56 4 Integration against rough paths

Transcription

1 56 4 Integration against rough paths comes to the definition of a rough integral we typically take W = LV, W ; although other choices can be useful see e.g. remark In the context of rough differential equations, with solutions in W = W, we actually need to integrate fy, which will be seen to be controlled by X for sufficiently smooth coefficients f : W LV, W. Definition 4.6. Given a path X C [, T ], V, we say that Y C [, T ], W is controlled by X if there exists Y C [, T ], LV, W so that the remainder term R Y given implicitly through the relation Y s,t = Y s X s,t + R Y s,t, 4.16 satisfies R Y 2 <. This defines the space of controlled rough paths, Y, Y D 2 X [, T ], W. Although Y is not, in general, uniquely determined from Y cf. Remark 4.7 and Section 6 below we call any such Y the Gubinelli derivative of Y with respect to X. Here, Rs,t Y takes values in W, and the norm 2 for a function with two arguments is given by 2.3 as before. We endow the space DX 2 with the semi-norm Y, Y def X,2 = Y + R Y As in the case of classical Hölder spaces, DX 2 is a Banach space under the norm Y, Y Y + Y + Y, Y X,2. This quantity also controls the -Hölder regularity of Y since, uniformly over X bounded in -Hölder seminorm, Y Y X + T R Y 2 Y X + T { Y X + R Y 2 } 1 + X Y + T Y, Y X,2 Y + T Y, Y X, Remark 4.7. Since we only assume that Y <, but then impose that R Y 2 <, it is in general the case that a genuine cancellation takes place in The question arises to what extent Y determines Y. Somewhat contrary to the classical situation, where a smooth function has a unique derivative, too much regularity of the underlying rough path X leads to less information about Y. For instance, if Y is smooth, or in fact in C 2, and the underlying rough path X happens to have a path component X that is also C 2, then we may take Y =, but as a matter of fact any continuous path Y would satisfy 4.16 with R 2 <. On the other hand, if X is far from smooth, i.e. genuinely rough on all small scales, uniformly in all directions, then Y is uniquely determined by Y, cf. Section 6 below. Remark 4.8. It is important to note that while the space of rough paths C is not even a vector space, the space DX 2 is a perfectly normal Banach space for any given X = X, X C. The twist of course is that the space in question depends in a

2 4.3 Integration of controlled rough paths 57 crucial way on the choice of X. The set of all pairs X; Y, Y gives rise to the total space C 2 def D = {X} D 2 X, X C with base space C and fibres DX 2. While this looks reminiscent of fibre-bundles like the tangent bundles of a smooth manifold, it is quite different in the sense that the fibre spaces are in general not isomorphic. Loosely speaking, the rougher the underlying path X, the smaller is DX 2, see Chapter 6. Remark 4.9. While the notion of controlled rough path has many appealing features, it does not come with a natural approximation theory. To wit, consider X, X C g [, T ], R d as limit of smooth paths X n : [, T ] R d in the sense of Proposition 2.5. Then it is natural to approximate Y = F X by the Y n = F X n, which is again smooth to the extent that F permits. On the other hand, there are no obvious approximations Y n, Y n D 2 X Y, Y D 2 X. n for an arbitrary controlled rough path We are now ready to extend Young s integral to that of a path controlled by X against X = X, X. Recall from Lemma 4.1 that Y = F X, with Y = DF X, is somewhat the prototype of a controlled rough path. The definition of the rough integral F XdX in terms of compensated Riemann sums, cf. 4.6, then immediately suggests to define the integral of Y against X by 6 1 Y dx = def lim P [s,t] P Ys X s,t + Y s X s,t, 4.19 where we took W = LV, W and used the canonical injection LV, LV, W LV V, W in writing Y s X s,t. With these notations, the resulting integral takes values in W. With these notations at hand, it is now straightforward to prove the following result, which is a slight reformulation of [Gub4, Prop.1]: Theorem 4.1 Gubinelli. Let T >, let X = X, X C [, T ], V for some [ 1 2, 1 3, and let Y, Y DX 2 [, T ], LV, W. Then there exists a constant C depending only on such that a The integral defined in 4.19 exists and, for every pair s, t, one has the bound t s Y r dx r Y s X s,t Y s X s,t C X R Y 2 + X 2 Y t s 3. b The map from D 2 X [, T ], LV, W to D 2 X [, T ], W given by Note the abuse of notation: we hide dependence on Y which in general affects the limit but is usually clear from the context.

3 58 4 Integration against rough paths Y, Y Z, Z := Y t dx t, Y, 4.21 is a continuous linear map between Banach spaces and one has the bound 7 Z, Z X,2 Y + Y X 2 + CT X R Y 2 + X 2 Y. Proof. Part a is an immediate consequence of Lemma 4.2, as already pointed out in the proof of Theorem 4.4. The estimate 4.2 was pointed out explicitly in It remains to show the bound on Z, Z X,2. Splitting up the left hand side of 4.2 after the first term, using the triangle inequality, gives immediately an Hölder estimate on t s Y rdx r = Z s,t, so that Z C. Z = Y C is trivial, by the very nature of Y since it is controlled by X. Similarly, splitting up the left hand side of 4.2 after the second term, gives a 2-Hölder type estimate estimate on t s Y rdx r Y s X s,t = Z s,t Z sx s,t =: R Z s,t, i.e. on the remainder term in the sense of The explicit estimate for Z, Z X,2 = Y + R Z 2 is then obvious. Remark As in the above theorem, assume that X, X C [, T ], V and consider Y and Z two paths controlled by X. More precisely, we assume Y, Y DX 2[, T ], L V, W and Z, Z DX 2[, T ], V, where of course V, V, W are all Banach spaces. Then, in terms of the abstract integration map I cf. the sewing lemma we may define the integral of Y against Z, with values in W, as follows, t s def Y u dz u = IΞ s,t, Ξ u,v = Y u Z u,v + Y uz ux u,v Here, we use the fact that Z u LV, V can be canonically identified with an operator in LV V, V V by acting only on the second factor, and Y u LV, L V, W is identified as before with an operator in LV V, W. The reader may be helped to see this spelled out in coordinates, assuming finite dimensions: using indices i, j in W, V respectively, and then k, l in V : Ξ u,v i = Y u i j Z u,v j + Y u i k,j Z u j l X u,v k,l. Note that, relative to the definition of Ξ in the previous proof, it suffices to replace X by Z and Y by Y Z. Making this substitution in δξ, as it appears in the aforementioned proof, then gives δξ s,u,t = R Z s,ux u,t Y Z s,u X u,t in the present situation. Clearly Y Z C and so δξ β is finite which allows the proof to go through mutatis mutandis. In particular, 4.2 is valid, with the above substitution, and reads 7 As in 4.18, this implies Z, Z X,2 Y + T Y, Y X,2, uniformly over bounded X.

4 6 4 Integration against rough paths 4.4 Stability I: rough integration Consider X = X, X, X = X, X C with Y, Y DX 2, Ỹ, Ỹ D 2 X. Although Y, Y and Ỹ, Ỹ live, in general, in different Banach spaces, the distance Y, Y ; Ỹ, Ỹ def = Y Ỹ + R Y RỸ 2 will be useful. Even when X = X, it is not a proper metric for it fails to separate Y, Y and Y + cx + c, Y + c for any two constants c and c. When X X, the assertion zero distance implies Y, Y = Ỹ, Ỹ does not even make sense. The two objects live in completely different spaces! That said, for every fixed X, X C, one has with Rs,t Y = Y s,t Y s X s,t as usual, a canonical map ι X : Y, Y C X Y, R Y C C 2 2. Given Y = ξ, this map is injective since one can reconstruct Y by Y t = ξ +Y X,t + R Y,t. From this point of view, one simply has = ι X. ι X.,2, and one is back in a normal Banach setting, where,,2 = + 2 is a natural semi-norm on C C2 2. In fact, it is a norm if one only considers elements in C started at. Elementary estimates of the form ab ã b a b b + a ã b 4.26 then lead to, with a constant C = C R, Y s,t Ỹs,t = Y,s Y Xs,t + Ỹ,s + Ỹ Xs,t + Rs,t Y RỸs,t C t s Y Ỹ + X X + Y, Ỹ, + R Y RỸ C t s Y Ỹ + X X + T Y Ỹ + R Y RỸ 2, provided Y, Y, X, and also with tilde, are bounded by R. It follows that Y Ỹ X C X + Y Ỹ + T Y, Y ; Ỹ, Ỹ An estimate of the proper -Hölder norm of Y Ỹ rather than its semi-norm is obtained by adding Y Ỹ to both sides. Theorem 4.16 Stability of rough integration. For 1 3, 2] 1 as before, consider X = X, X, X = X, X C, Y, Y DX 2, Ỹ, Ỹ D 2 in a bounded X set, in the sense Y + Y, Y X,2 M, ϱ, X X + X 2 M,

5 4.5 Controlled rough paths of lower regularity 61 with identical bounds for X, X, Ỹ, Ỹ, for some M <. Define Z, Z := Y dx, Y DX 2, and similarly for Z, Z. Then, the following local Lipschitz estimates holds true, Z, Z ; Z, Z C ϱ X, X + Y Ỹ and also Z Z C ϱ X, X + Y Ỹ + Y Ỹ where C = C M = CM, is a suitable constant. + T Y, Y ; Ỹ, Ỹ, T Y, Y ; Ỹ, Ỹ, 4.29 Proof. The reader is advised to review the proofs of Theorems 4.4, 4.1. We first note that 4.27 applied to Z, Z note: Z Z = Y Ỹ shows that 4.29 is an immediate consequence of the first estimate Thus, we only need to discuss the first estimate. By definition of, we need to estimate Z Z + R Z R Z 2 = Y Ỹ + R Z R Z 2. Thanks to 4.27, the first summand is clearly bounded by the right-hand side of For the second summand we recall R Z s,t = Z s,t Z sx s,t = t s Y dx Y s X s,t = IΞ s,t Ξ s,t + Y s X s,t where Ξ s,t = Y s X s,t + Y s X s,t and similar for R Z. Setting = Ξ Ξ, we use 4.11 with β = 3 and Ξ replaced by, so that R Z s,t R Z s,t = I s,t s,t + Y s X s,t Ỹ s X s,t C δ 3 t s 3 + Y s X s,t Ỹ s X s,t, where δ s,u,t = RỸs,u X u,t R Y s,ux u,t +Ỹ s,u X u,t Y s,ux u,t. We then conclude with some elementary estimates of the type 4.26, just like in the proof of Theorem Controlled rough paths of lower regularity Recall that we showed in Section 2.3 how an -Hölder rough path X could be defined as a path with values in the p-step nilpotent Lie group G p R d T p R d, with p = 1/. It does not seem obvious at all a priori how one would define a controlled

6 116 8 Solutions to rough differential equations M T Y, Y Ỹ M T, Ỹ X,2 1 Y 2 Ỹ, Y Ỹ X,2 and so M T admits a unique fixed point Y, Y B T, which is then the unique solution Y to 8.1 on the possibly rather small interval [, T ]. Noting that the choice of T can again be done uniformly in the starting point, the solution on [, 1] is then constructed iteratively as before. In many situations, one is interested in solutions to an equation of the type dy = f Y, t dt + fy, t dx t, 8.11 instead of 8.6. On the one hand, it is possible to recast 8.11 in the form 8.6 by writing it as an RDE for Ŷt = Y t, t driven by ˆX t = ˆX, ˆX where ˆX = X t, t and ˆX is given by X and the remaining cross integrals of X t and t, given by usual Riemann-Stieltjes integration. However, it is possible to exploit the structure of 8.11 to obtain somewhat better bounds on the solutions. See [FV1b, Ch. 12]. 8.6 Stability III: Continuity of the Itô Lyons map We now obtain continuity of solutions to rough differential equations as function of their rough driving signals. Theorem 8.5 Rough path stability of the Itô Lyons map. Let f Cb 3 and, for 1 3, ] 1 2, let Y, fy D 2 X be the unique RDE solution given by Theorem 8.4 to dy = fy dx, Y = ξ W. Similarly, let Ỹ, fỹ be the RDE solution driven by X and started at ξ where X, X C. Assuming X, X M < we have the local Lipschitz estimates Y, fy ; Ỹ, fỹ C M ξ ξ + ϱ X, X, and also Y Ỹ C M ξ ξ + ϱ X, X, where C M = CM,, f is a suitable constant. Remark 8.6. The proof only uses the apriori information that RDE solutions remain bounded if the driving rough paths do, combined with basic stability properties of rough integration and composition. Proof. Recall that, for given X C, the RDE solution Y, fy DX 2 constructed as the unique fixed point of is

7 8.7 Davie s definition and numerical schemes 117 M T Y, Y := Z, Z := ξ + fy s dx s, fy DX 2, and similarly for MT Ỹ, f Ỹ C X. Then, thanks to the fixed point property Y, fy = Y, Y = Z, Z = Z, fy, similarly with tilde and the local Lipschitz estimate for rough integration, Theorem 4.16, and writing Ξ, Ξ := fy, fy for the integrand, we obtain the bound Y, Y ; Ỹ, Ỹ = Z, Z ; Z, Z ϱ X, X + ξ ξ + T Ξ, Ξ ; Ξ, Ξ, Thanks to the local Lipschitz estimate for composition, Theorem 7.5, uniform in T 1, Ξ, Ξ ; Ξ, Ξ ϱ X, X + ξ ξ + Y, fy ; Ỹ, f Ỹ. In summary, for some constant C = C, f, M, we have the bound Y, fy ; Ỹ, f Ỹ C ϱ X, X + ξ ξ + T Y, fy ; Ỹ, f Ỹ. By taking T = T M,, f smaller, if necessary, we may assume that CT 1/2, from which it follows that Y, fy ; Ỹ, f Ỹ 2C ϱ X, X + ξ ξ, which is precisely the required bound. The bound on Y Ỹ then follows as in 4.29, and these bounds can be iterated to cover a time interval of arbitrary fixed length. 8.7 Davie s definition and numerical schemes Fix f Cb 2W, LV, W and X = X, X C β [, T ], V with β > 1 3. Under these assumptions, the rough differential equation dy = fy dx makes sense as well-defined integral equation. In Theorem 8.4 we used additional regularity, namely Cb 3, to establish existence of a unique solution on [, T ]. By the very definition of an RDE solution, unique or not, Y, fy D 2β X i.e. Y s,t = fy s X s,t + O t s 2β