RANDOM TIMES AND ENLARGEMENTS OF FILTRATIONS IN A BROWNIAN SETTING

By PATRICIA WILLIAMS

NOTATION AND CONVENTION PRELIMINARIES DOOB’S MAXIMAL IDENTITY BALAYAGE FORMULA PREDICTABLE COMPENSATORS ?-?ELDS ASSOCIATED WITH A RANDOM TIME ? INTEGRATION BY PARTS FORMULAE H1 AND BMO SPACES EXERCISES ENLARGEMENTS OF FILTRATIONS SOME GENERAL PROBLEMS OF ENLARGEMENTS PROGRESSIVE ENLARGEMENT DECOMPOSITION FORMULA PITMAN’S THEOREM ON 2S?B AND SOME GENERALIZATIONS VIA SOME PROGRESSIVE ENLARGEMENT OF FILTRATION INITIAL ENLARGEMENT FURTHER REFERENCES A APPENDIX: SOME ENLARGEMENTS FORMULAE TABLES 1? AND 1?: PROGRESSIVE ENLARGEMENTS TABLES 2? AND 2?: INITIAL ENLARGEMENTS COMMENTS ON THE TABLES STOPPING AND NON-STOPPING TIMES STOPPING TIMES AND DOOB’S OPTIONAL THEOREM THE KNIGHT-MAISONNEUVE CHARACTERIZATION OF STOPPING TIMES D WILLIAMS’ EXAMPLE OF A PSEUDO-STOPPING TIME A CHARACTERIZATION OF PSEUDO-STOPPING TIMES HOW BADLY ARE THE BDG INEQUALITIES A?ECTED BY A GENERAL RANDOM TIME? A GLOBAL APPROACH (COMMON TO ALL ?’S) AN “INDIVIDUAL” APPROACH (DEPENDING ON ?) LOCAL TIME ESTIMATES ON THE MARTINGALES WHICH VANISH ON THE SET OF BROWNIAN ZEROES SOME QUANTITIES ASSOCIATED WITH ? AZ´EMA SUPERMARTINGALE AND THE PREDICTABLE COMPENSATOR ASSOCIATED WITH ? PATH DECOMPOSITION RELATIVE TO ? BROWNIAN MEANDER SOME EXAMPLES OF MARTINGALES WHICH VANISH ON Z = {T; BT = 0} SOME BROWNIAN MARTINGALES WITH A GIVEN LOCAL TIME, OR SUPREMUM PROCESS A REMARKABLE COINCIDENCE BETWEEN E [X|F?] AND X? RESOLUTION OF SOME CONDITIONAL EQUATIONS UNDERSTANDING HOW E [X|F?] AND X? DI?ER PRP AND CRP FOR SOME REMARKABLE MARTINGALES DE?NITION AND FIRST EXAMPLE PRP AND EXTREMAL MARTINGALE DISTRIBUTIONS CRP: AN ATTEMPT TOWARDS A GENERAL DISCUSSION AN ATTEMPT TO UNDERSTAND THE CRP IN TERMS OF A GENERALIZED MOMENTS PROBLEM SOME SU?CIENT CONDITIONS FOR THE CRP THE CASE OF THE AZ´EMA MARTINGALE UNVEILING THE BROWNIAN PATH (OR HISTORY) AS THE LEVEL RISES ABOVE AND UNDER A GIVEN LEVEL FIRST COMPUTATIONS IN WILLIAMS’ FRAMEWORK FIRST COMPUTATIONS IN AZ´EMA-HU’S FRAMEWORK ROGERS-WALSH THEOREM ABOUT WILLIAMS’ FILTRATION SOME SPACE MARTINGALES WHICH ARE CONSTANT UP TO A FIXED LEVEL A DENSE FAMILY OF CONTINUOUS MARTINGALES A DISCUSSION RELATIVE TO (EA A, A ?R) HU’S RESULT ABOUT (EA A, A ?R)-MARTINGALES SOME MARKOV PROCESSES WITH RESPECT TO (EA A, A ?R) SOME SUB?LTRATIONS OF THE BROWNIAN FILTRATION WEAK AND STRONG BROWNIAN FILTRATIONS DE?NITIONS EXAMPLES OF WEAK BROWNIAN FILTRATIONS CHANGE OF PROBABILITY CHANGE OF TIME WALSH’S BROWNIAN MOTION AND SPIDER MARTINGALES INVARIANTS OF A FILTRATION SKETCHES OF SOLUTIONS FOR THE EXERCISES