Uniform factorial decay estimates for controlled differential equations *

We establish a uniform factorial decay estimate for the Taylor approximation of solutions to controlled differential equations in the p -variation metric. As part of the proof, we also obtain a factorial decay estimate for controlled paths which is interesting in its own right.


Introduction
For a controlled differential equation of the form  (1.3) where f •m : R e → L R d ⊗m , R e is defined inductively by The functions f •k can also be expressed in terms of iterative applications of the vector field f as differential operators [3]. The iterated integrals in (1.2) will appear numerous times and we shall use the shorthand X k s,t := s<s1<...<s k <t dX s1 ⊗ . . . ⊗ dX s k . (1.4) Since the 1−variation norm of X equals to the L 1 norm of the derivative of X, we have (see for example [4]) Estimates of the form (1.5) have application both as a theoretical tool for analysing the equation (1.1) and as a practical numerical scheme for constructing the solution. The estimate (1.5), when the 1-variation metric is replaced by the p-variation metric, has been shown in [2] (p < 3), [5] (p < 3) and [4] (all p ≥ 1) without the factorial decay factor.
We shall prove such estimate with the factorial decay factor. The estimates of Davie [2], Gubinelli [5], Friz and Victoir [4] as well as our estimates below gives a numerical scheme for approximating a solution to (1.1) in O (1) time steps.
Theorem 1.1. Let p ≥ 1. Let X = 1, X 1 , . . . , X p be a p-weak geometric rough path. Let f be a Lip(γ − 1) vector field where γ > p. Let Y be a solution to the differential defined in the sense of [3]. Then there exists a constant C p depending only on p such Lip(min(γ−m,1)) ; (1.9) We refer the readers to Definition 9.16 and Definition 10.2 in [3] for the definition of Lip (γ) vector fields and weak geometric rough paths respectively. We shall however recall the definition of p-variation and some basic notations in Section 2.  Taking the biggest γ may not yield the best estimate for the left hand side of (1.7). In general the term f •γ could grow factorially fast in γ. Since a Lip(γ) function is also Lip(γ ) for all γ < γ, we may choose γ which optimises the estimate (1.7).
The proof for (1.5) relies heavily on the relation between the 1-variation of the path and the L 1 norm of its derivative. Proving an estimate of the form (1.5) for the p-variation metric, even without the factorial decay factor, requires the clever idea of Young [9]. The integration with respect to a path can be expressed in terms of the limit of a Riemann sum as the size of partition converges to zero. Young's idea was to estimate the Riemann sum with respect to a partition by removing points from the partition successively. This idea had been used in [6] to show that, for p < 2, the n-th order iterated integral of a path X is uniformly bounded by where ζ is the classical zeta function. T. Lyons' proof for the p ≥ 2 case in [7] is slightly different and used the neoclassical inequality ( [7], [1] to obtain an uniform bound of the form where Γ is the Gamma function and β is as defined in (1.9).

Notations and basic definitions
For each k ∈ N, we equip a norm on R d ⊗k by identifying it with R d k If π k denotes the projection operator T N We first recall Lyons' extension theorem, which will be used repeatedly in the following form:  R d which extends X, X 0 = (1, 0 . . . , 0) and for all p ≤ l ≤ N , (2.2) Remark 2.3. We will denote X −1 s X t by X s,t and π l (X s,t ) by X l s,t . In particular, X s,u ⊗ X u,t = X s,t and so, for any s < u < t, Note that for paths with finite 1-variation, the X k k≥1 defined in this theorem are exactly the iterated integrals of X. Hence no confusion will arise by using the same notation as in (1.4).

Remark 2.4.
If r ≥ p , then for any m ≥ 0, where the limit is taken as the mesh size of the partition P = (s < t 1 < . . . < t n−1 < t) goes to zero. By convention, for any s < t, X 0 s,t = 1 and X m s,t = 0 if m < 0. In the case r = m, (2.4) follows directly from (2.3). For r < m, note that the sum over k from r + 1 to m in (2.4) vanishes after the taking of limit, due to (2.2). See [5] for details.

The proof
The following lemma is a factorial decay estimate for the Taylor remainder of a controlled path in the sense of Gubinelli [5]. This lemma is interesting in its own right. We interpret it as the dual counterpart of Fact 2.2.

5)
for all s ≤ t and for 0 ≤ m ≤ γ − p − 1, the limit where |P| → 0 denotes the limit as the mesh size of a partition P on [s, t] goes to zero, For l ≥ p + 1, let X l denote the projection to R d ⊗l of the unique extension of 1, X 1 , . . . , X p given in Fact 2.2. Then (2.5) holds for all 0 ≤ m ≤ γ .
Proof. We will carry out backward induction on k starting from γ − p and moving down to 0.
The base induction step of k = γ − p holds because of the assumption. We will assume from now onwards that k ≤ γ − p − 1. It is useful to bear in mind that For the induction step, note that by (2.4) and the equality of (2.6) and (2.7), where the limit is taken as the mesh size of the partition P = (s < t 1 < . . . < t n−1 < t) goes to zero. We first show that the term is in fact independent of the partition P.
Since (2.10) is independent of the partition, By induction hypothesis, (2.5) which holds for m > k and Theorem 2.2.1 in [7], (2.14) where the final line is obtained by the neoclassical inequality (1.11), proved in [1].
Let ω (s, t) = X p p−var, [s,t] . We now choose j such that, for |P| ≥ 2, and also that for all j. Then as γ − k ≥ p + 1, (2.14) is less than or equal to By removing points successively from P and using that Y where the final line follows from (1.9).
For the differential equation Proof. We will prove it by backward induction, starting from γ . The case m = γ is trivially true.
For the induction step, note first that by the fundamental theorem of calculus, Then by (2.16) and the induction hypothesis, Proof of Theorem 1. The only thing to prove is that Y, f •1 (Y ) , . . . , f •( γ ) (Y ) satisfies the assumptions of Lemma 2.5.
For each s ≤ t, let x s,t : [s, t] → R d be a continuous path with finite 1-variation such that for 1 ≤ l ≤ p , x s,t l s,t = X l s,t , (2.17) where we use the notation from (1.4) and  for a function c p of p which is specified in [3] along with the existence of x s,t .
Consider the differential equation