A remainder estimate for branched rough differential equations

Based on two isomorphisms of Hopf algebras, we provide a bound in the optimal order on the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths.


Introduction
In the seminal paper [1], Lyons builds the theory of rough paths. The theory solves rough differential equations (RDEs) of the form where x can be highly oscillating. Under a Lipschitz condition on the vector field, Lyons proves the unique solvability of the differential equation, and the solution obtained is continuous with respect to the driving signal in rough paths metric. The theory has an embedded component in stochastic analysis, and x can be Brownian motion, continuous semi-martingales, Markov processes, Gaussian processes [2] etc.
In 1972, Butcher [3] identifies a group structure in a class of integration methods including Runge-Kutta methods and Picard iterations, where each method can be represented by a family of real-valued functions indexed by rooted trees. In [4], Grossman and Larson describe several Hopf algebras associated with families of trees. One Hopf algebra of simple rooted trees, with product [4, (3.1)] and coproduct [4, p.199], is particularly relevant to our setting, which we refer to as the Grossman Larson Hopf algebra, denoted as H. In [5], Connes and Kreimer describe a Hopf algebra based on rooted trees [5, Section 2] to disentangle the intricate combinatorics of divergences in quantum field theory. We call this Hopf algebra the Connes Kreimer Hopf algebra, denoted as H R . The group identified by Butcher is the group of characters of H R [6]. Based on [7,8,9], H is isomorphic to the graded dual of H R .
Rough differential equations are originally driven by geometric rough paths over Banach spaces [1]. Geometric rough paths satisfy an abstract integration by parts formula, and take values in a nilpotent Lie group. The nilpotent Lie group can be expressed as a truncated group of characters of the shuffle Hopf algebra [10]. In [11], Gubinelli defines branched rough paths. Branched rough paths take values in a truncated group of characters of a labeled Connes Kreimer Hopf algebra. Both geometric and branched rough paths are of finite p-variation in rough paths metric, and encode information needed to construct solutions to differential equations. There exists a Hopf algebra homomorphism from the Connes Kreimer Hopf algebra onto the shuffle Hopf algebra, which induces an embedding of geometric rough paths into branched rough paths. On the other hand, the Grossman Larson algebra is freely generated by a collection of trees [12,13]. Based on the freeness of Grossman Larson algebra, Boedihardjo and Chevyrev construct an isomorphism between branched rough paths and a class of geometric rough paths [14]. As a result, a branched RDE can be expressed as a geometric RDE driven by a Π-rough path defined by Gyurkó [15].
Based on the isomorphism between H and the graded dual of H R [7,8,9], we clarify a relationship between rough paths taking values in the truncated group of characters of H R and rough paths taking values in the truncated group of grouplike elements in H (Proposition 2.3). Based on this relationship and the freeness of the Grossman Larson algebra, sub-Riemannian geometry [2, Section 7.5] and the neo-classical inequality [1,16], which are typical geometric rough paths tools, can be applied to branched rough paths. As an application, we provide an estimate for the remainder of the truncated Taylor expansion for controlled differential equations driven by branched rough paths (Theorem 2.5). The remainder estimate is in the optimal order (Remark 2.7), which is pleasantly surprising noting the rapid increase of the dimensions of simple rooted trees.

Notations and Results
A rooted tree is a finite connected graph that has no cycle with a distinguished vertex called root. We call a rooted tree a tree. We assume trees are non-planar, which means that the children trees of each vertex are commutative. A forest is a commutative monomial of trees. The degree |ρ| of a forest ρ is given by the number of vertices. For a given label set, a labeled forest is a forest for which each vertex is attached with a label.
Denote the label set L := {1, 2, . . . , d}. Let F L (T L ) denote the set of Llabeled forests (trees) of degree greater or equal to 1. Let F N L (T N L ) denote the set of elements in F L (T L ) of degree 1, . . . , N .
Let G N L denote the set of degree-N characters of the L-labeled Connes Kreimer Hopf algebra [5, p.214]. a is an element of G N L , if a is an R-linear map RF N L → R that satisfies (a, ρ 1 ) (a, ρ 2 ) = (a, ρ 1 ρ 2 ) for every ρ 1 , ρ 2 ∈ F N L , |ρ 1 | + |ρ 2 | ≤ N , where ρ 1 ρ 2 denotes the multiplication of commutative monomials of trees. Let △ denote the coproduct of the Connes Kreimer Hopf algebra based on admissible cuts [5, p.215]. Then G N L is a group with the multiplication given by for every ρ ∈ F N L . G N L is a labeled truncated Butcher group [3]. We equip G N L with the norm a := max With L = {1, 2, . . . , d}, let H L denote the L-labeled Grossman Larson Hopf algebra with product [4, (3.1)] and coproduct [4, p.199]. Denote the product and coproduct of H L as * and δ respectively. We consider H L as a Hopf algebra of labeled forests (by deleting the additional root in [4]). An element a ∈ H L is grouplike if δa = a ⊗ a. Let G L denote the group of grouplike elements in H L . For integer N ≥ 1, the set of series b = ρ∈FL,|ρ|>N (b, ρ) ρ form an ideal of H L . Let H N L denote the quotient algebra. Denote G N L := G L ∩ H N L . G N L is a group. We equip G N L with a continuous homogeneous norm. Let • a denote the labeled tree of one vertex with a label a ∈ L on the vertex. Let [τ 1 · · · τ k ] a denote the labeled tree with children trees τ 1 , . . . , τ k on the root and a label a ∈ L on the root. Define σ : F L → N as the symmetry factor given inductively by σ (• a ) := 1 and where τ i ∈ T L are different labeled trees (with labels counted). σ is the order of the permutation group on vertices in a tree that keeps the tree unchanged. Let △ denote the coproduct of the Connes Kreimer Hopf algebra, and let * denote the product of the Grossman Larson Hopf algebra. Based on [8, Theorem 43] and [9, Proposition 4.4], for ρ ∈ F L , For p ≥ 1, let [p] denote the largest integer that is less or equal to p.
L is continuous and of finite p-variation.
L , is continuous and of finite p-variation, and L . For integer N ≥ [p] + 1, there exists a unique extension of X resp.X to a continuous path of finite p-variation taking values in G N L resp. G N L . Still denote their extension as X resp.X. Then (3) holds for 0 ≤ s ≤ t ≤ T and ρ ∈ F N L .

Remark 2.4
Since the Grossman Larson algebra is free on a collection of trees [12,13],X acts as a bridge between X and geometric rough paths. In particular, sub-Riemannian geometry technique [2, Section 7.5] and the neo-classical inequality [1,16] can be applied toX. Then results are transferred back to X based on (3).
for τ i ∈ T L and a ∈ L, where d k f a denotes the kth Fréchet derivative of f a . Lipschitz functions and norms are defined as in [1, Definition 1.2.4, p.230]. For γ > 1, let ⌊γ⌋ denote the largest integer that is strictly less than γ.
L is a branched p-rough path over base space R d , and f : R e → L R d , R e is Lip (γ). Let y denote the unique solution of the branched rough differential equation Then with N := ⌊γ⌋, there exist two positive constants c 1 p,d,ω(0,T ) and c 2 p,d such that, The solution to branched RDEs is defined as in [11, (5) is in the optimal order (Remark 2.7).
Based on the fundamental theorem of calculus, for s ≤ t, for every s ≤ t and every N . On the other hand, consider f (t) = e −t . Then f Lip(n) = 1 on t ≥ 0 for n = 1, 2, . . . and The estimate (5) states that the remainder can be bounded similarly to that is in the optimal order even in the geometric case. The dimension of trees contributes a geometric increase factor 1 that is part of the control ω.
The proof of Theorem 2.5 is based on a mathematical induction that is an inhomogeneous analogue of [18]. The main estimate (5) is obtained by exploring the sub-Riemannian geometry of the truncated group of grouplike elements in the Grossman Larson Hopf algebra. The sub-Riemannian geometry structure is similar to that of the nilpotent Lie group [2, Theorem 7.32]. The factor ((N + 1) /p)! is obtained by the neo-classical inequality [1,16]. The tree neoclassical inequality is known to be false [19,Section 3]. Since the Grossman Larson algebra is free on a collection of trees, the analysis can be transferred back to the Tensor algebra where the neo-classical inequality holds. Our estimates rely critically on the simple fact that the number of words generated by a finite set of letters grows geometrically (Lemma 3.7).
where the infimum is taken over all continuous bounded variation paths The infimum in (6) can be obtained at a continuous bounded variation path x, which is called a geodesic associated with a ∈ G for every υ i ∈ B [p] L .
Proof. Define a norm on G inition of · ′ , equivalency of continuous homogeneous norms as in Proposition 3.4 and that X s,t , ρ = (X s,t , ρ) /σ (ρ), the proposed inequality holds. Notation 3.6 Let W denote the set of finite sequences t 1 · · · t k of t i ∈ B [p] L , including the empty sequence denoted as η. The degree |w| of w = t 1 · · · t k is |t 1 | + · · · + |t k |. The degree of η is 0.
Lemma 3.7 Let T n denote the number of elements in W of degree n. Then there exists K p ≥ 1 such that for n = 1, 2, 3, . . .
Proof. Recall that B denotes the collection of trees that freely generate the Grossman Larson algebra. . For p ≥ 1, let K p ≥ 1 be a number such that
For trees t i and a forest ρ, define (t 1 · · · t k ) ρ as the sum of |ρ| k forests that are obtained by linking each of the roots of t i , i = 1, . . . , k to a vertex of ρ by a new edge. Recall that * denotes the product in the Grossman Larson Hopf algebra (we delete the additional root in [4]). Then for trees t and t i , t * (t 1 · · · t k ) = tt 1 · · · t k + t (t 1 · · · t k ).
Lemma 3.11 For w ∈ W, |w| ≤ N , for y i ∈ R e . For w ∈ W, |w| < N and t ∈ B trees. Hence, for w ∈ W, the number of trees in F w is bounded by (|w| − 1)!. Each of these trees t is of degree |w| and corresponds to f (t) : R e → R e that is at least Lip (1 + {γ}) as |w| ≤ N . Then df (t) is a sum of |w| terms, as the differential d chooses a vertex in t. Hence, df (t) is bounded by |w|, because f and its derivatives of order up to N are uniformly bounded by 1 (we rescaled f by f −1 Lip(γ) ). As a result, for each tree t of degree |w|, f (t) (y 1 ) − f (t) (y 2 ) ≤ df (t) ∞ y 1 − y 2 ≤ |w| y 1 − y 2 . Then the first estimate follows, as there are at most (|w| − 1)! such trees in F w .
For w ∈ W, |w| < N and t ∈ B [p] L , the number of trees in F tw is bounded by |w|! ≤ (N − 1)!. Each tree corresponds to a map that is bounded on R e by 1.
Recall that B where t i , i = 1, 2, . . . range over elements in B [p] L . Proof. The equality can be obtained by iteratively applying the fundamental theorem of calculus.
Proof. Since K is the number of elements in B Recall that * denotes the product of the Grossman Larson Hopf algebra.
Notation 3.14 Define T X as T X s,t , t 1 · · · t k := X s,t , t 1 * · · · * t k for s ≤ t and t 1 · · · t k ∈ W for t i ∈ B [p] Proof. We prove the first estimate. The proof for the second estimate is similar.
Recall that T n denotes the number of elements in W of degree n. Based on Lemma 3.12, Lemma 3.11, Lemma 3.13, Lemma 3.5 and that T n ≤ (K p d) n in Lemma 3.7, we have Proof. According to T X s,t in Notation 3.14, T X s,t , t 1 · · · t k = X s,t , t 1 * · · · * t k for t i ∈ B [p] . Then based on the construction of the extension of T X andX [1, Theorem 2.2.1], it can be proved inductively that (7) holds for t i ∈ B [p] L , then X s,t , t 1 * · · · * t k = 0.
Based on the definition of T X in Notation 3.14, equivalency of continuous homogeneous norms on G [p] L as in Proposition 3.4 and X s,t , ρ = (X s,t , ρ) /σ (ρ), we have, for l ∈ W, |l| = 1, . . . , [p], with X s,t defined at (1), T X s,t , l Proof of Theorem 2.5. With K p in Lemma 3.7 and c p,d in Lemma 3.19, denote c 2 p,d := (K p d) p (c p,d ∨ 1) and setω := c 2 p,d ω. Denote Y w t := F w (y t ) for w ∈ W, |w| ≤ N and t ∈ [0, T ].