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On the Predictable Representation Property of Martingale Associated Marie Curie Initial Training Network, Call: FP7-PEOPLE-2007-1-1-ITN, no. 213841-2 Deterministic and Stochastic Controlled Systems and Applications On the Predictable Representation Property of Martingales Associated with L´evyProcesses Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) vorgelegt dem Rat der Fakult¨atf¨urMathematik und Informatik der Friedrich-Schiller-Universit¨atJena von M. Sc. Paolo Di Tella geboren am 3.3.1984 in Rom Gutachter 1. Prof. Dr. Hans-J¨urgenEngelbert (summa cum laude) 2. Prof. Dr. Yuri Kabanov (summa cum laude) 3. Prof. Dr. Ilya Pavlyukevich (summa cum laude) Tag der ¨offentlichen Verteidigung: 14. Juni 2013 A memoria del piccolo Vincenzo. A mia sorella Marta e a zio Paolo. i Contents Symbols and Acronyms v Acknowledgments vii Zusammenfassung ix Abstract xi Introduction xiii 1. Preliminaries 1 1.1. Measure Theory . .1 1.1.1. Measurability and Integration . .1 1.1.2. Monotone Class Theorems . .4 1.1.3. Total Systems . .5 1.2. Basics of Stochastic Processes . .6 1.2.1. Filtrations and Stopping Times . .8 1.2.2. Martingales . 10 1.2.3. Increasing Processes . 11 1.2.4. The Poisson process . 14 1.2.5. Locally Square Integrable Martingales . 15 1.2.6. The Wiener process . 15 1.2.7. Orthogonality of Local Martingales . 16 1.2.8. Semimartingales . 17 1.3. Spaces of Martingales and Stochastic Integration . 19 1.3.1. Spaces of Martingales . 19 1.3.2. Stochastic Integral for Local Martingales . 22 1.3.3. Stochastic Integral for Semimartingales . 25 1.4. Stable Subspaces of Martingales . 27 2. Poisson Random Measures and Levy´ Processes 33 2.1. Additive Processes and L´evyProcesses . 33 iii iv Contents 2.2. Poisson Random Measures . 37 2.2.1. The Jump Measure of a c`adl`agProcess . 37 2.2.2. Definition of Poisson Random Measures . 39 2.2.3. The Stochastic Integral for Poisson Random Measures . 41 2.2.4. Compensated Poisson Random Measures . 44 2.2.5. Construction of L´evyProcesses . 47 2.3. Generalities on L´evyProcesses . 51 2.3.1. The Jump Measure of a L´evyProcess . 51 2.3.2. It^o{L´evyDecomposition and Moments . 56 2.3.3. The Structure of the Natural Filtration of a L´evyProcess . 58 3. Martingale Problems for Levy´ Processes 61 3.1. The Gaussian Part . 62 3.2. The Jump Measure . 63 3.3. Uniqueness of the Solution of the Martingale Problem . 67 3.4. Representation of Martingales . 68 3.5. Square Integrable L´evyProcesses . 71 3.6. Open Problems . 74 3.6.1. Characterization of the Predictable Representation Property 74 3.6.2. A Stronger Representation . 75 4. Representation of Square Integrable Martingales 81 4.1. Compensated-Covariation Stable Families . 81 4.1.1. Continuous Gaussian Families . 97 4.1.2. Independent Families of Poisson Processes . 99 4.2. Martingales of the Natural Filtration of a L´evyProcess . 100 4.2.1. Complete Orthogonal Systems . 105 4.2.2. Square Integrable L´evyProcesses . 109 5. Examples 113 5.1. Compensated Poisson Processes . 113 5.2. Teugels Martingales . 114 5.3. L´evyMeasures Equivalent to the Lebesgue Measure . 117 5.3.1. Hermite Polynomials . 118 5.3.2. Haar Wavelet . 119 A. Complement to Additive Processes 121 A.1. Independence of Additive Processes . 121 A.2. It^o{L´evyDecomposition . 130 B. Denseness of Polynomials 137 Bibliography 141 Symbols and Acronyms v Symbols and Acronyms a.e. almost everywhere a.s. almost surely c`adl`ag continu ´adroite, limit`e´agauche cf. compare (abbreviation of Latin confer) e.g. for example (abbreviation of Latin exempli gratia) i.e. that is (abbreviation of Latin id est) p.d. pairwise disjoint PRP predictable representation property resp. respectively Norms and Numerical Sets Symbol Page Symbol Description k · kq; k · kLq(µ) 3 N natural numbers k · kH q 19 Q rational numbers k · kBMO 20 R real numbers ∗ k · kq 19 R+ nonnegative real numbers k · kLq(M) 22 Z integer numbers Classes of Processes Symbol Description Page A + integrable increasing processes 13 A processes with integrable variation 13 + Aloc locally integrable increasing processes 13 Aloc processes with locally integrable variation 13 BMO BMO-martingales 20 BMO0 BMO-martingales starting at 0 20 Cloc localized class C 9 H q q-integrable martingales 19 q H0 q-integrable martingales starting at 0 19 q q Hloc localization of H 19 q q Hloc;0 localization of H0 19 K integrands for semimartingales 25 q q L (X ) stable subspace generated in H0 by X 28 Lq(M) predictable integrands of order q for M 22 q Lloc(M) predictable integrands locally of order q for M 22 M uniformly integrable martingales 10 M0 uniformly integrable martingales starting at 0 10 Mloc local martingales 11 Mloc;0 local martingales starting at 0 11 c Mloc continuous local martingales 16 d Mloc purely discontinuous local martingales 16 V + increasing processes 11 V processes of finite variation 11 vi Symbols and Acronyms Other symbols Symbol Description Page σ(C ) σ-algebra generated by C 2 W i2I Fi σ-algebra generated by [i2I Fi 2 (Ω; F ) measurable space 2 B(A) Borel σ-algebra on the topological space A 2 µ(f) integral of f with respect to the measure µ 3 Lq(µ) Lq-space with respect to the measure µ 3 (·; ·)H scalar product on the Hilbert space H 3 −!; !; "; # convergence; monotone convergence 4 B := B(Ω; R) bounded functions on Ω with values in R 4 Span(T ) linear hull of T 5 (H ;k·k) T closure of T in the Banach space (H ; k · k) 5 (Ω; F ; P) probability space 6 N (P) null-sets of the probability measure P 6 E[·]; E[·|F ] expectation; conditional expectation 6 supK supremum over K 7 7−!; 7! application or functional 7 X0− := X0 left limit of the initial value of a c`adl`agprocess 7 X− left-hand limit process of the c`adl`agprocess X 7 ∆X jump process of the c`adl`agprocess X 7 F filtration 8 ~X ~X X X F ; F+ ; F+ ; F filtration generated by the process X,... 8 ~X X F1 ; F1 σ-algebra generated by the process X,... 8 O optional σ-algebra 9 P predictable σ-algebra 9 XT process X stopped at T 9 ^ minimum function 9 X1 terminal variable of the process X 10 ; ⊕ direct difference; direct sum 12; 29 Var(A) variation process of the process A 12 h·; ·i;[·; ·] point brackets process; covariation process 15; 18 1A indicator function of the set A 23 2 δxi ; δxixj partial derivative; second partial derivative 26 condition Cq 31 Q(·) set of the solutions of a martingale problem 32 (X; F) additive process relative to the filtration F 35 (L; F) L´evyprocess relative to the filtration F 35 (E; B(E)) measurable space (R+ × R; B(R+) ⊗ B(R)) 37 M Poisson random measure 39 m intensity measure 37 λ+ Lebesgue measure on (R+; B(R+)) 39 M compensated Poisson random measure 44 X(f) 49 M (α,β) compensated-covariation process 82 Acknowledgments I would like to express my deepest gratitude to my advisor Hans-J¨urgenEngelbert not only because this work would have been not possible without his help, but above all because in this years he taught me with passion and patience the art of being a mathematician. I thank Yuri Kabanov and Ilya Pavlyukevich because they agreed to spend their time for reading and evaluating my thesis. I thank Alberto Barchielli because he played a fundamental role in my education and he was always ready to help me every time I asked. To apply for a PhD position in the ITN was a suggestion of Carlo Sgarra and I thank him for this very good idea. I acknowledge the secretary of the Department of Stochastics of the faculty of Mathematics and Computer Science of \Friedrich-Schiller-University Jena" Nicole Hickethier for supporting my work with her competence. In this years I discussed of my work with several people. Among others I thank Monique Jeanblanc, Monique Pontier, Francesco Russo and Martina Z¨ahle. I thank all my colleagues of the ITN, in particular Andrii Andrusiv, Matteo Be- dini, Christine Gr¨un,Elena Issoglio and Holger Metzler. Among my colleagues at \Friedrich-Schiller-University Jena" I specially acknowledge Stefan Blei and Lev Markhasin. Ringrazio i miei genitori Maria e Antonio e mia sorella Marta per il grande sostegno e affetto che mi hanno dato. Ich danke Ortrun, das Beste und Sch¨onstewas mir in diesen Jahren passiert ist. Zusammenfassung Im Rahmen von L´evy-Prozessen, untersuchen wir die vorhersagbare Darstellbarkeits- eigenschaft, die wir mit PRP abkurzen¨ (von dem englischen Ausdruck \predictable representation property"). Um eine allgemeine Definition der PRP zu geben, ver- wenden wir die Theorie der stabilen R¨aume. Sei L ein L´evy-Prozessbezuglich¨ seiner L vervollst¨andigten erzeugten Filtration F und sei ν das zugeh¨orige L´evy-Maß.Wir L konstruieren Familien von F -Martingalen, welche die PRP besitzen. Die Martingale, die wir betrachten, werden mittels der stochastischen Integration von deterministi- schen Funktionen bezuglich¨ des kompensierten Poissonschen zuf¨alligen Sprungmaßes von L erzeugt. Als N¨achstes erkl¨aren wir das Obenstehende im Detail. Wir betrachten einen L L´evy-Prozessbezuglich¨ seiner vervollst¨andigten erzeugten Filtration F . Sei N ein L lokales Martingal bezuglich¨ F und q = 1; 2. Wir definieren die Norm 1 q kNkH q := E[supt≥0 jNtj] L und bezeichnen den Raum aller F -adaptierten Martingale N, fur¨ die N0 = 0 und q q kNkH q < +1, mit H0 .
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