Multidimensional Stochastic Processes as Rough Paths

Abstract

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.

Yet another introduction to rough paths

“…For example, many trajectories of stochastic processes have been dealt with (Brownian motion [Sip93], semi-martingales [CL05], fractional Brownian motion [CQ02, MSS05, Cou07], Wiener process [LLQ02], Gaussian processes [FV07b,FV07a], free Brownian motion [Vic04], ... The book [FV06a] contains a large number of these constructions). All these results are done in general in connection with an approximation results of Wong-Zakai type.…”

Stochastic Taylor expansions and heat kernel asymptotics

“…More precisely considering the closure, with respect to a convenient topology, of the algebra of Chen series of absolutely continuous paths leads to the notion of geometric rough paths. So, the results of this section admits a counterpart in rough paths theory (See [22]). …”

“…Related discussions for the Taylor expansion of solutions of equations driven by fractional Brownian motions may be found in Baudoin-Coutin [6]. For the case of rough paths we refer to Inahama [27] and Friz-Victoir [21,22].…”

“…Chen's theorem can actually be extended to Brownian paths (see Baudoin [4], Ben Arous [10], Castell [15], Fliess [19]) and even to rough paths (see Lyons [32], Friz-Victoir [22]). …”

“…Kunita, in [29], proved an extension of this theorem to stochastic flows generated by stochastic differential equations driven by Lévy processes. Also, the result may be stated in the context of rough paths theory (see [22]). …”