Multidimensional Stochastic Processes as Rough Paths: Theory and Applications
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.
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Contents
Continuous paths of bounded variation
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19 |
Variation and Hölder spaces
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77 |
Young integration
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112 |
Free nilpotent groups
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125 |
Variation and Hölder spaces on free groups
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165 |
1
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168 |
2
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182 |
1
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211 |
Stochastic differential equations and stochastic flows
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503 |
Stochastic Taylor expansions
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528 |
Support theorem and large deviations
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533 |
Malliavin calculus for RDES
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545 |
A Sample path regularity and related topics
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571 |
Kolmogorovtype corollaries
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582 |
Comments
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596 |
Comments
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602 |
Rough differential equations
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212 |
2
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226 |
smoothness
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281 |
RDES with drift and other topics
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302 |
Continuous paths of bounded variation on metric spaces
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324 |
Brownian motion
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327 |
13
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343 |
29
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352 |
39
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360 |
44
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366 |
Continuous semimartingales
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386 |
Gaussian processes
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402 |
Markov processes
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454 |
Malliavin calculus
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613 |
Comments
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614 |
E Analysis on local Dirichlet spaces
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615 |
Symmetric Markovian semigroups and Dirichlet forms
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617 |
Doubling Poincaré and quasiisometry
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620 |
Parabolic equations and heatkernels
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623 |
Symmetric diffusions
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625 |
Stochastic analysis
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627 |
Comments
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635 |
Frequently used notation
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636 |
References
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638 |
652 | |
Common terms and phrases
a-Hölder Assume assumption bounded sets bounded variation c₁ Carnot-Caratheodory Chen's theorem continuous path convergence Corollary Cp-var define definition denotes differential equations dissection enhanced Brownian motion estimate Exercise finite p-variation fixed follows full RDE function g¹ Rd G² Rd Gaussian process geodesic geometric p-rough path GN Rd hence Hölder homogenous norms implies inequality large deviation Lemma lift Lipschitz Lipschitz continuous martingale metric metric space path topology piecewise linear piecewise linear approximations Proof Proposition Rd)-valued RDE solution resp Riemann-Stieltjes integral rough path S₂ sample paths Section sequence SN+1 space step-N stochastic Theorem ti+1 TN Rd topology triangle inequality Ts,t uniformly uniformly continuous unique vector fields wx,p Xs,t Young integral Ys,t