Diffusions, Markov Processes, and Martingales

Transcription

1 Diffusions, Markov Processes, and Martingales Volume 2: ITO 2nd Edition CALCULUS L. C. G. ROGERS School of Mathematical Sciences, University of Bath and DAVID WILLIAMS Department of Mathematics, University of Wales, Swansea CAMBRIDGE UNIVERSITY PRESS

2 Contents Some Frequently Used Notation xiv CHAPTER IV. INTRODUCTION TO ITO CALCULUS TERMINOLOGY AND CONVENTIONS R-processes and L-processes Usual conditions, etc. Important convention about time 0 1. SOME MOTIVATING REMARKS 1. Ito integrals 2 2. Integration by parts 4 3. Ito's formula for Brownian motion 8 4. A rough plan of the chapter 9 2. SOME FUNDAMENTAL IDEAS: PREVISIBLE PROCESSES, LOCALIZATION, etc. Previsible processes 5. Basic integrands Z(S, 7] Previsible processes on (0, oo), ^*, bj*, W 11 Finite-variation and integrable-variation processes 7. FV 0 and IV 0 processes Preservation of the martingale property 14 Localization 9. H(0, r], X T Localization of integrands, lb^* Localization of integrators, J? 0,\ x, FVj? 04oc etc Nil desperandum! Extending stochastic integrals by localization Local martingales, ~# loc, and the Fatou lemma 21 Semimartingales as integrators 15. Semimartingales,^ Integrators 24 Likelihood ratios 17. Martingale property under change of measure 25

3 IX 3. THE ELEMENTARY THEORY OF FINITE-VARIATION PROCESSES 18. Ito's formula for FV functions The Doleans exponential r (x.) 29 Applications to Markov chains with finite state-space 20. Martingale problems Probabilistic interpretation of Q Likelihood ratios and some key distributions STOCHASTIC INTEGRALS: THE L 2 THEORY 23. Orientation Stable spaces of Jt\, cjt\, &M\ Elementary stochastic integrals relative to M in M\ The processes [Af ] and [M,,/V] Constructing stochastic integrals in L The Kunita-Watanabe inequalities STOCHASTIC INTEGRALS WITH RESPECT TO CONTINUOUS SEMIMARTINGALES 29. Orientation Quadratic variation for continuous local martingales Canonical decomposition of a continuous semimartingale Ito's formula for continuous semimartingales APPLICATIONS OF ITO'S FORMULA 33. Levy's theorem Continuous local martingales as time-changes of Brownian motion Bessel processes; skew products; etc Brownian martingale representation Exponential semimartingales; estimates Cameron-Martin-Girsanov change of measure First applications: Doob /i-transforms; hitting of spheres; etc Further applications: bridges; excursions; etc Explicit Brownian martingale representation Burkholder-Davis-Gundy inequalities Semimartingale local time; Tanaka's formula Study of joint continuity Local time as an occupation density; generalized Ito-Tanaka formula The Stratonovich calculus Riemann-sum approximation to Ito and Stratonovich integrals; simulation 108

4 X CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSIONS 1. INTRODUCTION 1. What is a diffusion in 1^"? FD diffusions recalled SDEs as a means of constructing diffusions Example: Brownian motion on a surface Examples: modelling noise in physical systems Example: Skorokhod's equation Examples: control problems PATHWISE UNIQUENESS, STRONG SDEs, AND FLOWS 8. Our general SDE; previsible path functionals; diffusion SDEs Pathwise uniqueness; exact SDEs Relationship between exact SDEs and strong solutions The Ito existence and uniqueness result Locally Lipschitz SDEs; Lipschitz properties of a 1 ' Flows; the diffeomorf -,n theorem; time-reversed flows CarverhilFs noisy North-South flow on a circle The martingale optimality principle in control WEAK SOLUTIONS, UNIQUENESS IN LAW 16. Weak solutions of SDEs; Tanaka's SDE 'Exact equals weak plus pathwise unique' Tsirel'son's example MARTINGALE PROBLEMS, MARKOV PROPERTY 19. Definition; orientation Equivalence of the martingale-problem and'weak'formulations Martingale problems and the strong Markov property Appraisal and consolidation: where we have reached Existence of solutions to the martingale problem The Stroock-Varadhan uniqueness theorem Martingale representation 173 Transformation of SDEs 26. Change of time scale; Girsanov's SDE Change of measure Change of state-space; scale; Zvonkin's observation; the Doss- Sussmann method Krylov's example OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY 30. Introduction; some key ideas; Stratonovich-to-Ito conversion Brownian motion on a submanifold of B N 186

5 XI 32. Parallel displacement; Riemannian connections Extrinsic theory of BM h0r (O( )); rolling without slipping; martingales on manifolds; etc Intrinsic theory; normal coordinates; structural equations; diffusions on manifolds ;etc.(!) Brownian motion on Lie groups Dynkin's Brownian motion of ellipses; hyperbolic space interpretation; etc Khasminskii's method for studying stability; random vibrations Hormander's theorem; Malliavin calculus; stochastic pullback; curvature ONE-DIMENSIONAL SDEs 39. A local-time criterion for pathwise uniqueness The Yamada-Watanabe pathwise uniqueness theorem The Nakao path wise-uniqueness theorem Solution of a variance control problem A comparison theorem ONE-DIMENSIONAL DIFFUSIONS 44. Orientation Regular diffusions The scale function, s The speed measure, m; time substitution Example: the Bessel SDE.' Diffusion local time : Analytical aspects Classification of boundary points Khasminskii's test for explosion An ergodic theorem for 1-dimensional diffusions Coupling of 1-dimensional diffusions 301 CHAPTER VI. THE GENERAL THEORY 1. ORIENTATION 1. Preparatory remarks Levy processes DEBUT AND SECTION THEOREMS 3. Progressive processes Optional processes, (?; optional times The 'optional' section theorem Warning (not to be skipped) 318

6 Xll 3. OPTIONAL PROJECTIONS AND FILTERING 7. Optional projection X of X The innovations approach to filtering The Kalman-Bucy filter The Bayesian approach to filtering; a change-detection filter Robust filtering CHARACTERIZING PREVISIBLE TIMES 12. Previsible stopping times; PFA theorem Totally inaccessible and accessible stopping times Some examples Meyer's previsibility theorem for Markov processes Proof of the PFA theorem The ff-algebras f(p-),f(p), J*"(P + ) Quasi-left-continuous nitrations DUAL PREVISIBLE PROJECTIONS 19. The previsible section theorem; the previsible projection P X of A" Doleans' characterization of FV processes Dual previsible projections, compensators Cumulative risk Some Brownian motion examples Decomposition of a continuous semimartingale Proof of the basic (/i, -4) correspondence Proof of the Doleans 'optional' characterization result Proof of the Doleans 'previsible' characterization result Levy systems for Markov processes THE MEYER DECOMPOSITION THEOREM 29. Introduction The Doleans proof of the Meyer decomposition Regular class (D) submartingales; approximation to compensators The local form of the decomposition theorem An L 2 bounded local martingale which is not a martingale The <M> process Last exits and equilibrium charge STOCHASTIC INTEGRATION: THE GENERAL CASE 36. The quadratic variation process [M] Stochastic integrals with respect to local martingales Stochastic integrals with respect to semimartingales Ito's formula for semimartingales Special semimartingales Quasimartingales 396

7 Xlll 8. ITO EXCURSION THEORY 42. Introduction Excursion theory for a finite Markov chain Taking stock Local time L at a regular extremal point a Some technical points: hypotheses droites, etc The Poisson point process of excursions Markovian character of n Marking the excursions Last-exit decomposition; calculation of the excursion law n The Skorokhod embedding theorem Diffusion properties of local time in the space variable; the Ray-Knight theorem Arcsine law for Brownian motion Resolvent density of a 1-dimensional diffusion Path decomposition of Brownian motions and of excursions An illustrative calculation Feller Brownian motions Example: censoring and reweighting of excursion laws Excursion theory by stochastic calculus: McGill's lemma REFERENCES 449 INDEX 469