随机过程一般理论综述
An essay on the general theory of stochastic processes
Ashkan Nikeghbali
英文版

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
2 Basic notions of the general theory . . . . . . . . . . . . . . . . . . . . 347
2.1 Stopping times . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
2.2 Progressive, Optional and Predictable σ-fields . . . . . . . . . . . 348
2.3 Classification of stopping times . . . . . . . . . . . . . . . . . . . 351
2.4 D´ebut theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
3 Section theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
4 Projection theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
4.1 The optional and predictable projections . . . . . . . . . . . . . . 357
4.2 Increasing processes and projections . . . . . . . . . . . . . . . . 360
4.3 Random measures on (R+ ×
) and the dual projections . . . . . 362
5 The Doob-Meyer decomposition and multiplicative decompositions . . 371
6 Multiplicative decompositions . . . . . . . . . . . . . . . . . . . . . . . 372
7 Some hidden martingales . . . . . . . . . . . . . . . . . . . . . . . . . 375
8 General random times, their associated σ-fields and Az´ema’s supermartingales
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
8.1 Arbitrary random times and some associated sigma fields . . . . 381
8.2 Az´ema’s supermartingales and dual projections associated with
random times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
8.2.1 The case of honest times . . . . . . . . . . . . . . . . . . . 385
8.2.2 The case of pseudo-stopping times . . . . . . . . . . . . . 391
8.3 Honest times and Strong Brownian Filtrations . . . . . . . . . . 395
9 The enlargements of filtrations . . . . . . . . . . . . . . . . . . . . . . 396
9.1 Initial enlargements of filtrations . . . . . . . . . . . . . . . . . . 397
9.2 Progressive enlargements of filtrations . . . . . . . . . . . . . . . 401
9.2.1 A description of predictable and optional processes in (Gρ
t )and􀀀FLt. . . . . . . . . . . . . . . . . . . . . . . . . . . 402
9.2.2 The decomposition formula before ρ . . . . . . . . . . . . 403
9.2.3 The decomposition formula for honest times . . . . . . . . 405
9.3 The (H) hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . 407
9.4 Concluding remarks on enlargements of filtrations . . . . . . . . 408
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408