Topics in Gaussian rough paths theory vorgelegt von Diplom-Mathematiker Sebastian Riedel Hannover Von der Fakult¨atII - Mathematik und Naturwissenschaften der Technischen Universit¨atBerlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Dr.rer.nat. genehmigte Dissertation Promotionsausschuss: Vorsitzende: Prof. Dr. Gitta Kutyniok Berichter/Gutachter: Prof. Dr. Peter K. Friz Berichter/Gutachter: Prof. Dr. Martin Hairer Tag der wissenschaftlichen Aussprache: 23. April 2013 Berlin 2013 D 83 Berlin, May 5, 2013 Acknowledgement At first, I would like to express my gratitude to my PhD advisor, Professor Peter Friz, who constantly supported me during the time of my doctorate. In particular, I would like to thank Peter for the time he always found for discussing with me and for the patience he had. His enduring encouragement laid the basis for the current work. Next, I would like to thank Professor Martin Hairer for being my second examiner, and Professor Gitta Kutyniok who kindly agreed to be the chair of the examination board. I am indebted to all my collaborators who worked with me during the last three years. Namely, I would like to thank Doctor Christian Bayer, Doctor Benjamin Gess, Professor Archil Gulisashvili, Professor Peter Friz, PD Doctor John Schoenmakers and Weijun Xu. Furthermore, I would like to thank Professor Terry Lyons for inviting me to Oxford during my PhD and for the valuable discussions we had. This work could have not been written without the financial support of the International Research Training Group \Stochastic models of complex processes" and the Berlin Mathematical School (BMS). I would like to thank all the people working there for their helpfulness and kindness they showed to me during the last years. Special thanks go to Joscha Diehl, Cl´ement Foucart, Birte Schr¨oderand Maite Wilke Berenguer for reading parts of this thesis and giving valuable comments. At this point, I would like to mention my colleagues and the friends I met in the mathematical institute of the Technische Universit¨atBerlin who gave me a very warm welcome and provided an open and friendly atmosphere during the time of my doctorate. In particular, I would like to thank Professor Michael Scheutzow, my BMS mentor, who gave me a lot of helpful advices concerning my PhD. I am also more than thankful to the following people: Michele, who made me laugh uncountably many times and who introduced me to the dark secrets of pasta and facebook. To Joscha for the possibility to ask the really relevant questions about rough paths. Thank you, Simon, for many fruitful discussions about and not about math. Maite, thank you for making me get up on Monday before 7:00 by offering coffee and for the joint exercise sessions. Thanks to Cl´ement for having many beers with me after work and for the first part of Brice de Nice. Last, thank you, Stefano, for B.F.H., the 1st of May and the 2nd chaos. At the end, I would like to thank my family, in particular my parents, who always had faith in what I am doing. Finally, my biggest thanks go to Birte for encouraging me during the last years, for sharing successes and defeats, joy and sorrow, and for chasing math away when it should not be there. ii To Birte iv Contents Introduction 1 Notation and basic definitions 17 1 Convergence rates for the full Brownian rough paths with applications to limit theorems for stochastic flows 21 1.1 Rates of Convergence for the full Brownian rough path . 24 2 Convergence rates for the full Gaussian rough paths 35 2.1 Iterated integrals and the shuffle algebra . 39 2.2 Multidimensional Young-integration and grid-controls . 43 2.3 The main estimates . 47 2.4 Main result . 66 3 Integrability of (non-)linear rough differential equations and integrals 73 3.1 Basic definitions . 74 3.2 Cass, Litterer and Lyons revisited . 76 3.3 Transitivity of the tail estimates under locally linear maps . 81 3.4 Linear RDEs . 83 3.5 Applications in stochastic analysis . 85 4 A simple proof of distance bounds for Gaussian rough paths 91 4.1 2D variation and Gaussian rough paths . 93 4.2 Main estimates . 98 4.3 Applications . 103 5 Spatial rough path lifts of stochastic convolutions 107 5.1 Main Result . 109 5.2 Conditions in terms of Fourier coefficients . 113 5.3 Lifting Ornstein-Uhlenbeck processes in space . 120 6 From rough path estimates to multilevel Monte Carlo 131 6.1 Rough path estimates revisited . 133 6.2 Probabilistic convergence results for RDEs . 137 6.3 Giles' complexity theorem revisited . 141 6.4 Multilevel Monte Carlo for RDEs . 149 Appendix 151 A Kolmogorov theorem for multiplicative functionals . 151 Contents vi Introduction \What I don't like about measure theory is that you have to say 'almost everywhere' almost everywhere." { Kurt Friedrichs \What is Ω? Cats?" { Michele Salvi In order to describe correctly the research contributions of this thesis, we begin by a brief history of the theory of stochastic integration with a focus on the attempt to define a pathwise integral. We will explain the very first goal of the precursors as well as the generalisations made by some of the leaders in the field. In particular, we will highlight the link between pathwise stochastic integration and Gaussian analysis. The introduction closes with an outline of our results. A basic problem in stochastic calculus is to give a meaning to differential equations of the form Y_t = f(Yt) X_ t; Y0 = ξ 2 W; (1) Y taking values in some Banach space W , X : [0;T ] ! V being some random signal with values in a Banach space V and f taking values in the space of linear maps from V to W . In a deterministic setting, these equations are also called controlled differential equations. In many cases in stochastics, it is natural to assume that X_ denotes some \noise" term which can formally be written as the differential of a Brownian motion B. However, this causes problems when we try to give a rigorous meaning to (1). In fact, a famous property of the trajectories t 7! !(t) of the Brownian motion, i.e. its sample paths, is their non-differentiability on a set of full measure. Therefore, we cannot apply the deterministic theory of controlled differential equations. One approach is to rewrite the differential equation (1) as an integral equation: Z t Yt = Y0 + f(Ys) dXs: (2) 0 By doing this, we shift the problem of defining (1) to the problem of how to define the (stochas- tic) integral in (2). More generally, we may ask the following question: How can we define a stochastic integral of the form Z t Ys dXs (3) 0 where X and Y are stochastic processes taking values in V resp. L(V; W )? There are basically two strategies we can follow. The first one ignores all the probabilistic structure the processes X and Y might have and tries to build up a deterministic theory of integration which is rich enough Introduction in order to integrate all sample paths of X and Y with respect to each other. We will call this the pathwise approach. The second strategy uses the probabilistic properties of the processes under consideration in order to define the integral, and we will call this the probabilistic approach. We will see that Lyons' rough paths theory can be seen as a pathwise approach, whereas the classical It¯otheory is rather a probabilistic approach. In the following, we will summarise the most important attempts of defining stochastic integrals of the form (3) in order to better understand the contribution of rough paths theory in the context of stochastic integration. This permits us to explain the notion of Gaussian rough paths which provides the framework for this thesis. Young's approach The first and probably most \natural" pathwise approach is to define the integral (3) as the limit (at least in probability) of Riemann-sums: Z t X Ys dXs = lim Yti (Xti+1 − Xti ); (4) 0 jΠj!0 ti2Π where the Π are finite partitions of the interval [0; t]. It is commonly known that this limit exists (pathwise) if the sample paths of X have bounded variation (which is the same as to say that the sample paths have finite length): X lim jXti+1 − Xti j < 1 a.s. jΠj!0 ti2Π Unfortunately, this is not the case for the Brownian motion and the above quantity will be infinite almost surely in this case. A more elaborated approach was developed by Laurence C. Young, starting from the article [You36] and further developed in a series of papers. Recall the notion of p-variation, a generalisation of the concept of bounded variation: If x: [0;T ] ! V is a path and p ≥ 1, the p-variation of x is defined as 1 0 1 p X p sup @ jxti+1 − xti j A : Π ti2Π The main theorem of Young can be stated as follows: If x and y are paths of finite p- resp. 1 1 1 q-variation with p + q > 1, the limit in (4) exists and can be bounded in terms of the p- and 1 1 q-variation of x and y. Let us note that the condition p + q > 1 is necessary; indeed, Young gives a counterexample by constructing paths x and y which have finite 2-variation only and for which the Riemann sums (4) diverge. Recall now that our initial aim was to solve integral equations of the form Z t yt = y0 + f(ys) dxs: (5) 0 If this equation has a solution, we expect (at least for smooth f) that the solution y has a regularity (on small scales) which is similar to the regularity of x.