目录
目 录 Some Frequently Used Notation 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. Integration by parts 3. Ito''''s formula for Brownian motion 4. A rough plan of the chapter 2. SOME FUNDAMENTAL IDEAS: PREVISIBLE PROCESSES,LOCALIZATION, etc. Previsible processes 5. Basic integrands Z[S, T] 6. Previsible processes on 0, , b, b Finite-variation and integrable-variation processes 7. FVo and IVo processes 8. Preservation of the martingale property Localization 9. H[O, T], XT 10. Localization of integrands, 1b 11. Localization of integrators, FV etc. 12. Nil desperandum! 13. Extending stochastic integrals by localization 14. Local martingales, and the Fatou lemma Semimartingales as integrators 15. Semimartingales, 16. Integrators Likelihood ratios 17. Martingale property under change of measure 3. THE ELEMENTARY THEORY OF FINITE-VARIATION PROCESSES 18. Ito''''s formula for FV functions 19. The Doleans exponential x. Applications to Markov chains with finite state-space 20. Martingale problems. 21. Probabilistic interpretation of Q. 22. Likelihood ratios and some key distributions 4. STOCHASTIC INTEGRALS: THE L2 THEORY 23. Orientation 24. Stable spaces of 25. Elementary stochastic integrals relative to M in 26. The processes [M] and [M, N] 27. Constructing stochastic integrals in L2 28. The Kunita-Watanabe inequalities 5. STOCHASTIC INTEGRALS WITH RESPECT TO CONTINUOUS SEMIMARTINGALES 29. Orientation 30. Quadratic variation for continuous local martingales 31. Canonical decomposition of a continuous semimartingale 32. Ito''''s formula for continuous semimartingales 6. APPLICATIONS OF ITO''''S FORMULA 33. Levy''''s theorem 34. Continuous local martingales as time-changes of Brownian motion. 35. Bessel processes; skew products; etc 36, Brownian martingale representation 37. Exponential semimartingales; estimates 38. Cameron-Martin-Girsanov change of measure 39. First applications: Doob h-transforms; hitting of spheres; etc, 40. Further applications: bridges; excursions; etc. 41. Explicit Brownian martingale representation 42. Burkholder-Davis-Gundy inequalities 43. Semimartingale local time; Tanaka''''s formula 44. Study of joint continuity 45. Local time as an occupation density; generalized It6-Tanaka formula 46. The Stratonovich calculus 47. Riemann-sum approximation to It6 and Stratonovich integrals; simulation CHAPTER V. STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSIONS 1. INTRODUCTION 1. What is a diffusion in Rn 2. FD diffusions recalled 3. SDEs as a means of constructing diffusions 4. Example: Brownian motion on a surface 5. Examples: modelling noise in physical systems 6. Example: Skorokhod''''s equation 7. Examples: control problems 2. PATHWISE UNIQUENESS, STRONG SDEs, AND FLOWS 8. Our general SDE; previsible path functionals; diffusion SDEs 9. Pathwise uniqueness; exact SDEs 10. Relationship between exact SDEs and strong solutions 11. The It6 existence and uniqueness result 12. Locally Lipschitz SDEs; Lipschitz properties of am 13. Flows; the diffeomor n theorem; time-reversed flows 14. Carverhill''''s noisy North-South flow on a circle 15. The martingale optimality principle in control 3. WEAK SOLUTIONS, UNIQUENESS IN LAW 16. Weak solutions of SDEs; Tanaka''''s SDE 17. ''''Exact equals weak plus pathwise unique'''' 18. Tsirel''''son''''s example 4. MARTINGALE PROBLEMS, MARKOV PROPERTY 19. Definition; orientation 20. Equivalence of the martingale-problem and ''''weak'''' formulations 21. Martingale problems and the strong Markov property 22. Appraisal and consolidation: where we have reached 23. Existence of solutions to the martingale problem 24. The Stroock-Varadhan uniqueness theorem 25. Martingale representation Transformation of SDEs 26. Change of time scale; Girsanov''''s SDE. 27. Change of measure 28. Change of state-space; scale; Zvonkin''''s observation; the Doss- Sussmann method 29. Krylov''''s example 5. OVERTURE TO STOCHASTIC DIFFERENTIAL GEOMETRY 30. Introduction; some key ideas; Stratonovich-to-Ito conversion 31. Brownian motion on a submanifold of RN 32. Parallel displacement; Riemannian connections 33. Extrinsic theory of BMhor O ; rolling without slipping; martin- gales on manifolds; etc. 34. Intrinsic theory; normal coordinates; structural equations; diffu- sions on manifolds; etc. ! 35. Brownian motion on Lie groups 36. Dynkin''''s Brownian motion of ellipses; hyperbolic space interpret- ation; etc. 37. Khasminskii''''s method for studying stability; random vibrations. 38. H6rmander''''s theorem; Malliavin calculus; stochastic pullback; curvature 6. ONE-DIMENSIONAL SDEs 39. A local-time criterion for pathwise uniqueness 40. The Yamada-Watanabe pathwise uniqueness theorem 41. The Nakao pathwise-uniqueness theorem 42. Solution of a variance control problem 43. A comparison theorem 7. ONE-DIMENSIONAL DIFFUSIONS 44. Orientation 45. Regular diffusions 46. The scale function, s 47. The speed measure, m; time substitution 48. Example: the Bessel SDE 49. Diffusion local time 50. Analytical aspects 51. Classification of boundary points 52. Khasminskii''''s test for explosion 53. An ergodic theorem for 1-dimensional diffusions 54. Coupling of 1-dimensional diffusions CHAPTER VI. THE GENERAL THEORY 1. ORIENTATION 1. Preparatory remarks 2. Levy processes 2. DEBUT AND SECTION THEOREMS 3. Progressive processes 4. Optional processes, optional times 5. The ''''optional'''' section theorem 6. Warning not to be skipped 3. OPTIONAL PROJECTIONS AND FILTERING 7. Optional projection X of X 8. The innovations approach to filtering 9. The Kalman-Bucy filter 10. The Bayesian approach to filtering; a change-detection filter 11. Robust filtering. 4. CHARACTERIZING PREVISIBLE TIMES 12. Previsible stopping times; PFA theorem 13. Totally inaccessible and accessible stopping times. 14. Some examples. 15. Meyer''''s previsibility theorem for Markov processes 16. Proof of the PFA theorem 17. The a-algebras p- , p , p 18. Quasi-left-continuous filtrations 5. DUAL PREVISIBLE PROJECTIONS 19. The previsible section theorem; the previsible projection oX of X 20. Doleans'''' characterization of FV processes 21. Dual previsible projections, compensators 22. Cumulative risk 23. Some Brownian motion examples 24. Decomposition of a continuous semimartingale 25. Proof of the basic u, A correspondence 26. Proof of the Doleans ''''optional'''' characterization result 27. Proof of the Doleans ''''previsible'''' characterization result 28. Levy systems for Markov processes 6. THE MEYER DECOMPOSITION THEOREM 29. Introduction. 30. The Doleans proof of the Meyer decomposition 31. Regular class D submartingales; approximation to compensators 32. The local form of the decomposition theorem 33. An L2 bounded local martingale which is not a martingale 34. The <M> process 35. Last exits and equilibrium charge 7. STOCHASTIC INTEGRATION: THE GENERAL CASE 36. The quadratic variation process [M] 37. Stochastic integrals with respect to local martingales 38. Stochastic integrals with respect to semimartingales 39. It6''''s formula for semimartingales 40. Special semimartingales 41. Quasimartingales
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