximum Likelihood Estimation in Directly Observed Fractional Diffusions .....................................109 6.4 Maximum Likelihood Estimation in Partially Observed Fractional Diffusions .....................................113 6.5 Examples...............................................118 Part II Discrete Sampling 7 Approximate Maximum Likelihood Estimation in Nonhomogeneous Diffusions.............................125 7.1 Introduction ............................................125 7.2 Model, Assumptions and Definitions .......................127 7.3 Accuracy of Approximations of the Itoˆ and FS Integrals......135 7.4 Accuracy of Approximations of the Log-likelihood Function...142 7.5 Accuracy of Approximations of the Maximum Likelihood Estimate ...............................................146 7.6 Example: Chan-Karloyi-Longstaff-Sanders Model ............148 7.7 Summary of Truncated Distributions.......................155 Contents XI 8 Rates of Weak Convergence of Estimators in the Ornstein-Uhlenbeck Process ........................159 8.1 Introduction ............................................159 8.2 Notations and Preliminaries ..............................160 8.3 Berry-Esseen Type Bounds for AMLE1.....................162 8.4 Berry-Esseen Type Bounds for AMLE2.....................173 8.5 Berry-Esseen Type Bounds for Approximate Minimum Contrast Estimators .....................................178 8.6 Berry-Esseen Bounds for Approximate Bayes Estimators .....192 9 Local Asymptotic Normality for Discretely Observed Homogeneous Diffusions ...................................201 9.1 Introduction ............................................201 9.2 Model, Assumptions and Preliminaries .....................202 9.3 Weak Convergence of the Approximate Likelihood Ratio Random Fields..........................................207 9.4 Asymptotics of Approximate Estimators and Bernstein-von Mises Type Theorems....................................221 9.5 Example: Logistic Diffusion...............................223 10 Estimating Function for Discretely Observed Homogeneous Diffusions ...................................225 10.1 Introduction ............................................225 10.2 Rate of Consistency......................................233 10.3 Berry-Esseen Bound .....................................238 10.4 Examples...............................................240 References.....................................................245 Index..........................................................263 Basic Notations (Ω,F,P) probability space R real line C complex plane I indicator function of a set A A D→[P] convergence in distribution under the measure P →P convergence in probability P a.s. [P] almost surely under the measure P P-a.s. almost surely under the measure P aann ==oO((bbnn)) aabbnnnn →is b0ounded Xn =oP(bn) Xbnn →P 0 Xn =OP(bn) Xbnn is stochasticaXlly bounded, i.e., lim supP{| n|>A}=0 A→∞ n bn end of a proof A:=B A is defined by B A=:B B is defined by A ≡ identically equal (cid:3) absolute continuity of two measures i.i.d. independent and identically distributed N(a,b) normal distribution with mean a and variance b Φ(.) standard normal distribution function X ∼F X has the distribution F w.r.t. with respect to r.h.s. right hand side l.h(cid:1).s. left hand side a(cid:2)b maximum of a and b a b minimum of a and b 1 Parametric Stochastic Differential Equations Stochastic differential equations (SDEs) are a natural choice to model the timeevolutionofdynamicsystemswhicharesubjecttorandominfluences(cf. Arnold (1974), Van Kampen (1981)). For example, in physics the dynamics of ions in superionic conductors are modelled via Langevin equations (cf. Dieterichetal.(1980)),andinengineeringthedynamicsofmechanicaldevices are described by differential equations under the influence of process noise as errorsofmeasurement(cf.Gelb(1974)).Otherapplicationsareinbiology(cf. Jennrich and Bright (1976)), medicine (cf. Jones (1984)), econometrics (cf. Bergstrom (1976, 1988)), finance (cf. Black and Scholes (1973)), geophysics (cf. Arato (1982)) and oceanography (cf. Adler et al. (1996)). It is natural that a model contains unknown parameters. We consider the model as the parametric Itˆo stochastic differential equation dX = µ (θ, t, X ) dt + σ (ϑ, t, X ) dW , t≥0, X =ζ t t t t 0 where {W ,t ≥ 0} is a standard Wiener process, µ : Θ ×[0,T]×R → R, t called the drift coefficient, and σ : Ξ ×[0,T]×R → R+, called the diffu- sion coefficient, are known functions except the unknown parameters θ and ϑ, Θ ⊂ R, Ξ ⊂ R and E(ζ2) < ∞. The drift coefficient is also called the trend coefficient or damping coefficient or translation coefficient. The diffu- sion coefficient is also called volatility coefficient. Under local Lipschitz and the linear growth conditions on the coefficients µ and σ, there exists a unique strong solution of the above Itoˆ SDE, called the diffusion process or simply a diffusion,whichisacontinuous strongMarkov semimartingale. Thedriftand the diffusion coefficients are respectively the instantaneous mean and instan- taneousstandarddeviationoftheprocess.Notethatthediffusioncoefficientis almostsurelydeterminedbytheprocess,i.e.,itcanbeestimatedwithoutany error if observed continuously throughout a time interval (see Doob (1953), Genon-Catalot and Jacod (1994)). We assume that the unknown parameter in the diffusion coefficient ϑ is known and for simplicity only we shall assume that σ =1 and our aim is to estimate the unknown parameter θ.