General Rough Path Theory With Its Application In Linear And Nonlinear Expectations

In many areas of engineering, finance and mathematics one encounters equations of the form dyt = f(yt)dxt, (0.0.25) where x is a multi-dimensional driving signal,f a collection of driving vector fields.2 For x ? C1 or C1-var, i.e. continuous paths of locally finite 1-variation, this can be written as y(t) = f(y(t))x(t). In the later case, more specifically, it means that For x (?)C1-var, it is helpful to distinguish between low regularity (e.g. x?CP-var for some p > 1) and lack of continuity (e.g. x ? V1) before tackling the case of general (possibly discontinuous) driver with finite p-variation, i.e.x ? Vp,for arbitrary p < ?. The first case, x ? Cp-var includes the important class of continuous semimartingales (with p > 2), and here already one encounters a fundamental ambiguity in the interpretation of (0.0.25), with Ito- and Stratonovich interpretation being the most popular choices. The literature on Ito stochastic integration against - and differential equations driven by general semimartingale is vast, see e.g. [40, 62,46, 2]. Without semimartingale (or in fact:any probabilistic) structure, rough paths [53] provide a satisfactory substitute,that deals with all p <?, provided x ? Cqp, the space of continuous, geometric p-rough paths: any ambiguity is then resolved by the additional information contained in x,by the very nature of a rough path. (Probability is still used to construct a - random - rough path over some given stochastic process, see e.g. [18],[17].) we note that continuous semimartingales as continuous rough path were studied in [10, 20, 18]. In [15] the authors consider piecewise linear, axis-directed approximations of continuous semimartingale and so obtain an Ito SDE in the Wong-Zakai limit; in [52] Ito SDEs are obtained as averaged Stratonovich solutions. Concerning the history of rough differential equations with jumps: the Young case p ? [1,2) was already studied in[73]. Rough integration against cadlag rough path, p ? [2,3), was introduced in [22], as was the notion of cadlag geometric p-rough path, any p < ?. In the latter setting, limit theorems and applicability to semimartingales (canonical solutions in the sense of Marcus) was obtained in [7].A different interesting phenomena, purely deterministic (and unrelated to rough path consid-erations), arises when one drops continuity of the driving signal, even in the case of finite 1-variation. To wit,take x ? D1?D?V1,where D denotes the space of cadlag paths(so that dx can be interpreted as Lebesgue-Stieltjes measure). We may consider the possible integral equations with the meaning of (0.0.29) best desribed by words: "replace every jump ?xs = xs - xs-by a straight line over some artificial extra time interval and solve the resulting continuous differential equations; finally disregard all extra time for the solution" (this is Marcus' canon-cial solution in the finite variation context; it has the advantage of preserving the chain-rule.)The price we need to pay for this transformation is that we indeed change the integration and equation. The reverse procedure is highly nontrivial, so it may be difficult to study equation(0.0.28) through (0.0.29).The first equation (0.0.27) deserves no further attention, as in general it does not admit solutions (a simple counter-example y0= 1, f(y) = y, dx= ?1 easy to obtain y1 = 1 + y1).So we are left with (0.0.28), (0.0.29), solutions to which have been called [73] forward and geometric respectively. In absence of jumps forward and geometric solutions coincide. We also note that the (to stochastic analysts familiar) structure of (0.0.28), with caglad integrand and cadlag integrator, is by no means necessary and it is only consequent in a purely deterministic development to discard of this assumption: it is enough to have x?V1 to study a generalization of (0.0.28), based on a left-point Riemann-Stieltjes integral, which we write as One can go beyond 1-variation regularity by suitable rough path considerations. At least conceptually, this is easier in the geometric case, i.e. equation 0.0.29, see recent work of Chevyrev-Friz [7]. The "forward theory" on the other hand,topic of the present paper,comes with different challenges. First, the lack of a chain-rule makes it impossible to work in the geometric rough path setting. For p ? [2,3) this just means that the state space can be identified with Rd ?(Rd)(?)2, dropping the well-known geometricity (a.k.a. "first order calculus") condition [17, (2.5)]. For general p < oo, however, we need to use the full Hopf algebraic formalism of branched rough paths. That is,x?Vp =Vp([0,T],G|p|(H*)),the space of path on [0, T] with values in the step-[p] Butcher group, again with a suitable p-variation condition. For example, as pointed out in the works of Gubinelli and Hairer-Kelly, [26, 28], the seemingly suitable state space Rd? (Rd)(?)2 ?(Rd)?3 is not sufficient to understand level-3 rough paths. Secondly, the analysis in our case, starting with rough integration, has to be carried out from first principles, i.e. cannot be reduced to the well-developed continuous case. To this end,we develop some variants of the "sewing lemma"(e.g. [17, Ch. 4] and the references therein) with non-continuous a.k.a. non-regular controls-which in turn offers (pathwise) expansions of integrals and solutions to (possibly stochastic) differential equations. This also implies that our notion of solution delicately depends on the fine-structure of the jumps (left-right limits exist everywhere). On a technical level, the presence of (big) jumps puts an a priori stop to the interval on which Picard iteration can be used to construct maximal solutions, this is dealt with by treating (finitely many) big jumps by hand. Our analysis comes with stability estimates in terms of p-variation rough path metrics,which readily can be framed in (purely deteministic) limit theorems with respect to p-variation rough path variants of the Skorokhod J1 metric. An immediate application then concerns discrete approximations (higher order Euler or Milstein-type schemes) for rough differential equations, underpinning the numerical nature of the entire theory, as previously done by Davie [11], 2 < p < 3 and then [18], any p < ?. Moreover, now we allow for jumps in the limiting rough paths.Furthermore, we also study the case of random p-rough paths. General (cadlag) semimartin-gales, with their Ito-lift, are seen to give rise to such random rough paths, and important estimates such as the BDG inequality remain valid for the homogenous p-rough path norm. This allows us to finally reconcile the "general" theory of (cadlag) semimartingales and their Ito theory, with a matching theory of general (and then cadlag) rough path. The abstract main result of that section is Theorem 5.1. Amongst others, we shall see that many classical stability results ("limit theorems") in the general Ito semimartingale theory, notably the classical UCV/UT conditions (Kurtz-Protter [47], [48] and Jakubowski, Memin and Pages [41]) thus admit a perfect explanation from a rough path point of view, see also [10]. In essence,UCV/UT implies tightness of p-variation rough path norms, which is the key condition (in a random rough path setting, but without any implicit semimartingale assumptions) for limit theorem of random rough differential equations. This is especically interesting in the case of Ito SDEs (interpreted as random RDEs) driven by a convergent sequence of semimartingales which does not satisfy UCV/UT. This is a typical situation in homogenization theory, see e.g.the works of Kelly and Melbourn. Checking the desired rough path p-variation tightness in Theorem 5.1 can be non-trivial. In Section Five we thus present a catalogue of the relevant (general) techniques we are aware of. Amongst others, we present a Besov-type criterion for discrete rough path approximation that presents a definite improvement of the main result in [42], namely a "2+ instead of 6+ moments assumption" relative to the Holder scale used in Kelly [42] (cf. also Erhard-Hairer [14] applied in the rough path setting). This is almost optimal, as seen from martingale examples in which the UCV/UT conditions exhibits "2 moments" as optimal. (It would be desirable, though this is by no means the purpose of this paper,to build a similar Besov theory for discretizations of regularity structures in the sense of [14].)We note however that Besov spaces are ill-suited to deal with (limiting) differential equations with jumps, a point also made in [61]. In particular, the recent Besov ramifications [27, 29] would not be able to cover our result.In the last part of this paper, we study the application of rough paths in G-expectation theory.G-expectation theory was introduced by Peng in [59], [58]. G-expectation is a time consistent sublinear expectation, which is obtained from a fully nonlinear parabolic PDE, called G-heat equation, with the canonical process Bt as G-Brownian motion. Stochastic analysis and the corresponding BSDEs in G-framework are established in [59], [58], [60], [34], [35]. More recent work concerning G-expectation theory would be referred to Denis-Hu-Peng [12], Hu-Peng [37, 36], Li-Peng [50], Soner-Touzi-Zhang [64], Song [65, 67, 68, 69]?A natural question is what is the relation between rough integrals and stochastic integrals with respect to G-Brownian motion. Such questions are first studied in Geng-Qian-Yang [24]. In this paper we study the rough path properties based on the ?-Holder continuity of G-Brownian motion, of which the enhancement could be completed by a generalized Kolmogorov's criterion for rough paths under G-expectation framework, which is more direct and probabilistic compared with [24]. The relation among rough integral, Ito integral and Stratonovich integral with respect to G-Brownian motion is established. Then we turn to the pathwise approximation of Wong-Zakai kind to SDEs driven by Brownian motion. At last, the roughness of G-Brownian motion is calculated and then the Norris lemma for stochastic integral with respect to G-Brownian motion is obtained.Now we present main results of this article.1. Sewing lemma and calculus with left-right jumps under bounded variation and Young's caseDefinition 0.1. Let X : [0, T]? (E, ?·?),where,E is a Banach space. Define Xs,t: Xt-Xs.We call X regulated,denoted as X ? V?, if X has both left and right limits on [0,T]?We say X has finite p- variation, p ? [1, ?), denoted as X ? VP, if the following is finite,where P is a partition of [0, T]?Specially, if X is right-continuous, we denote X ? Dp,Definition 0.2. We call a function ?=?(s,t) defined on {0 ? s ? t ? T} with values in[0, oo) a control, if it satisfies ?(s, s) = 0; ?(s, t) + ?(t, u) ??(s, u). In particular, givenX ?Vp,?X,p(s,t) := ?X?p,[s,t]p is a control.Definition 0.3. We call a function ?(s,t) from the simplex {(s, t) : 0 ? s ? t ? T} to nonnegative real numbers a mild control,if it is increasing in t, decreasing in s,null on the diagonal, and for any ? > 0, there exist only finite u? [0, T] such that We call ?(s, t) a mild control for some path x if d(xs,xt) ??(s, t)for any s, t? [0, T]. I Now we introduce two kinds of convergence. In the following,one may take ?s,t ?YsXs,t?Definition 0.4. (MRS v.s. RRS) For a partition P of [0,T] and any [u,u] ?P, ?u,v(e.g. yuxu,v)takes value in Rd. We call the Riemann sum ?[u,v]?P?u,v converges to K in the sense of·Mesh Riemann-Stieltjes (MRS) sense( classical one): if for any ? > 0, there exists ?> 0,such that any |P| < ?, one has |?[u,v]?P?u,v-K| < ?.·Refinement Riemann-Stieltjes (RRS) sense (Hildebrandt[71],[72]): if for any ?> 0, there exists P? such that for any refinement P(?)P?, one has |?[u,v]?P?u,v-K|<?.We denote this limit K by (MRS resp. RRS) I?0,T·Here is the main result of this section, which could be known as an abstract version of integration theory without continuity assumption.Theorem 0.1. (mild sewing) Suppose ?s,t is a mapping from simplex {(s,t) : 0 ? s < t ? T}to a Banach space (E,?·?). Let ??s,u,t:=?s,t-?s,u-?u,t.Assume ?? satisfies???s,u,t???(s,u)?(u,t),where a is a mild control and ? is a control. Then the following limit exists in the RRS sense,and one has the following local estimate:?I?s,t-?s,t???(s,t-)?(s+,t),Furthermore,if ?(s,t) is right-continuous in s,i.e.or ?(s,t) is left-continuous in t, the convergence holds in the MRS sense.As an immediate application of the above theorem, we have the following.Proposition 0.1. (integration in bounded variation case)·If x?V1([0, T],Rd), y?V?([0, T],L(Rd, Re)), the Riemann sum ?(u,v)?P yuxu,v?Re converges in the RRS-sense and we write Furthermore,one has·If x ? V1([0,T],Rd),y ? V?([0,T],?(Rd,Re)), and furthermore y is caglad, or x cadlag, the Riemann sum ?(u,v)?P yuxu,v converges in the MRS-sense.·If x?D1([0,T],Rd), y? V?([0,T],L(Rd,Re)), then·If x ?D1([0,T],Kd),y ?V? ([0,T],L(Rd,Rd)),,then one has with the righthand-side uses Lebsgue-Stieltjes integration,?x([a,b]) = x(b) - x(a).Similarly, in Young's case, i.e. x?Vp, p ? [1,2), we also have the following sewing lemma.Theorem 0.2. (Young sewing) Suppose ?s,t is a mapping from simplex {(s, t) : 0 ? s?t ?T} to a Banach space (E, ?·?). Define ??s,u,t:=?s,t-?s,u-?u,t.Assume ?? satisfies???s,u,t???1?1(s,u)?2?2(u,t).where ?1??2 are controls on the same simplex and ?1 + ?2 >1. Then the following limit exists uniquely in the RRS sense,and one has the following local estimate:?I?s,t-?s,t??C??1(s,t-)?2?2(s+,t)with C depending only on ?1+?2. Furthermore,if ?2(s,t) is right-continuous in s,i.e.or ?1(s, t) is left-continuous in t, the convergence holds in the MRS sense.As an application of the above theorem, we have the following integration theory in Young's case.Proposition 0.2. (integration in Young 's case)Let x ? Vp([0, T],Rd),y?Vq([0,T],L(Rd, Rn)) with 1/p + 1/q > 1. Then the limit (P paritition of[0,T])exists in RRS sense and we have the local estimate If x is right-continuous (i.e. cadlag), then the above limit exists in MRS sense and we will write ? y- dx for the integral,notation consistent with the BV case.With left-right jumps, we also have the following version of Taylor expansion and integration by part formula.Proposition 0.3. (general integration by parts) Let x, y belong to Vp, Vq respectively , where either p = 1,q = ? or 1/p +1/q > 1. Then one has where ?t-y :=yt- yt-, ?t+y:=yt+-yt,?T+y := 0. In particular, if x and y are right-continuous, we have the following familiar form,Theorem 0.3. (Taylor expansion)Suppose x? Vp([0,T),Rd) with p? [1,2),f?Liploc1+?(Rd, L( with ? > p - 1. Then we have the following Taylor's formula In particular,if x is right-continuous,we have the classical result,For a differential equation driven by x which is bounded variation or in Young's case, i.e.x?Vp, p ?[1,2),we have the following existence and uniqueness result?Theorem 0.4. (differential equations in Young's case) Suppose x ? Vp([0,T],Rd), p ?[1,2), f? C2(Re,L(Rd ,Re)). Thenn for any y0?Re, there exists t0 > 0, such that there exists a unique yt on [0, t0], such that where ?0t f(yr)ldxr is defined as in Proposition 1.3. Furthermore, if f ?Cb2, then the solution exists on [0,T]. In particular,if x is right-continuous,according to Lemma 2.1,y is also continuous?2. Calculus and differential equations driven by general second order rough pathIn the part we consider the case of p ? [2,3), which tells the story of second order rough paths.Here are some notations.Definition 0.5. (second order rough path) Suppose X : [0, T] ?Rd,X: [0, T]2?(?)Rd satisfying·(algebraic condition) Xs,t - Xs,u - Xu,t= Xs,u (?)Xu,t·(analytic condition) ?X?p,[0,T]<?; ?X?p/2,[0,T]p/2:=supP?[u,v]?P |Xu,v|p/2?,p?[2,3).We call (X,X)a second order rough path?We denote the set of rough paths as Vp([0,T],Rd)?Now we introduce the integrand for rough integral, which is called controlled rough path?Definition 0.6. (controlled rough path) (Y, Y') is called a X-controlled rough path, if there exist Y, Y' defined on [0, T] with values in Rl,L(Rd,Rl) respectively, with finite p-variation, satisfying that R : [0, T]2 ?Rl defined by has finite p /2-variation?We call the collection of controlled rough paths as VXp(0,T],Rl)?Similar as Young's case, we have the following abstract version of integration theory.Theorem 0.5. (general sewing lemma)? is defined on {(s,t) : 0 ? s < t ? T} with values in a Banach space (E,?·?). Let ??s,u,t:=?s,t-?s,u-?u,t and suppose ?? satisfies where ?1,j,?2,j are controls and ?1,j + ?2, j > 1 for j=1,...,N. Then the following limit exists uniquely in the RRS sense,and one has the following estimate:with C depending only on min j{?1,j+?2,j} and N. Furthermore, if ?1,j(s,t), j=1,...,N are left-continuous in t or ?2,j(s,t), j = 1,...,N are right-continuous in s, then the convergence holds in the MRS sense.According to the above theorem, we have the following proposition.Proposition 0.4. (integral of second order rough paths) Suppose X = (X, X) is a rough path with finite p-variation for p ? [2,3), and (Y, Y') is a controlled rough path in the following sense,Y,Y' ? Vp, Rs,t :=Ys,t-Ys'Xs,t?Vp/2.Define ?s,t = YsXs,t+Ys'Xs,t. Then one has the following convergence and estimate? C(?R?p/2,[s,t)?X?p,(s,t]+?Y'?p,[s,t)?X?p/2,(s,t]), (0.0.34)where C depends only on p. In particular, if X is cadlag, then the convergence in (0.0.32)holds in MRS-sense and we write For second order rough paths,we also have Taylor's formula (or say Ito's formula)?Definition 0.7.(reduced rough paths)X=(X,S)is called a reduced rough path if Xt?Rd,Ss,t?Sym(Rd (?) Rd) which is null on the diagonal and furthermore satisfies·X has finite p-variation, S has finite p/2-variation, p ? [2,3).In particular, if X' = (X, X) is a second order rough path, then X := (X, Sym(X)) defines a reduced rough path?Definition 0.8. (bracket of reduced rough path) Suppose X = (X, S) is a reduced rough path. Define [X]s,t :{(s, t)|0 ?s ?t ?T} ? Sym(Rd (?) Rd) which satisfies [X]s,t:= Xs,t(?)Xs,t-2Ss,t.Indeed,according to "Chen's" condition (algebraic condition) in the definition of reduced rough paths, we have [X]s,t=[X]0,t - [X]0,s,which means [X]t := [X]0,t actually is a path with values in Sym(Rd (?) Rd)?Furthermore, by definition, [X] has finite p/2-variation?Now we give the Ito's formula of rough paths?Theorem 0.6. (Ito's formula of rough path) Given a (reduced) p-rough path X and F ?Liplocp+?,with ? > 0, one has the following identity,where ?0T D2F(Xt)ld[X]t is Young's integral,and In particular, if X is cadlag, then one has the following form,Furthermore, if X = (X, X) is a rough path, then the above formula holds with ?0T F(Xt)ldXt considered as rough integral and [X]t = X0,t(?)X0,t-2Sym(X) .As for the differential equations driven by second order rough paths, we have the following theorem.Theorem 0.7. (global solutions for differential equations driven by second order rough paths)(i). Suppose X ? Vp is a second order rough path, F?C3? Then there exists [0,t0], such that there exists a unique Yt ? Vp([0, t0], Re) solving the following rough differential equation where (Y,F(Y)) is a controlled rough path?Furthermore, if F ?Cb3,then the solution is global?(ii). Let ?X?p,[0,T],?X?p,[0,T]<L, and Y, Y are solutions driven by X, X respectively?Then we have the following local estimate ?Y-Y?p,[0,T]?M2(CP,F(1?L))M+1(?X;X?p,[0,T]+ |Y0-Y0|),where M is a constant bounded by Cp,F Lp+ 1?3. Integration and differential equations driven by branched rough pathDenote (G(H*),*) the Butcher group,with its inverse mapping given by S*?More detailed definition can be obtained in Chapter Three. Here is the definition of general branched rough path?Definition 0.9. (branched p-rough path) A branched p-rough path is a mapping X : [0, T]?G[p](H), such that for any f ?F[p],where Xs,t := Xs-1 * Xt and P is any partition of[0, T]. Denote the following as the control of the branched rough path,We also use the following norm,Similar as second order rough path,now we introduce the controlled rough path?Definition 0.10. (controlled rough paths)X is a branched p-rough path. Z : [0,T] ?H|p|-1 is called a -controlled rough path if for any f*?(F|p|-10)*:= (F|p|-1)*?{1*},(f*,Zt) =<Xs,t*f*,Zs>+Rs,tZ,f, (0.0.37) or in a explicit form,where RZ,f are functions on the simplex and satisfy ?f?|p|-1F0 ?RZ,f?p/(?-|f|),[0,T]p/(?-|f|)??Z,?(0,T).for some control ?Z,? with ? ? [p]. In particular, denote Zsh:= <h*, Zs> and zt := <1*, Zt>,and one has and one calls Z a controlled rough path above z. In the following, we usually take ?=[p] for simplicity.For the integration of branched rough path,we have the following result?Theorem 0.8. (integration of branched rough paths) Suppose X is a branched p-rough path and Z is a X-controlled rough path. Let Then one has exists,and with C depending on ? and p. In particular,if ?X,p(s+,t) = ?X,p(s,t) or ?Z,?(s,t-)=?z,?(s,t) the convergence holds in MRS sense.For differential equations driven by branched rough paths, we also obtained the existence and uniqueness result.Theorem 0.9. (global solution for differential equations driven by branched roug paths)Suppose X ? Vp, F ? Cb|p|+1. Then there exists a unique solution on [0,T] satisfying Furthermore, if Y is the solution of differential equation driven by X, then we have the following local Lipschitz estimate,?Y-Y?p,[0,t] ? C(|y0 - y0| +?X;X?p,[0,T]).where C depends F, y0,p, L, ?X?p, ?X?p< L?4. Cadlag RDE stability under Skorokhod type metricsWe recall the Skorokhod topology for cadlag paths space in some metric space E. Denote ?[0,T] the set of increasing bijective functions from [0, T] to [0, T]. For any x,y ? D([0,T],E),the Skorokhod metric is given by where |?|:= supt?[0,T] ?(t) -t|. We can define a p-variation variant of this metric. To this end, let E be the Butcher group GN(H*) as introduced in Section Three, equipped with leftinvariant metric (see appendix). For any cadlag branched rough paths X, Z with finite pvariation,where we recall that In particular, for the level-2 rough path case, one has E?Rd ?Rd×d, and with ?X - Z??,[0,T]:= sup0?s<t?T |Xs,t- Zs,t|.According to interpolation inequality of general rough paths, i.e. Lemma 4.1, one has the following result.Theorem 0.10. (convergence of RDEs under uniform and Skorokhod topology) Suppose X, Xn are cadlag p-rough paths with, as above, supn ?Xn?p,[0,T] = L < ?.Let Yn be the (unique) solution to dYn = F(Yn)dXn, Yn = y0.Then, for any p' > p with [p'] = [p], one has the following estimate in the uniform version,with M = Cp,FLp as before, and then also in Skorokhod rough metric,In particular, if Xn converge to X uniformly or in Skorokhod topology, with a uniform pvariation bound, the RDE solutions converge uniformly or in Skorokhod topology, also with uniform p-variation bounds.An important application of general rough paths is the discrete approximation, which could be taken as high level Euler approximation. For a partition P of [0,T], we define a cadlag rough path as following: for any [s, t) ? P,XuP=Xs if u ? [s,t), XTn?XT.e.g. in level two rough path case,X0,uP = (X0,s, X0,s) if u ? [s,t), X0,TP?(X0,T,X0,T). For differential equations driven by rough paths, we consider the above approximation by discrete rough paths, which is indeed our familiar Euler approximation.Theorem 0.11. (Higher order Euler schemes for cadlag RDEs)Given a cadlag p- rough path X, consider the RDE Suppose (Pn) is a sequence of partitions of [0,T],with vanishing mesh-size. For any [sn ,tn) ?Pn, define piecewise constant path Yn by where F?: Re ?Re is defined in the argument before Theorem 3.2. In particular, for the level-2 rough path case,Ytnn: Ysnn+F (Ysnn) Xsn,tn+DF(Ysnn)Xsn,tn.Then Yn converges to Y in the Skorokhod sense, with uniform bounded p-variation. In particular, the convergence holds in p'-variation metric of Skorokhod type, for any p' > p.5. General rough paths in linear expectationSuppose (Xn) = (Xn(?)) is a sequence of random p-rough paths. For example,given a semimartingale X,then its Ito integral gives a lifted level two rough path?For random Rough differential equations (RDEs), we have the following convergence theorem?Note that random rough paths are not limited in semimartingale framework.Theorem 0.12. (convergence for random RDEs)Let 1 ? p <? and F?Cb[p]+1 Consider random, cadlag p-rough paths Xn? X weakly(or in probability) under the uniform (or Skorokhod) metric, with {?Xn?p,[0,T](?)} tight. Let Yn solve random RDEs dYn = F(Yn)-dXn (0.0.41)and Y solve the same one driven by X with the same initial value y0. Then the random RDE solution Yn converges weakly (or in probability) to Y under the uniform (or Skorokhod) sense.Moreover, {?Yn?p,[0,T](?): n?1} is tight and one also has the weakly (or in probability) convergence in p'-variation uniform (or Skorokhod) metric for any p'> p.For stochastic differential equations driven by semimartingales, we have the following equivalence with random RDEs.Proposition 0.5. (equivalence of RDEs and SDEs )Suppose X is a semimartingale and X is its Ito's lift as above. F ?Cb3. Then the solution for the random RDE dYt = F(Yt)ldXt, Y0 = y0,agrees,with probability one,with the Ito's SDE dYt = F(Yt-)dXt, Y0 = y0.As for the compact assumption in Theorem 0.12, it may be nontrivial for checking such condition. In the last part of Chapter Five, we provide some criterions known to us.6. Rough paths in nonlinear expectationsIn this part we consider Holder continuous rough paths in G-expectation?For a continuous path X defined on [0, T] with values in Rd, its ?-Holder norm is where Xs,t =Xt-Xs?We denote C?([0, T],Rd) the space of ?-Holder continuous paths. Similarly, X is defined on [0, T]2 with values in Rd(?)Rd We define norm In this part, we always assume ?? (1/3,1/2), since we only consider stochastic calculus driven by G-Brownian motion.Definition 0.11. For fixed ?, the space of continuous rough path C?([0,T],Rd) includes the following elements (X, X), which satisfies·Xs,t -Xs,u -Xu,t = Xs,u(?)Xu,t,·X has finite a-Holder norm, X has finite 2?-Holder norm?For such pare X := (X,X) ? C?([0,T], Rd], we define the following semi-norm,?X?C?:=?X?? +?X?2?.Definition 0.12. A path Y E C?([0, T],Rm) is called a X-controlled rough path, if there exists Y'?Ca([0,T],L(Rn,Rm)),such the remainder term exists Y'?C?([0, T], L(Rn, Rm)), such the remainder term satisfies ?RY?2? < ?. We denote the space of rough paths as CX2?([0,T],Rm)?For any(Y, Y') ? CX2?([0, T], Rm), we define its semi-norm as ?Y, Y'?X,2? := ?Y'??+ ?RY?2??First we need to lift G-Brownian motion to level-2 rough paths?An obvious choice is (B, B) (?):=(B,?st Bs,rdBr)(?)?One may check that by Kolmogorov theorem for rough paths under G-expectation,it belongs to rough path space C? quasi-surely?Furthermore,we have the following equivalence of stochastic integral with respect to G-Brownian motion and rough integral.Proposition 0.6. (G-Ito integral equivalent with rough integral) Suppose (Y,Y')(?))?CB(?)2?([0,T],R),c-q.s, and Y, Y' ?MG2(0, T), with Yt,Yt'?LG2(?t),for any t ? [0, T]. Furthermore, suppose ??Y???L2,??Y'???L2< ??Then one has the following identity,In particular,?(u,v)?P(YuBu,v +Yu'Bu,v) converges to ?0T YrdBr under LG2 norm?The following theorem describes the roughness of G-Brownian motion?Definition 0.13. Given ?? (0,1), A path X ? C?([0,T],Rd) is said to be ?-Holder rough for some given ?? (0,1), on the scale of ?0> 0, if there exists a constant L >0, such that for any a ? Rd, s ? [0, T], and ?? (0, ?0], there always exists t ? [0, T], satisfying |t-s|<?,and |a · Xs,t| ?L??|a|.The largest value of such L is called the modulus of ?-Holder roughness of X, denoted by L?(X). It is obvious that the modulus L?(X) has the following expression:Proposition 0.7. (Holder roughness for G-Brownian motion) Let B be a d-dimensional G-Brownian motion. Then for any ?? (1/2, 1), B (?) is ?-Holder rough, c - q.s. with scale T/2.More precisely, there exist positive constants K,l, depending on T,?,?, such that for any ?? (0,1/2T?), one has the bound c(L?(B) <?)? Kexp(-l?-2). (0.0.44) In particular, by Norris lemma of rough paths, we have the following corollary which distin-guishes the stochastic integral driven by G-Brownian motion with smooth integrals?Corollary 0.1. Let B=(B,B), (Y, Y')(?)?DB(?)2?([0,T],L(Rd,Rn)),and Z? C?([0,T],Rn),c-q.s.. Furthermore, suppose (Y, Y') satisfies assumptions in Proposition 6.3. Then denote It=?0tYsdBs+?0tZsds,and R=1+L?(B)-1+?B?l?+?Y,Y'?B,2?+|Y0|+|Y0'|+?Z??+|Z0|.One has the inequality ?Y??+?Z???MRq?I??r c-q.s.,for some constants M, q, r, depending only on ?, ?, T.In particular, if it holds that Y?Y',Z?Z',c-q.s..According to the continuity theorem of rough differential equations, one may obtain the Wong-Zakai theory for differential equations driven by G-Brownian motion. Furthermore, we can also obtain the pathwise convergence rate of such approximation. We consider approximation of G-Brownian motion by piecewise linear paths. Suppose B is a d-dimensional G-Brownian motion on [0,1]?{tj(n)}j=0n is the partition of [0,T] with mesh size 1/n,Bt(n) is the piecewise linearized G-Brownian motion according to {tj(n)}j=0n and y0?Rm is the initial condition for following ODEs,dYt(n) = f(Yt(n))dBt(n) + g(Yt(n))d<B>t + h(Yt(n))dt. (0.0.45)Then we have the following Wong-Zakai approximation in G-framework.Theorem 0.13. Suppose Yt(n) solves ODEs (0.0.45) c-quasi surely, and Xt solves the following G-SDE of Stratonovich's kind,with f ? Cb2,g,h?Cb1. Then for any t ? [0,1],Yt(n) converges to Xt in LG2-norm sense.Furthermore, for any t ? [0,1], one has the following inequality,where K depends on f, g, h and ?.Now we provide the pathwise convergence rate for Wong-Zakai approximation?Theorem 0.14. Suppose f, g,h? Cb3, and Y(n) defined as in (0.0.45). Also, suppose X solves the following G-Stratonovich SDE and Y solves the following RDE driven by G-Stratonovich rough paths,dYt = f(Yt)dBstrat + g(Yt)d(B)t + h(Yt)dt, (0.0.48) with initial condition x0. Then for any ? < 1/2-?, one has the following inequality ?Y-Y(n)???M(?)1/n?,c-q.s..In particular, X = Y, c- q.s., and ?Y-Y(n)???M(?)1/n?,c-q.s..