Topics in Gaussian rough paths theory

Transcription

1 Topics in Gaussian rough paths theory vorgelegt von Diplom-Mathematiker Sebastian Riedel Hannover Von der Fakultät II - Mathematik und Naturwissenschaften der Technischen Universität Berlin zur Erlangung des akademischen Grades Doktor der Naturwissenschaften Drrernat 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: 3 April 013 Berlin 013 D 83

2 Berlin, May 5, 013

3 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, Clément Foucart, Birte Schröder and 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 Universität Berlin 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 Clément for having many beers with me after work and for the first part of Brice de Nice Last, thank you, Stefano, for BFH, the 1st of May and the nd 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

4 ii

5 To Birte

6 iv

7 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 1 11 Rates of Convergence for the full Brownian rough path 4 Convergence rates for the full Gaussian rough paths 35 1 Iterated integrals and the shuffle algebra 39 Multidimensional Young-integration and grid-controls 43 3 The main estimates 47 4 Main result 66 3 Integrability of non-linear rough differential equations and integrals Basic definitions 74 3 Cass, Litterer and Lyons revisited Transitivity of the tail estimates under locally linear maps Linear RDEs Applications in stochastic analysis 85 4 A simple proof of distance bounds for Gaussian rough paths D variation and Gaussian rough paths 93 4 Main estimates Applications Spatial rough path lifts of stochastic convolutions Main Result Conditions in terms of Fourier coefficients Lifting Ornstein-Uhlenbeck processes in space 10 6 From rough path estimates to multilevel Monte Carlo Rough path estimates revisited Probabilistic convergence results for RDEs Giles complexity theorem revisited Multilevel Monte Carlo for RDEs 149 Appendix 151 A Kolmogorov theorem for multiplicative functionals 151

8 vi Contents

9 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 Ẏ t = fy t Ẋt; Y 0 = ξ 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 Ẋ 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 ωt of the Brownian motion, ie 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: Y t = Y 0 + t 0 fy s dx s By doing this, we shift the problem of defining 1 to the problem of how to define the stochastic integral in More generally, we may ask the following question: How can we define a stochastic integral of the form t 0 Y s dx s 3 where X and Y are stochastic processes taking values in V resp LV, 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

10 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 Itō theory 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: t 0 Y s dx s = lim Π 0 t i Π Y ti X ti+1 X ti, 4 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: lim Π 0 t i Π X ti+1 X ti < as 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 sup Π x ti+1 x ti p ti Π The main theorem of Young can be stated as follows: If x and y are paths of finite p- resp q-variation with 1 p + 1 q > 1, the limit in 4 exists1 and can be bounded in terms of the p- and q-variation of x and y Let us note that the condition 1 p + 1 q > 1 is necessary; indeed, Young gives a counterexample by constructing paths x and y which have finite -variation only and for which the Riemann sums 4 diverge Recall now that our initial aim was to solve integral equations of the form y t = y 0 + t 0 1 p fy s dx s 5 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 In other words, if x has finite p-variation, also y and fy should have finite p-variation This means that as long as x has finite p-variation for some p <, equation 5 should be solvable That this is indeed the case was first rigorously worked out - to the author s knowledge - by Terry Lyons in [Lyo94] for finite dimensional Banach spaces, see also [LCL07] for the general case Lyons solves equation 5 by a Picard iteration scheme and shows that the solution y varies continuously in x with respect 1 Note here that Young considers the case of complex valued paths only; however, the same proof works also in Banach spaces, cf [LCL07]

11 to p-variation topology Let us go back to stochastics now If we consider again the Brownian motion B, it turns out that sup Π B ti+1 B ti = as t i Π for a proof, cf [FV10b, Section 139], hence the sample paths of the Brownian motion do not have finite p-variation almost surely for p In other words, the regularity of the trajectories slightly fail to fulfill the necessary regularity condition and therefore the theory of Young integration cannot be applied in the Brownian motion case This, of course, is a remarkable drawback of Young s theory However, it still can be used to solve equations of the form if the trajectories of the driving signal X are not too rough ; for instance, it applies when X is a fractional Brownian motion with Hurst parameter H > 1/ the precise definition of a fractional Brownian motion will be given below Itō s theory of stochastic integration We will only consider finite dimensional Banach spaces in this paragraph In the seminal work [Itō44], Kiyosi Itō was the first who gave a satisfactory definition of the stochastic integral 3 in the case where X is a Brownian motion In [Itō51], he used this definition to solve differential equations driven by a Brownian motion Since his approach differs very much from the pathwise approach, we decided to sketch it briefly In modern language, the Itō integral is constructed by first identifying a family of simple processes which are piecewise constant, leftcontinuous and adapted with respect to the Brownian filtration Adaptedness can be understood as that at time t, the process does not have more information about the Brownian motion than it provides up to time t for instance, it cannot look in the future This is of course a probabilistic notion The integral is then defined in a natural way with respect to these processes One realizes that these simple processes and the stochastic integral both belong to a certain space of processes called martingale spaces, and the stochastic integral defines an isometry between these spaces Taking the closure in the space of integrable processes then defines the stochastic integral In contrast to the pathwise approach, the Itō integral is now defined as an element in some space of processes via an isometry In a second step, one can show that also the Riemann sums 4 converge to this object, but in general only in probability which is weaker than almost sure convergence The theory of Itō integration had an enormous success and is now widely used in stochastic calculus Together with a change of variable formula, called Itō s Lemma, it provides a powerful tool to solve stochastic differential equations of the form even for more general driving processes X However, the theory has certain constraints We list two of them: i The class of driving processes is essentially limited to semi-martingales, ie to processes which have the probabilistic properties of a fair game It is not hard to imagine models eg in finance for which the driving signal does not have this structure ii Since the integral is defined in a global way, it is a priori not clear what happens on the level of trajectories Recall that Lyons proved a pathwise continuity for the map x I f x, ξ := y, y being the solution of 5, when x is a path of finite p-variation for p < However, for Brownian trajectories ω, we do not know which regularity properties the map ω I f ω, ξ enjoys We will actually see that it is not and cannot be continuous The Itō integral has some unexpected properties For instance, when replacing the Riemann sums in 4 by Ỹ ti X ti+1 X ti, t i Π 3

12 Introduction where Ỹt i = Y ti + Y ti+1 / and then passing to the limit Π 0, we still have convergence in probability, but not to the same object, which we call the Stratonovich integral in this case This phenomenon does not occur for the Riemann-Stieltjes or the Young integral Moreover, the change-of-variable formula or Itō formula for the Itō integral contains an additional, unexpected term, the Itō correction term This term does not occur for the Stratonovich integral This already indicates that stochastic integration is very different from the usual integration theory we know, and that one has some freedom when defining the integral Föllmer s Itō formula without probability An interesting contribution in the direction of a pathwise approach was made by Hans Föllmer in the year 1981 in the work [Föl81] Föllmer considers the quadratic variation [x] of a continuous, real valued path x with respect to a sequence of partitions Π n for which the mesh-size tends to 0 for n, defined by lim x ti+1 x ti =: [x] t n t i Π n; t i <t if the right-hand side exists Föllmer shows that for these paths, the Itō-type change of variable formula fx t = fx 0 + t 0 f x s dx s + 1 t 0 f x s d[x] s holds for functions f C Note that the definition of the integral on the right hand side does not cause any problems since t [x] t is increasing, hence has bounded variation and the integral exists as a Riemann-Stieltjes integral The formula thus shows the existence of the integral t 0 f x s dx s as the limit of Riemann sums along the sequence of partitions Π n In the case of the Brownian motion, it is well-known that for its trajectories ω we have [ω] t = t almost surely for any sequence of nested partitions Π n n, hence the integral t 0 f B s db s can be defined in a pathwise manner The work of Föllmer is interesting for us since he found a sufficient criterion, finiteness of the quadratic variation, to define a stochastic integral in a pathwise sense Lyons key insights and the birth of rough paths theory In the humble opinion of the author, the final breakthrough in the task of defining stochastic integrals in a pathwise manner was made by Terry Lyons For a better understanding of the issues, we will first give some negative results which show what will not work Then we will try to sketch the main ideas of Lyons which constitute what is known today under the term of rough paths theory In the work [Lyo91], Lyons proves the following result: Let C C[0, T ], R be a class of paths for which the Itō or Stratonovich integral µ dν exists as a limit of Riemann sums for all µ, ν C Then C has Wiener measure 0 This result shows that even if we managed to extend the definition of the Young integral to a wider class of paths, using, for instance, a finer notion than the notion of p-variation, we would never be able to integrate all Brownian paths with respect to each other One could hope that a different and maybe more sophisticated definition of the integral might help us getting out of this trouble That this is not the case is 4

13 shown by Lyons in [LCL07, Proposition 19]: Let B be a Banach space on which the Wiener measure can be defined in a natural way Then there is no bilinear, continuous functional I : B B R for which Iµ, ν = 1 0 µ t dν t when µ, ν B are trigonometric polynomials This implies that whatever we choose as a linear 3 subspace C C[0, T ], R where C should be at least rich enough to handle Brownian paths, we will never be able to define a bilinear, continuous functional I : C C R which should be thought of our integral which fulfills the basic requirement that Iµ, ν = 1 0 µ t dν t holds for all µ, ν {cosπn, sinπn n N} On the level of controlled differential equations, Lyons proves the following cf [LCL07, Section 15]: The map x I f x, ξ is not continuous in -variation topology The way Lyons proves this gives us a first hint what goes wrong for p and how one might overcome this issue Lyons defines f in such a way that the solution y = I f x, ξ is given explicitly as y t = yt 1, yt = x t x 0, dx u1 dx u V V V, 0<u 1 <u <t ie, y consists of the increment of x and its first iterated integral What he actually proves is that the map x 0<u 1 <u < dx u 1 dx u is not continuous in -variation topology Loosely speaking, what we observe here is that a sufficiently rough path does not contain enough information to define its second and higher order iterated integral However, the above map I f, ξ will trivially be continuous if we include the information the iterated integral gives us, ie if we consider the map x x 0, dx u1 dx u =: x I f x, ξ 0<u 1 <u < instead We will see that this kind of continuity indeed holds in much greater generality The key insight of Lyons is that the path alone does not provide enough information in order to build up a satisfactory integration theory, but the path together with some extra information which compensates its roughness does That this extra information is indeed encoded in the iterated integrals which, we repeat, have to be defined since they are not intrinsically given in the path can be seen, for example, from numerical considerations The solution of a linear controlled differential equation can be written down formally as a power series: y t = ξ + fx t x 0 ξ + f dx u1 dx u ξ 0<u 1 <u <t + + f n 0<u 1 <<u n<t dx u1 dx un ξ + For smooth x, this series indeed converges to the solution y It is quite natural to generalise this solution concept for non-smooth paths x In stochastics, the importance of iterated integrals is well-known and widely used for numerical schemes which are known as Taylor schemes cf [KP9] Lyons first gave a systematic approach to the treatment of rough differential equations in the seminal work [Lyo98] He begins with the definition of iterated integrals in an abstract setting Of course, they cannot be just the limit of Riemann sums, but are defined to be objects which behave like iterated integrals in an algebraic and analytic way One could say that these objects mimic the iterated integrals of a path x In his First theorem or Extension theorem, cf [Lyo98, Theorem 1], he proves that the number of iterated integrals one needs to consider actually depends on the roughness of the path: If x is a path of finite p-variation and its first p iterated integrals are known, then the higher order iterated integrals are uniquely determined cf [LCL07, Proposition 19] for the precise definition here 3 We will see that the condition of a linear subspace will be crucial; indeed, our integration theory for rough paths will not be linear 5

14 Introduction Consequently, he defines a p-rough path x to be the path x together with his first p iterated integrals One should note at this point that rough paths spaces are not vector spaces they actually cannot be linear if we want to be able to integrate Brownian paths as we saw before but are metric spaces The distance of two rough paths x and y takes into account the distance in p-variation between the paths x and y and the higher order iterated integrals Lyons then defines a notion of an integral along a rough path x Note that in general it will not be possible to integrate two rough paths x and y with respect to each other since the joint integral would necessarily contain mixed integrals of x and y and hence information which is not included in the respective rough paths Instead, Lyons first defines the integral over sufficiently smooth 1-forms α: V LV, W : If x is a p-rough path, αx dx, is defined to be another p-rough path and we have continuity in rough paths topology of the map x αx dx If we can make sense of x, y as a joint rough path 4, we will be able to define the integral fy dx for sufficiently smooth functions f : W LV, W as a rough integral It turns out that in the situation of controlled differential equations we can indeed follow this strategy Lyons solves the equation dy t = fy t dx t ; y 0 = ξ 6 via a Picard iteration for a p-rough path x and sufficiently smooth f The solution y is again a p-rough path, and the map x y =: I f x, ξ is seen to be continuous in rough paths topology These results are quite technically involved, but now well understood and outlined in several monographs cf [LQ98], [LCL07] Before we come to the application of rough paths theory in the field of stochastic analysis, we will give some further remarks concerning the deterministic theory i If x n is a sequence of smooth paths, we can solve equation 6 and obtain smooth solutions y n If the iterated integrals of x n converge to a rough path x, continuity of the map x I f x, ξ implies that also the iterated integrals of y n converge to the solution y and the limit does not depend on the choice of the initial sequence This Theorem is known as the Universal limit theorem It can be seen as a deterministic analogue of the well-known Wong-Zakai theorem for Stratonovich stochastic differential equations ii The statement that the necessary extra information to define a rough path is encoded in its iterated integrals is slightly misleading In fact, the information is encoded in all iterated integrals indexed by rooted trees cf Gubinelli s work [Gub10] for a clarification However, the original statement is correct when we define the product of two iterated integrals in such a way that the algebra of iterated integrals is isomorphic to the shuffle algebra In this case, the rough path x has a nice geometric feature; namely, it is seen to take values in a Lie group, the free nilpotent group of step p over V Such paths are called weakly geometric rough paths Iterated integrals of smooth paths are also taking values in this Lie group and taking the closure with respect to the p-variation metric defines the space of geometric rough paths Every geometric rough path is also weakly geometric, but the converse is false, cf Friz and Victoir [FV06a] The geometric point of view of rough paths theory is worked out in great detail by Friz and Victoir in the monograph [FV10b] 4 This is, of course, stronger than just defining x and y as rough paths; the situation can be compared to the fact that the distribution of two random variables X and Y do not determine the joint distribution of X, Y 6

15 iii Although the space of rough paths is not a linear space, one can show that for a fixed reference rough path x, there is a linear space of paths for which all elements can be integrated with respect to x These spaces are called spaces of controlled paths and were introduced by Gubinelli in [Gub04] The integration theory for controlled paths is often more flexible and easier to handle than Lyons original integration theory and is now widely used, see also the forthcoming monograph [FH] iv Rough paths theory was, from the very beginning, closely related to numerical approximation schemes In the work [Dav07], Davie showed that deterministic Euler- and Milstein schemes converge to the solution of the respective rough differential equation This was generalised to step-n Taylor schemes for geometric rough paths by Friz and Victoir in [FV08b], see also [FV10b, Chapter 10] v The map x, f, ξ I f x, ξ, which we will call the Itō-Lyons map in the following, is even more regular than we already stated In fact, it can be seen that it is locally Lipschitz continuous in every argument cf [FV10b, Chapter 10] Moreover, the map x I f x, ξ is even Fréchet differentiable cf Li and Lyons [LL06] for the case p < and Friz and Victoir [FV10b, Theorem 116] for the general case of geometric rough paths Rough paths theory applied to stochastic analysis Let us go back now to the initial problem of solving controlled differential equations driven by some random signal X If we want to apply rough paths theory, we have to say what the iterated integrals of X should be On the level of trajectories, it is not clear what a natural choice of an iterated integral is 5 We will see that taking into account the probabilistic properties of the process helps to find a natural candidate for an iterated integral From now on, we will only consider finite dimensional Banach spaces In the case of the Brownian motion with independent components, the natural choices for the iterated integrals are the usual Itō and Stratonovich integrals: db u1 db u, db u1 db u 0<u 1 <u <t 0<u 1 <u <t Together with the Brownian process, this indeed defines a process with values in a rough paths space, cf [LQ0], [FV05] or [FV10b, Chapter 13], and we can apply all the machinery provided by rough paths theory One can also show that the rough path solution one obtains when solving considering the Itō integral coincides with the solution given by Itō s classical theory; the same is true for the Stratonovich integral In a similar fashion, it is possible to find natural lifts for continuous Markov processes cf [FV08c], [FV10b, Chapter 16] and semimartingales [CL05], [FV08a] and [FV10b, Chapter 14], but we will concentrate on Gaussian processes in the following As an example, consider a fractional Brownian motion B H with Hurst parameter H 0, 1, ie a centered Gaussian process with independent components and each component has the covariance function Rs, t = 1 s H + t H t s H For H = 1/, Rs, t = s t and we obtain the usual Brownian motion, hence the fractional Brownian motion can be considered as a natural generalisation There are many attempts to define a stochastic calculus with respect to a fractional Brownian motion; cf [BHOZ08] for a nice summary The process is interesting for us since it is not a semimartingale or a Markov process for H 1/, thus Itō s theory of integration does not apply The sample paths of B H 5 However, it can be seen that every path of finite p-variation can be lifted to a geometric rough path, cf Lyons and Victoir [LV07] The problem is that this lift is not and cannot be unique 7

16 Introduction are seen to be α-hölder continuous for every α < H, hence we can apply Young s integration theory for H > 1/ For H 1/, the sample paths fail to be α-hölder for α > 1/ The question now is: are there still natural choices of iterated integrals with respect to a fractional Brownian motion in the case H < 1/? The first article which gave an answer to this question is the work of Coutin and Qian [CQ0] The authors consider the sequence Π n of dyadic partitions of the interval [0, t] and define the process B H n to be the process B H for which the sample paths are piecewise linear approximated at the points Π n Considering the process Bt H n B0 H n, dbu H 1 n dbu H n, 0<u 1 <u <t where the integral exists as a usual Riemann-Stieltjes integral, they show that for H > 1/3, this is a Cauchy sequence almost surely in rough paths topology The limiting object is defined to be the rough path lift of B H The authors also consider the third iterated integral and prove convergence, hence there is also a lift for H > 1/4 However, for H = 1/4, they can show that the second iterated integral diverges This notion of a rough path lift of a fractional Brownian motion is indeed quite natural at least for two reasons First, for H = 1/, we find the usual Stratonovich integral Second, it generalises the Wong-Zakai theorem for the fractional Brownian motion: If Y n denotes the solution of the random controlled differential equation dy n t = fy n t db H t n; Y n = Y ξ, by the universal limit theorem, Y n converges almost surely in rough paths topology to a limit which is precisely the solution Y of the corresponding rough differential equation projected to the first tensor level We would like to mention at this point that there are also different approaches to define a rough path lift to a fractional Brownian motion cf eg [Unt09] and the following articles by the same author, but we will not comment on these approaches here Gaussian rough paths in the sense of Friz Victoir In [FV10a], Friz and Victoir generalise the method of Coutin and Qian and give a sufficient criterion on the covariance function R under which a given Gaussian process can be lifted in a natural way to a process with sample paths in a rough paths space In this thesis, we will always work in their framework, therefore we decided to sketch their main ideas here Let X = X 1,, X d be a d-dimensional Gaussian process with independent and identically distributed 6 components The main problem is to make sense of the integral t 0 X i s X i 0 dx j s for i j If the trajectories of X are differentiable, we can formally calculate the second moment: t E Xs i X0 i dxs j t t = E Xs i X0X i u i X0 i s Xs j u Xu j ds du = E[Xs i X0X i u i X0] i s u E[XsX j u] j ds du [0,t] = Rs, u Rs, 0 R0, u + R0, 0 drs, u, [0,t] where R denotes the covariance function and the right hand side is a suitable version of a D Young integral Fortunately, there is indeed a theory for dimensional Young integration developed by Towghi in [Tow0] and we can bound the right hand side in terms of the 6 The assumption that the components should have the same distribution is not really necessary and only assumed for the sake of simplicity 8

17 dimensional ρ-variation of R provided ρ < Natural approximations of the sample paths of the process X such as a piecewise linear approximation or the convolution with a smooth function yield approximations of the covariance function for which the ρ-variation is seen to be uniformly bounded The following result should therefore not come as a surprise: Assume that the covariance function of every component of X has finite ρ-variation for some ρ < Then there exists a natural lift of X to a process with values in a rough paths space The lift of the process will be denoted by X in the following The results of Friz and Victoir are sharp in the sense that the covariance function of a fractional Brownian motion is seen to have finite ρ-variation for ρ = 1 H, but not better The threshold ρ = therefore corresponds to the Hurst parameter H = 1 4 for which Coutin and Qian already showed that the natural approximation of the second integral diverges Once the existence of a Gaussian rough paths lift is established under this very general condition, it can be shown that many theorems from stochastic analysis proven for the Brownian motion generalise to Gaussian rough paths For instance, in the article [FV10a], Friz and Victoir prove Fernique estimates for the lift X see also the work of Friz and Oberhauser [FO10] for a different proof of this result A support theorem for Gaussian rough paths is proven in [FV10a, Theorem 55] A large deviation principle for the lift of a fractional Brownian motion was proven by Millet and Sanz-Solé in [MSS06] and later generalised for Gaussian rough paths by Friz and Victoir in [FV10b, Theorem 1555] A Malliavin-type calculus was established cf Cass, Friz and Victoir [CFV09], Friz and Victoir [FV10b, Chapter 0], Cass and Friz [CF11] and a Hörmander-type theorem for Gaussian rough paths can be proven cf Cass, Friz [CF10], Cass, Litterer, Lyons [CLL], Cass, Hairer, Litterer and Tindel [CHLT1] The results of this thesis We will now summarise the main contributions of this thesis More details to the respective results may be found in the beginning of the corresponding chapters In Chapter 1, we consider the Brownian rough paths lift B, seen as the Brownian increments of a Brownian motion B : [0, 1] R d together with its iterated Stratonovich integrals By Lyons extension theorem, we can lift the sample paths of B to any p-rough paths space provided p > If we approximate the trajectories of the underlying Brownian motion piecewise linear at the points {0 < 1/n < /n < < 1}, we obtain another process B n with piecewise linear trajectories This process can be lifted to a process B n with sample paths in a p-rough paths space using Riemann-Stieltjes theory The first result is the following Theorem I For all p > and η < 1 1 p, ρ 1 p HölB, B n C 1 η n almost surely for all n N where C is a finite random variable ρ 1 Höl, denotes a rough paths metric here Note that the convergence rate increases for p large p but does not exceed 1 From the local Lipschitz continuity of the Itō Lyons map, we immediately obtain convergence rates for the Wong Zakai theorem Moreover, for sufficiently smooth vector fields, the solution flow of a rough differential equation is differentiable and our convergence rates for the Wong Zakai theorem also hold true on the level of flows: Theorem II Let f = f 0, f 1,, f d be smooth vector fields and consider the random flow y 0 U Bn,t 0 y 0 on R e defined by dy = f 0 y dt + d f i y dbn, i y 0 = y 0 i=1 9

18 Introduction Then as U Bn,t 0 y 0 converges uniformly as do all its derivatives in y 0 on every compact subset K [0, R d ; and the limit U B,t 0 y 0 := lim n U B n,t 0 y 0 solves the Stratonovich SDE dy = f 0 y dt + d f i y db i, y 0 = y 0 i=1 Moreover, for every η < 1/ and every k {1,, } and K [0, R d, there exists an as finite random variable C such that 1 η max α U B, 0 α U Bn, 0 ;K C α=α 1,,α e n α =α 1 + +α e k for all n N Note that this implies an almost sure Wong Zakai convergence rate of almost 1, which is known to be sharp modulo possible logarithmic corrections The results in this chapter were obtained in collaboration with Prof Peter Friz and are published in the journal Bulletin des Sciences Mathématiques, see [FR11] In Chapter, we generalise the results of Chapter 1 to lifts of Gaussian processes X in the sense of Friz Victoir Again, X n denotes the process with piecewise linear approximated trajectories Our main theorem can be stated as follows: Theorem III Assume that the covariance of X has finite ρ-variation in D sense and that the ρ-variation over every square [s, t] [0, 1] can be bounded by a constant times t s 1 ρ Then for all η < 1 ρ 1 and p > ρ 1 ρη, ρ 1 p HölX, X n 0 for n almost surely and in L q for any q 1, with rate η Note again that a good convergence rate forces p to be chosen large Note also that our theorem holds for much more general approximations than piecewise linear approximations As a consequence, we obtain almost sure convergence rates for the Wong Zakai theorem for Gaussian rough paths Corollary IV Let f = f 0, f 1,, f d be smooth vector fields and consider the random controlled differential equation Then as dy n = f 0 Y n dt + d f i Y n dxn; i Y n 0 = ξ i=1 Y n Y uniformly for n with rate η for any η < 1 ρ 1 and the limit solves the random rough differential equation dy = fy dx; Y 0 = ξ 10

19 Recall that Davie presented a step- Taylor scheme for solving rough differential equations RDEs and computed the convergence rate cf [Dav07] Step-N schemes with convergence rates are considered in Friz and Victoir [FV10b, Chapter 10] In [DNT1], Deya, Neuenkirch and Tindel present a simplified Milstein-type scheme for solving rough differential equations driven by a fractional Brownian motion The advantage of this numerical scheme is that the iterated integrals which are hard to simulate numerically are replaced by a product of increments Our results imply sharp convergence rates for these schemes in a general Gaussian setting Corollary V The approximation Y n obtained by running a simplified step-3 Taylor scheme 7 with mesh size 1/n for solving the random rough differential equation dy = fy dx; Y 0 = ξ converges almost surely uniformly to the solution Y with rate η for any η < 1 ρ 1 This proves a conjecture stated by Deya, Neuenkirch and Tindel in the work [DNT1] The results in this chapter were obtained in collaboration with Prof Peter Friz and are accepted for publication by the journal Annales de l Institut Henri Poincaré Probabilités et Statistiques, see [FR] In Chapter 3, we consider the work [CLL] of Cass, Litterer and Lyons Before we describe their results, it will be useful to make the following definition Recall that for each Gaussian process X, there is an associated Cameron Martin space or reproducing kernel Hilbert space Definition VI We say that complementary Young regularity holds for the trajectories of a Gaussian process and its Cameron Martin paths if the Cameron Martin space is continuously embedded in the space of paths which have finite q-variation, the trajectories of X have finite p-variation almost surely and 1 p + 1 q > 1 The condition assures that we can make sense of the Young integral between the Cameron Martin paths and the trajectories of the process In [FV06b, Corollary 1], Friz and Victoir show that complementary Young regularity holds for the fractional Brownian motion with Hurst parameter H > 1 4 and from their work [FV10a, Proposition 17] it follows that complementary Young regularity holds for a Gaussian process X for which the covariance has finite ρ-variation for ρ < 3 The aim of the article [CLL] is to prove that the Jacobian of a Gaussian RDE flow has finite L q moments 8 for every q 1 They introduce a map which assigns an integer N α x to a p-rough path x which equals the number the p-th power of the p-variation of x exceeds the barrier α The main work of [CLL] is to show that if we replace the rough paths x by the lift of a Gaussian process X, this number has tails which are strictly better than exponential tails More precisely, N α X is seen to have Weibull tails with shape parameter strictly greater than 1 provided the trajectories of the underlying Gaussian process X and its Cameron Martin paths have complementary Young regularity Our first contribution is the identification of so-called locally linear maps Ψ, mapping from one rough paths space to another, under which the tail estimates remain valid Our result is purely deterministic 7 In the case ρ = 1, a step- scheme converges with the same rate 8 The motivation for this is that this result can be used to prove that the solution of a Gaussian rough differential equation has a smooth density with respect to Lebesque measure at every fixed time point t, cf also [CHLT1] 11

20 Introduction Theorem VII Let Ψ be a locally linear map Then there is a α such that N α Ψx N α x Since the p-variation of x can be bounded by a constant times N α x, the tail estimates obtained for N α x remain valid for the p-variation of x Rough integration and the Itō-Lyons map are examples of locally linear maps; hence we immediately obtain Corollary VIII Assume that complementary Young regularity holds for the trajectories of X and its Cameron Martin paths Then the following objects have exponential tails 9 : i The rough integral αx dx where α is a suitable one-form ii The p-variation of Y where Y solves the random rough differential equation dy = fy dx; Y 0 = ξ for smooth and bounded vector fields f For linear vector fields f LW, LV, W = LV, LW, W, the situation is different Theorem IX If y solves the linear rough differential equation then dy = fy dx = fdxy; y0 = ξ, 7 N α y C1 + ξ p expcn α x for a constant C In particular, if N α X has Weibull tails with shape parameter strictly greater than 1 which is the case if complementary Young regularity holds for the trajectories of X and its Cameron Martin paths, the p-variation of the solution Y of the random linear rough differential equation 7 has finite L q momemts for any q 1 Our estimates particularly imply that the Jacobian of a Gaussian RDE flow has finite L q moments for any q 1, which was the main result of the work [CLL] The estimates are also robust in the sense that they can be used to prove uniform tail estimates As an example, we show that a certain rough integral over a family of Gaussian processes has uniformly Gaussian tails, a technical result needed by Hairer in [Hai11] The results in this chapter were obtained in collaboration with Prof Peter Friz and are published in the journal Stochastic Analysis and Applications, see [FR13] In Chapter 4, we apply the methods from Lyons and Xu presented in [LX1] to bound the distance between two Gaussian rough paths in p-variation topology Our estimates are very similar to the ones needed for proving Theorem III, but we show how to avoid the algebraic machinery presented in Chapter and still get optimal bounds in the case ρ = 1 Our main theorem states the following: 9 In fact, our tail estimates are sharper and can be expressed in terms of Weibull tails, cf Chapter 3 We restrict ourselves to exponential tails for the sake of simplicity 1

21 Theorem X Let X, Y = X 1, Y 1,, X d, Y d be a jointly Gaussian process and let X i, Y i and X j, Y j be independent for i j Assume that there is a ρ [1, 3 such that the ρ- variation of R X i,y i is bounded by a constant K for every i = 1,, d Let γ ρ such that 1 γ + 1 ρ > 1 Then, for every p > γ, q 1 and δ > 0 small enough, there exists a constant C K such that 1 i if γ + 1 ρ > 1, then 1 ii if γ + 1 ρ 1, then ρ p var X, Y L q C sup X t Y t 1 ρ γ, 8 L t ρ p var X, Y L q C sup X t Y t 3 ρ δ 9 L t In our theorem, ρ p var, denotes a p-rough path metric Note that for ρ = 1, we can always use the estimate 8 Inequality 8 is actually valid for all γ ρ provided 1 γ + 1 ρ > 1 which can be seen by using the techniques developed in Chapter, but one aim of Chapter 4 is to show that we can avoid a bulk of calculations and still obtain estimate 9 which is not sharp though One can show that our results imply convergence rates as in Theorem III, and for ρ = 1 we obtain optimal convergence Another application of Theorem X appears in the field of stochastic partial differential equations In [Hai11], Hairer considers the stationary solution ψ of the equation dψ = xx 1ψ dt + σ dw 10 where σ is a positive constant, the spatial variable x takes values in [0, π], xx is equipped with periodic boundary conditions and dw is space-time white noise, ie a standard cylindrical Wiener process on L [0, π], R Hairer shows that for every fixed time point t, the Gaussian process ψ t obtained by taking d independent copies of the spatial processes ψ t can be lifted to a process Ψ t with values in a p-rough paths space for any p > He also shows that there is a continuous modification of the map t Ψ t Our results imply optimal time regularity Corollary XI There is an α-hölder continuous modification of the map t Ψ t for every α < p Note that the Hölder exponent increases for large p and is bounded by 1/4 which is known to be a sharp bound The results in this Chapter were obtained in collaboration with Weijun Xu and are available online, see [RX1] In Chapter 5, we reinvestigate the solution of the modified heat equation 10 The space regularity of ψ t essentially depends on two factors: the smoothing effect of the operator xx and the colouring of the noise dw We have already seen that the crucial condition for lifting ψ t to a process in a rough paths space in the sense of Friz Victoir is a sufficiently regular covariance function R ψt in terms of dimensional ρ-variation The parameter ρ should therefore also depend on the smoothing effect of the operator and the colouring of the noise Our main theorem determines the parameter ρ for which the ρ-variation of R ψt is finite in terms of the spectrum of the operator and the noise For simplicity, we only state the result for the fractional heat equation here 13

22 Introduction Theorem XII Let ψ be the stationary solution of the fractional, modified heat equation dψ = xx α 1ψ dt + σ dw ; α 1/, 1] 11 with periodic boundary conditions where dw is space-time white noise Then the ρ-variation of R ψ is finite for ρ = 1 α 1 In particular, if we set ψα := ψ 1,, ψ d where the ψ i are independent copies of ψ, for every fixed t we can lift the trajectories of ψα t to p-rough paths in the sense of Friz Victoir for all p > α 1 provided α > 3/4 Moreover, there is a Hölder continuous modification of the lifted process t Ψ α t In addition, our results imply uniform bounds of the ρ-variation for viscosity and Galerkin approximations of 11 which can be used in a future work for numerical considerations We also give a new and easy criterion on the covariance of a Gaussian process with stationary increments to have finite ρ-variation If the process is given as a random Fourier series as in the situation above, these conditions translate into conditions on the Fourier coefficients The results in this chapter were obtained in collaboration with Prof Peter Friz, Dr Benjamin Gess and Prof Archil Gulisashvili and are available online, see [FGGR1] In Chapter 6 we come back to numerical considerations Let Y be the solution of a random rough differential equation dy = fy dx; Y 0 = ξ, 1 X being the lift of a Gaussian process whose covariance is of finite ρ-variation In Corollary V, we saw that there is an easy implementable numerical scheme which converges almost surely to the solution of 1 Let Y n denote an approximation of Y using such a scheme with mesh-size 1/n Assume now that we are interested in evaluating a quantity of the form EgY where g is a functional which may depend on the whole path of Y The first obstacle is that we do not know, even for smooth g, with which rate EgY n converges to EgY for n since we only proved almost sure convergence, not L 1 convergence which would imply a convergence rate when g is at least Lipschitz This is our first result Theorem XIII Assume that complementary Young regularity holds for the trajectories of X and its Cameron Martin paths Then the Wong-Zakai approximation in Corollary IV and the simple Taylor scheme in Corollary V both converge in L q for any q 1 with the same convergence rate We would like to mention here that the proof of this theorem is more involved than one might expect at a first sight In fact, we improve the estimate for the Lipschitz constant of the Itō Lyons map slightly, using similar estimates as in Chapter 3 for the case of linear rough differential equations, and then use the results of Cass, Litterer and Lyons in [CLL] to prove the assertions As an immediate corollary of Theorem XIII, we obtain strong convergence rates for our numerical scheme in the case when g is Lipschitz However, at the present stage, we can only bound the weak convergence rate from below with the strong rate, whereas the weak rate might be better, at least for smooth g If we want to evaluate EgY, a Monte Carlo evaluation would be a possible and easy method In the seminal work [Gil08b], Giles showed that one can reduce the computational complexity more precisely: its asymptotics for a given mean squared error dramatically when using a multilevel Monte Carlo method For us, the multilevel method is also interesting because the strong convergence rate plays a more important role here than the weak one which would be used calculating the complexity of the usual Monte Carlo evaluation Indeed, we can prove an abstract, more general complexity theorem as in [Gil08b] which fits our purposes Applied to the evaluation of EgY, we can prove the following result Theorem XIV Assume that complementary Young regularity holds for the trajectories of X and its Cameron Martin paths and that g is Lipschitz Then the Monte Carlo evaluation of 14

23 a path-dependent functional of the form EgY, to within a mean squared error of ε, can be achieved with computational complexity O ε θ θ > ρ ρ In the case of a Brownian motion, the asymptotics of the computational complexity is bounded by O ε θ for any θ > which is known to be sharp modulo logarithmic corrections, cf [Gil08a, Gil08b] Compared to a usual Monte Carlo method, we see that indeed a multilevel method decreases the computational complexity in the general Gaussian setting The results in this chapter were obtained in collaboration with Dr Christian Bayer, Prof Peter Friz and PD Dr John Schoenmakers 15

24 16 Introduction

25 Notation and basic definitions In this chapter, we introduce the most important concepts and definitions from rough path theory For a detailed account, we refer to [FV10b], [LCL07] and [LQ0] Fix a time interval [0, T ] For all s < t [0, T ], we define the n-simplex sup D [s,t] n s,t := {u 1,, u n s < u 1 < u n < t} We will simply write s,t instead of s,t and instead of 0,T Let E, d be a metric space and x C[0, T ], E For p 1 and α 0, 1] we define 1 x p var;[s,t] := p dx ti, x ti+1 p d x u, x v and x α Höl;[s,t] := sup u,v s,t v u α t i,t i+1 D where D [s, t] means that D is a finite dissection of the form {s = t 0 < < t M = t} of the interval [s, t] We will use the short hand notation p var and α Höl for p var;[0,t ] resp α Höl;[0,T ] which are easily seen to be semi-norms Given a positive integer N, the truncated tensor algebra of degree N is given by the direct sum T N R d = R R d R d R d R d N = N R d n n=0 and we will write π n : T N R d R d n for the projection on the n-th tensor level T N R d is a finite-dimensional R-vector space For elements g, h T N R d, we define g h T N R d by n π n g h = π n i g π i h i=0 One can easily check that T N R d, +, is an associative algebra with unit element e := We call it the truncated tensor algebra of level N A norm is defined by g T N R d = max π n g n=0,,n which turns T N R d into a Banach space A continuous map x: T N R d is called multiplicative functional if for all s < u < t one has x s,t = x s,u x u,t For a path x = x 1,, x d : [0, T ] R d and s < t, we will use the notation x s,t = x t x s If x has bounded variation or finite 1-variation, we define its n-th iterated integral by x n s,t = dx dx = n s,t 1 i 1,,i n d n s,t dx i 1 dx in e i1 e in R d n