Rough path analysis provides a fresh perspective on Ito's important theory of stochastic differential equations. Key theorems of modern stochastic analysis (existence and limit theorems for stochastic flows, Freidlin-Wentzell theory, the Stroock-Varadhan support description) can be obtained with dramatic simplifications. Classical approximation results and their limitations (Wong-Zakai, McShane's counterexample) receive 'obvious' rough path explanations. Evidence is building that rough paths will play an important role in the future analysis of stochastic partial differential equations and the authors include some first results in this direction. They also emphasize interactions with other parts of mathematics, including Caratheodory geometry, Dirichlet forms and Malliavin calculus. Based on successful courses at the graduate level, this up-to-date introduction presents the theory of rough paths and its applications to stochastic analysis. Examples, explanations and exercises make the book accessible to graduate students and researchers from a variety of fields.
Author(s): Peter K. Friz, Nicolas B. Victoir
Series: Cambridge Studies in Advanced Mathematics 120
Edition: 1
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 672
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 15
Introduction......Page 17
1 From ordinary to rough differential equations......Page 20
2 Carnot-Caratheodory geometry......Page 24
3 Brownian motion and stochastic analysis......Page 29
Part I Basics......Page 33
1.1 Continuous paths on metric spaces......Page 35
1.2.1 Bounded variation paths and controls......Page 37
1.2.2 Absolute continuity......Page 42
1.2.3 Lipschitz or 1-Holder continuity......Page 44
1.3 Continuous paths of bounded variation on Rd......Page 45
1.3.1 Continuously differentiable paths......Page 46
1.3.2 Bounded variation......Page 47
1.3.3 Closure of smooth paths in variation norm......Page 49
1.3.4 Lipschitz continuity......Page 52
1.4.1 Paths of Sobolev regularity on Rd......Page 55
1.4.2 Paths of Sobolev regularity on metric spaces......Page 58
1.5 Comments......Page 60
2.1 Basic Riemann-Stieltjes integration......Page 61
2.2 Continuity properties......Page 65
2.3 Comments......Page 68
3.1 Preliminaries......Page 69
3.2 Existence......Page 71
3.3 Uniqueness......Page 75
3.4 A few consequences of uniqueness......Page 76
3.5.1 Limit theorem for 1-variation signals......Page 78
3.5.2 Continuity under 1-variation distance......Page 81
3.6 Comments......Page 83
4.1.1 Directional derivatives......Page 84
4.1.2 Frechet differentiability......Page 89
4.2 Comments......Page 92
5.1.1 Definition and first properties......Page 93
5.1.2 On some path-spaces contained in Cp-var ([0,T], E)......Page 101
5.2 Approximations in geodesic spaces......Page 104
5.3.1 Holder and p-variation Banach spaces......Page 108
5.3.2 Compactness......Page 110
5.3.3 Closure of smooth paths in variation norm......Page 111
5.4.1 Definition and basic properties......Page 115
5.4.2 Some explicit estimates for…......Page 119
5.5.1 Definition and basic properties......Page 120
5.5.2 Approximations to 2D functions......Page 123
5.6 Comments......Page 127
6.1 Young-Loeve estimates......Page 128
6.2 Young integrals......Page 131
6.3 Continuity properties of Young integrals......Page 134
6.4 Young-Loeve-Towghi estimates and 2D Young integrals......Page 135
6.5 Comments......Page 138
Part II Abstract theory of rough paths......Page 139
7.1 Motivation: iterated integrals and higher-order Euler schemes......Page 141
7.2.1 Definition of SN......Page 144
7.2.2 Basic properties of SN......Page 147
7.3.1 The group 1+tN ( Rd)......Page 150
7.3.2 The Lie algebra tN ( Rd) and the exponential map......Page 151
7.3.3 The Campbell-Baker-Hausdorff formula......Page 153
7.4 Chow's theorem......Page 156
7.5.1 Definition and characterization......Page 158
7.5.2 Geodesic existence......Page 160
7.5.3 Homogenous norms......Page 162
7.5.4 Carnot-Caratheodory metric......Page 163
7.5.5 Equivalence of homogenous norms......Page 165
7.5.6 From linear maps to group homomorphisms......Page 169
7.6.1 Quantitative bound on SN......Page 172
7.6.2 Modulus of continuity for the map SN......Page 175
7.7 Comments......Page 179
8 Variation and Holder spaces on free groups......Page 181
8.1.1 Homogenous p-variation and Holder distances......Page 182
8.1.2 Inhomogenous p-variation and Holder distances......Page 185
8.1.3 Homogenous vs inhomogenous distances......Page 188
8.2 Geodesic approximations......Page 190
8.4 The d0/d…......Page 191
8.5 Interpolation and compactness......Page 193
8.6 Closure of lifted smooth paths......Page 194
8.7 Comments......Page 197
9 Geometric rough path spaces......Page 198
9.1.1 Quantitative bound on SN......Page 199
9.1.2 Definition of the map SN on Cop-var ([0,T], G[p] (Rd))......Page 201
9.1.3 Modulus of continuity for the map SN......Page 204
9.2 Spaces of geometric rough paths......Page 207
9.3 Invariance under Lipschitz maps......Page 212
9.4 Young pairing of weak geometric rough paths......Page 213
9.4.2 The space C(p,q)-var ([0,T], Rd…......Page 214
9.4.3 Quantitative bounds on SN......Page 216
9.4.4 Definition of Young pairing map......Page 220
9.4.5 Modulus of continuity for the map SN......Page 221
9.4.6 Translation of rough paths......Page 225
9.5 Comments......Page 227
10.1 Preliminaries......Page 228
10.2 Davie's estimate......Page 231
10.3.1 Passage to the limit with uniform estimates......Page 237
10.3.2 Definition of RDE solution and existence......Page 240
10.3.3 Local existence......Page 241
10.3.4 Uniqueness and continuity......Page 243
10.3.5 Convergence of the Euler scheme......Page 254
10.4.1 Definition......Page 257
10.4.2 Existence......Page 258
10.4.3 Uniqueness and continuity......Page 261
10.5 RDEs under minimal regularity of coefficients......Page 264
10.6 Integration along rough paths......Page 269
10.7 RDEs driven along linear vector fields......Page 278
10.8 Appendix: p-variation estimates via approximations......Page 284
10.9 Comments......Page 295
11.1 Smoothness of the Ito-Lyons map......Page 297
11.1.1 Directional derivatives......Page 298
11.1.2 Frechet differentiability......Page 303
11.2 Flows of diffeomorphisms......Page 305
11.3 Application: a class of rough partial differential equations......Page 310
11.4 Comments......Page 317
12.1 RDEs with drift terms......Page 318
12.1.1 Existence......Page 320
12.1.2 Uniqueness and continuity......Page 327
12.2.1 (Higher-)area perturbations and modified drift terms......Page 332
12.2.2 Limits of Wong-Zakai type with modified area......Page 337
12.3 Comments......Page 340
Part III Stochastic processes lifted to rough paths......Page 341
13.1.1 Brownian motion......Page 343
13.1.2 Levy's area: definition and exponential integrability......Page 344
13.1.3 Levy's area as time-changed Brownian motion......Page 348
13.2.1 Brownian motion lifted to a G2( Rd) -valued path......Page 349
13.2.2 Rough path regularity......Page 352
13.3 Strong approximations......Page 355
13.3.2 Nested piecewise linear approximations......Page 356
13.3.3 General piecewise linear approximations......Page 359
13.3.4 Limits of Wong-Zakai type with modified Levy area......Page 363
13.3.5 Convergence of 1D Brownian motion and its…......Page 367
13.4.1 Donsker's theorem for enhanced Brownian motion......Page 370
13.5 Cameron-Martin theorem......Page 373
13.6 Large deviations......Page 375
13.6.1 Schilder's theorem for Brownian motion......Page 376
13.6.2 Schilder's theorem for enhanced Brownian motion......Page 378
13.7.1 Support of Brownian motion......Page 383
13.7.3 Support of enhanced Brownian motion......Page 385
13.8.1 Brownian motion conditioned to stay near the origin......Page 386
13.8.2 Intermezzo on rough path distances......Page 393
13.8.3 Enhanced Brownian motion under conditioning......Page 394
13.9 Appendix: infinite 2-variation of Brownian motion......Page 397
13.10 Comments......Page 399
14.1 Enhanced continuous local martingales......Page 402
14.2 The Burkholder-Davis-Gundy inequality......Page 404
14.3 p-Variation rough path regularity of enhanced martingales......Page 406
14.4 Burkholder–Davis–Gundy with p-variation rough path norm......Page 408
14.5 Convergence of piecewise linear approximations......Page 411
14.6 Comments......Page 417
15.1 Motivation and outlook......Page 418
15.2.1 ρ-Variation of the covariance......Page 420
15.2.2 A Cameron-Martin/variation embedding......Page 424
15.2.3 Covariance of piecewise linear approximations......Page 426
15.2.4 Covariance of mollifier approximations......Page 429
15.2.5 Covariance of Karhunen-Loeve approximations......Page 430
15.3 Multidimensional Gaussian processes......Page 432
15.3.1 Wiener chaos......Page 433
15.3.2 Uniform estimates for lifted Gaussian processes......Page 437
15.3.3 Enhanced Gaussian process......Page 444
15.4 The Young-Wiener integral......Page 449
15.5.1 Piecewise linear approximations......Page 452
15.5.2 Mollifier approximations......Page 453
15.5.3 Karhunen-Loeve approximations......Page 454
15.6.1 Tightness......Page 458
15.6.2 Convergence......Page 459
15.7 Large deviations......Page 461
15.8 Support theorem......Page 464
15.9 Appendix: some estimates in G3(Rd)......Page 467
15.10 Comments......Page 468
16.1 Motivation......Page 470
16.2 Uniformly subelliptic Dirichlet forms......Page 473
16.3 Heat-kernel estimates......Page 479
16.4 Markovian rough paths......Page 480
16.5.1 Geodesic approximations......Page 483
16.5.2 Piecewise linear approximations......Page 485
16.6.2 Convergence......Page 496
16.7 Large deviations......Page 499
16.8.1 Uniform topology......Page 500
16.8.2 Holder topology......Page 501
16.8.3 Holder rough path topology: a conditional result......Page 503
16.9.1 Haar measure......Page 509
16.9.2 Jerison's Poincare inequality......Page 511
16.9.3 Carnot-Caratheodory metric as intrinstic metric......Page 514
16.10 Comments......Page 515
Part IV Applications to stochastic analysis......Page 517
17.1.2 Integration......Page 519
17.1.3 Differential equations......Page 520
17.1.4 Differential equations with drift......Page 521
17.2.1 Stratonovich integration as rough integration......Page 522
17.2.2 Stratonovich SDEs as RDEs......Page 525
17.3 Stochastic differential equations driven by non-semi-martingales......Page 531
17.4.1 Strong limit theorems......Page 533
17.4.2 Weak limit theorems......Page 536
17.5 Stochastic flows of diffeomorphisms......Page 537
17.6.1 Anticipating vector fields and initial condition......Page 539
17.6.2 Stochastic delay differential equations......Page 540
17.7 A class of stochastic partial differential equations......Page 541
17.8 Comments......Page 542
18.1 Azencott-type estimates......Page 544
18.2 Weak remainder estimates......Page 547
18.3 Comments......Page 548
19.1 Support theorem for SDEs driven by Brownian motion......Page 549
19.2 Support theorem for SDEs driven by other stochastic processes......Page 552
19.3 Large deviations for SDEs driven by Brownian motion......Page 554
19.4 Large deviations for SDEs driven by other stochastic processes......Page 557
19.5 Support theorem and large deviations for a class of SPDEs......Page 558
19.6 Comments......Page 560
20.1 H-regularity of RDE solutions......Page 561
20.2 Non-degenerate Gaussian driving signals......Page 565
20.3 Densities for RDEs under ellipticity conditions......Page 566
20.4 Densities for RDEs under Hormander's condition......Page 569
20.4.1 Conditions on the Gaussian process......Page 570
20.4.2 Taylor expansions for rough differential equations......Page 574
20.4.3 Hormander's condition revisited......Page 577
20.4.4 Proof of Theorem 20.12......Page 579
20.5 Comments......Page 582
Part V Appendices......Page 585
A.1.1 Generalities......Page 587
A.2.1 Garsia–Rodemich–Rumsey on metric spaces......Page 589
A.2.2 Garsia–Rodemich–Rumsey on GN( Rd) -valued paths......Page 593
A.3.1 Holder regularity and tightness......Page 598
A.3.2 Lq-convergence for rough paths......Page 602
A.4 Sample path regularity under Gaussian assumptions......Page 603
A.5 Comments......Page 612
B.1 Preliminaries......Page 613
B.2 Directional and Frechet derivatives......Page 614
B.3 Higher-order differentiability......Page 617
B.4 Comments......Page 618
C.1 Definition and basic properties......Page 619
C.2 Contraction principles......Page 620
D.1 Preliminaries......Page 622
D.2 Isoperimetry and concentration of measure......Page 624
D.4 Wiener–Ito chaos......Page 626
D.5 Malliavin calculus......Page 629
D.6 Comments......Page 630
E.1 Quadratic forms......Page 631
E.2 Symmetric Markovian semi-groups and Dirichlet forms......Page 633
E.3 Doubling, Poincare and quasi-isometry......Page 636
E.4 Parabolic equations and heat-kernels......Page 639
E.5 Symmetric diffusions......Page 641
E.6.1 Fernique estimates......Page 643
E.6.2 Schilder's theorem......Page 645
E.6.3 Support theorem......Page 648
E.7 Comments......Page 651
Frequently Used Notation......Page 652
References......Page 654
Index......Page 668
