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A Brief Introduction to Rough Paths

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A Brief Introduction to Rough Paths A brief introduction to rough paths Giovanni Zanco September 2, 2016 These notes are based on a short course I taught at the University of Pisa in April, 2016, and originated from a course taught by Jan Maas at IST Austria in autumn 2015, for which I have been the teaching assistant. They provide a brief introduction to the theory of rough paths of Hölder regularity between 1=3 and 1=2, with some hints at the theory for rough paths of arbitrary regularity. The exposition and the results presented are heavily based on the book [Friz and Hairer, 2014], which I suggest as a reference for a first study of the subject. Some of the material presented here is also adapted from [Baudoin, 2013], [Friz and Victoir, 2010] and from personal handwritten notes by Jan Maas. Further references are given throughout the text. The proofs given here follow very closely the cited references. These notes have been developed as a handout, with the goal of organising and extending the material presented during the mentioned courses, thus giving a very concise overview of the basic results and of the lines along which the theory of rough paths developed. They are not supposed to be exhaustive, and many important topics are omitted, together with many details. For all sections but the last one basic knowledge in real analysis and stochastics is required, but nothing more. The last section contains more advanced topics, but it is kept at a very informal level, with few rigorous proofs. I warmly thank Jan Maas, for his course provided me with a general and comprehensive outline upon which these short notes developed. The current shape of this material has also benefited from the comments given and the question raised during the course in Pisa; I thank all the participants for their interest and help, and in particular I acknowledge Franco Flandoli for his useful suggestions about the exposition of some topics discussed herein. Contents 1 Introduction 2 2 Elements of Young integration 3 3 Rough Paths 6 1 4 Some comments on the general theory 11 5 Rough integration 13 6 Rough differential equations 23 7 Stochastic processes as rough paths 25 8 Stochastic differential equations 30 9 Applications to a stochastic partial differential equation 31 1 Introduction The central objects of these lectures will be differential equations of the form dYt = f (Yt) dXt (1) where X : [0;T ] ! E is a driving signal, Y : [0;T ] ! V is the output (unknown) and f : V !L(E; V ) is a smooth function, E and V are Banach spaces and L(E; V ) denotes the space of continuous linear maps from E to V . A common choice is E = Rd and V = Rn, so that f : Rn ! Rn×d. We expect X and Y to be continuous functions. The usual way to interpret equation (1) together with some initial datum y0 is in its integral form Z t Yt = y0 + f (Ys) dXs (2) 0 and a standard scheme to solve (2) is (i) to give meaning to the integral; (ii) to apply some fixed point result. To deal with item (i) we need an integration theory that is satisfactory in the sense that it allows to work with signals and unknowns of suitable regularity, depending on the problem we are in- terested in. To deal with item (ii) we need to space of solutions to (2) to have some nice metric structure. If f is smooth and X is differentiable, then the classical theory applies: equation (2) is inter- preted as Z t Yt = y0 + f (Ys) X_ s ds 0 and the solution Y can be found as the fixed point of the map M defined on continuous functions by Z t M(Y )t := y0 + f (Ys) X_ s ds : 0 We will be interested here in situations in which Y is a α-Hölder function (and hence t 7! f (Yt) is α-Hölder as well) and X is a β-Hölder function. If α + β > 1 we can interpret the integral 2 appearing in (2) as a Young integral (see section 2) and find the solution as the unique fixed point of the map M above in the space Cα. If α + β ≤ 1 the question is more tricky; rough paths theory provides a convenient answer. Of course there are well known probabilistic results that allow to define the so-called stochastic integrals and to solve stochastic differential equations like (2). However they typically do not provide pathwise solutions (if X is a Brownian motion one cannot fix a Brownian path X(!) and solve (2) for that particular realisation of X), rather solutions in a probabilistic sense; in- deed all stochastic integrals require some probabilistic property of X and Y to be well defined (semimartingale structure, adaptedness, etc.) rather than some regularity property of the typical paths of X and Y . We will see that many classical results about stochastic differential equations are recovered in the theory of rough paths. What does the study of rough paths then add to the classical theories of stochastic analysis? Among many interesting answers to these question, we will focus mainly on the following one. A celebrated result by T. Lyons (see [Lyons, 1991]) states the following: Theorem 1.1. There exists no separable Banach space B ⊂ C ([0; 1]; R) with the properties: (i) sample paths of Brownian motion belong to B almost surely; R · _ (ii) The map (g; h) 7! 0 g(t)h(t) dt extends from smooth functions to a continuous map on B × B taking values in C ([0; 1]; R). For example the solution map B 7! Y of the Stratonovich differential equation dYt = f (Yt) ◦ dBt is measurable but not continuous, in general, with respect to any reasonable topology (and in- deed not all smooth approximations to B give convergence to the Stratonovich solution for the above equation). Rough paths provide a framework in which continuity of the solution map can, to a certain extent, be restored. Here we will consider driving signals with Hölder regularity α 2 (1=3; 1=2]. The original theory prefers to work with functions of finite p-variation rather than with Hölder functions, and allows to consider signals of arbitrarily low regularity, but requires heavy algebraic methods that would need too long to be introduced. Restricting to α 2 (1=3; 1=2] allows to avoid the study of signatures, to avoid many algebraic difficulties and is anyway interesting enough to see many features of the theory and tackle some interesting problems. Some hints of the general theory will however be given hereinafter; the interested reader can refer to [Friz and Victoir, 2010] and [Lyons et al., 2007]. 2 Elements of Young integration We recall the following definitions that will be frequently used in these notes. 3 Definition 2.1 (Hölder continuous functions). Let α > 0. A function X defined on an interval [0;T ] ⊂ R and taking values in a Banach space E is α-Hölder continuous (often α-Hölder for brevity) if the quantity jXt − XsjE kXkα := sup α t6=s jt − sj is finite. The space of all α-Hölder continuous functions from [0;T ] into E is denoted by Cα ([0;T ]; E). When no confusion can arise on the domain of the functions at hand we will simply write Cα(E), or even Cα if the co-domain is clear as well. Hölder continuous functions are continuous, and any α-Hölder function with α > 1 is constant. The quantity k·kα is a semi-norm (it does not separates constant functions); however the quantity kXkCα := jX0jE + kXkα is a norm that makes Cα(E) a Banach space (in general not separable). This norm is equivalent β α to k · k1 + k · kα. If α < β then obviously C ⊂ C . We say that a E-valued function X belongs to Ck,α(E) if it is k times differentiable with k,α its k-th derivative being α-Hölder. The space C (E) endowed with the norm kXkCk,α = (k) kXkCk + kX kα is also a Banach space. By partition of an interval I, in the sequel, we will mean a finite family Π of (essentially) disjoint sub-intervals [s; t] of I such that [[s;t]2Π[s; t] = I. Therefore choosing a partition is equivalent to choosing a finite number of points t0 = 0 < t1 < ··· < tN = T and dividing I into the sub-intervals [ti; ti+1]. The one-point overlap between adjacent intervals will cause no trouble. The mesh of a partition Π is defined as jΠj := max[s;t]2Π jt − sj. Definition 2.2 (Finite p-variation functions). Let p > 0. A function X defined on an interval [0;T ] ⊂ R and taking values in a Banach space E has finite p-variation if 1 0 1 p X p kXkp−var := @sup jXt − XsjEA < 1 Π [s;t]2Π where the supremum is taken over all partitions Π of [0;T ]. The space of all continuous functions from [0;T ] into E with finite p-variation is denoted by Cp−var ([0;T ]; E). As above we will write Cp−var(E) or even Cp−var when spaces are clear from the context. Any function with finite p-variation for some p < 1 is constant, and functions of finite 1-variation are known as bounded variation (BV) functions. The quantity k · kp−var is a semi-norm on Cp−var, but kXkCp−var := jX0jE + kXkp−var p−var is a norm (equivalent to k·k1 +k·kp−var) that turns C into a Banach space (not separable, in general).
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