A moment estimate of the derivative process in rough path theory

In this paper we prove the derivative process of a rough differential equation driven by Brownian rough path has finite $L^r$-moment for any $r /ge 1$. Thanks to Burkholder-Davis-Gundy's inequality, this kind of problem is easy in the usual SDE theory. In the context of rough path theory, however, it does not seem so obvious.


Introduction
In stochastic analysis, the derivative process of a given stochastic differential equation (or equivalently, equation of the stochastic flow) has been studied extensively, because it plays a very important role in various situations. On the other hand, in rough path theory, the derivative process was not studied very much. One reason could be that it has unbounded coefficients, for existence of solution for a rough differential equation (RDE) is in general difficult. The aim of this paper is to prove L r -integrability for the first level path of the derivative process for any r ≥ 1. Now we will give a more detailed explanation. We consider the following RDE in a Banach setting. Here, W is Brownian rough path and a is an initial value.
Its derivative equation is given by Roughly speaking, J t is the derivative of a map a → Y t = Y (a) t . It is known among experts of this research field that a unique solution (Y, J) in the rough sense exists, although there seems to be no published paper which proves it.
Our main result is as follows (see Theorem 3.7 below for details); Let 2 < p < 3 and let W be Banach space-valued Brownian rough path. Then, under a suitable condition on the coefficient σ, 1/p-Hölder norm of the first level path J 1 is L r -integrable for any r ≥ 1.
This kind moment estimate appears in various occasions. In the author's case, when he and H. Kawabi try to prove a stationary phase for solutions RDEs in a forthcoming paper [7] (which can be regarded as a rough path version of [2]), this type of integrability of the derivative process J is needed.
To the author's knowledge, the only exposition which explicitly discusses the derivative equation is Aida's unpublished manuscript [1], in which he established T. Lyons' continuity theorem for (Y, J) and proved the following estimate; If W is controlled by a control function ω, i.e., |W i s,t | ≤ ω(s, t) i/p for all 0 ≤ s ≤ t ≤ 1 and i = 1, 2, then sup t |J 1 0,t | ≤ C exp(Cω(0, 1) γ ) for some constants C > 0 and γ ≥ 1. This kind of deterministic argument is of great importance. However, it is not suitable for our purpose, becsuse even if we take ω(s, t) = W 1 p p−var;[s,t] + W 2 p/2 p/2−var;[s,t] , the right hand side is not integrable. Here, · p−var;[s,t] is p-variation norm on the subinterval [s, t]. Therefore, we need to take a closer look at the argument in [1]. Remark 1.1 (i) the same results holds for an RDE with a drift term. In such a case, we only need to consider the lift of a "space-time" process t → (w t , t) instead of Brownian rough path. Here, w is Brownian motion (i.e., the first level path of W ). (ii) The author does not know whether the main result is true or not when the driving rough path is a lift of fractional Brownian motion. See Lemma 3.4 and remark 3.5 below for details.

Setting
Let (V, H, µ) be an abstract Wiener space and let (w t ) 0≤t≤1 be Brownian motion on V associated to µ, which starts at 0. Let 2 < p < 3 and let GΩ p (V) be the geometric rough path space over V with p-variation norm. (When given the 1/p-Hölder norm, the geometric rough path space is denoted by GΩ 1/p−Hld (V).) In this article, the time interval is always [0, 1] and tensor spaces of Banach spaces are equipped with the projective tensor norm, unless stated otherwise. We basically use the original formulation of rough path theory as in Lyons and Qian [10], although there are a few variants of the theory now.
We denote by w(m) be the mth dyadic approximation of w, i.e., the piecewise linear approximation associated to the partition {0 < 1/2 m < 2/2 m < · · · < (2 m − 1)/2 m < 1}. Its lift, i.e., the smooth rough path above w(m), is denoted by W (m) as usual. Unlike the finite dimensional case, existence of Brownian rough path is not known. So we set the following assumption: Here, the norm is Hölder norm of index i/p. The limit is denoted by W and is called Brownian rough path. (This formulation is used in Deriech [3]. The well-known sufficient condition "Exactness " as in Definition 4.6.1, [10] implies (A1). So, we will work under this assumption.) Interestingly, (A1) implies almost sure convergence of W (m), too (see [3]).
Let W be another real Banach space and σ : W → L(V, W) be C 4 b in the Fréchet sense (i.e., ∇ j σ is bounded for j = 0, 1, . . . , 4). Here, L(V, W) denotes the set of bounded linear maps from V to W, which is equipped with the operator norm. Consider the following RDE; for X ∈ GΩ p (V), Under this regularity condition for σ, this RDE has a unique solution. So, X → Y (=: Φ(X)) defines a map, which is called Itô map. By T. Lyons' continuity theorem, Φ : Aida [1] gave a rahter quantative estimate for the growth of the solution and the local Lipschitz constant for two solutions. If X is controlled by a control function ω (that is, |X i s,t | ≤ ω(s, t) i/p for all s ≤ t and i = 1, 2), then Z = (X, Y ) is controlled by a control functionω of the formω(s, t) = C(1+ω(0, 1) γ )ω(s, t) with certain positive constants C, γ, which is independent of the initial value a.
Adding to RDE (2.3), we also consider the following "derivative equation." Notice that a solution J takes its values in L(W) := L(W, W) and that ∇σ(Y t ) is a bounded bilinear map from W × V to W. Formally, J t is the derivative of a map a ∈ W → a + Y 1 0,t ∈ W. RDEs (2.1) and (2.2) are obviously equivalent to consider the following RDE; where M an L(W)-valued path, which is given by is C 3 b and the right hand side of (2.5) is well-defined as rough path integral. So, if X is controlled by ω, then the right hand side of (2.6) is a rough integral and M is controlled by ω ′ (s, t) = C ′ (1 + ω(0, 1) γ ′ )ω(s, t), with certain positive constants C ′ , γ ′ , which is independent of the initial value a.
RDEs ( In solving an RDE with an unbounded coefficient, the most difficult part is always how to control the first level path of a solution. In this case, however, thanks to the special form in (2.5) and the series representation (2.7) below, it is possible to prove existence of a unique solution (Y, J).
Let us recall how this is solved in [1]: (Step 1) let us first consider the case when X is a smooth rough path lying above x ∈ C 1−var 0 ([0, 1], V). Then, ODEs (2.3) and (2.4) has a unique solution t → (y t , j t ) in 1-variational sense. Moreover, it is well-known that j t can be written explicitly as follows; where A k is given by and M t on the right hand side is given by (2.6) in 1-variational sense (with X and Y being replaced with x and y, resp.). Notice the order of the product of M t j 's on the right hand side. ( Step 2) Fortunately, A k is written in the form of interated integral. So, the series representaion in (2.7)-(2.8) fits well with rough path theory. The following argument is quite similar to the "fundamental theorem of rough path theory" (Theorem 3.1.2, [10]), which states that one can obtain the ith level path (i ≥ 3) from the first and the second level paths. Note also that, in the same way as in the fundamental theorem, the map x → M → j extends continuously with respect to the topology of GΩ p (V).

(Step 3)
Suppose that x ∈ C 1−var 0 ([0, 1], V) satisfies that X 1 p p−var + X 2 p/2 p/2−var ≤ R for R > 0. Then, it is shown that sup 0≤t≤1 |j t | ≤ C R < ∞ for some positve constant C R . So, when we try to solve RDEs (2.3)-(2.4) for such x, the (local) solution (Y, J) coincides with (the lift of) the solution (y, j) in the usual sense and it (=its first level path) does not get out a large ball of radius C ′ R > 0. It is also shown in [1] that, for such an x and its lift X, Lipschits property for the map holds; Therefore, this map naturally extends to a one from {X ∈ GΩ p (V) | X 1 p p−var + X 2 p/2 p/2−var ≤ R} to GΩ p (W ⊕ L(W)). Since R > 0 is arbitrary, this map is defined for any X ∈ GΩ p (V).
Summing up, we have the following proposition in [1].

Proposition 2.1 RDEs (2.3)-(2.2) has a unique solution for any X and the map
is (locally Lipschitz) continuous. Moreover, J 1 i.e., the first level path of J, admits a series representation as in (2.7)-(2.8).

Moment estimate of J 1
In this section we prove that L r -momoent of J 1 is finite when X = W , by using the series in is finite by using the series representation in (2.7)-(2.8).
Let M = (M 1 , M 2 ) ∈ GΩ p (L(W)). Then, by the fundamental theorem of rough path theory, we can construct M 3 , M 4 , . . .. When M is a smooth rough path lying above m, then M k coincides with the iterated Stieltjes integral, i.e., for a smooth rough path M = (M 1 , M 2 ) ∈ GΩ p (L(W)) lying above m. Then, M → A k:s,t extends to a continuous map from GΩ p (L(W)) and the following inequality holds; |A k:s,t | ≤ |M k s,t | for all k and s ≤ t Proof. Set T : L(W) ⊗k → L(W) by T (a 1 ⊗ · · · ⊗ a k ) = a k · · · a 1 . A basic propety of the projective tensor norm, the operator norm of T is 1. Noting that A k:s,t = T (M k s,t ), we can easily prove the lemma.
It is well-known how fast L r -norm of W i (i = 1, 2) grows as r → ∞, from which one can obtain growth of M i (i = 1, 2).
We will show that, in general, if a random variable Z ≥ 0 defined on a certain probability space with Fernique-type integrability condition E[e βZ 2 ] < ∞, then Z L r = O( √ r). (In the following, we assume β > 1/2 for simplicity. Otherwise, we take constant multiple of Z instead of Z itself.) By Chebychev's inequality, there exists a constant C > 0 such that P(Z ≥ η) ≤ Ce −βη 2 for all η > 0. Let ε > 0 be sufficiently small so that α := β − ε > 1/2.
So, it suffices to show that f (n, r) := − r log r 2 + r log(n + 1) − αn 2 is bounded from above. ∂f /∂n(n, r) = −2αn + r/(n + 1). It is easy to see that, as a function of n, f takes it maximum at n = (−1 + √ 1 + 2α −1 r)/2. Hence, It is easy to see that, as r → ∞, Take δ > 0 so small that log(1 + δ) < 1/4. There exists r 0 > 0 such that, for all r ≥ r 0 , So, the right hand side of (3.2) is dominated by which implies that the right hand side of (3.2) is bounded from above. Lemma 3.3 When X = W (i.e., Brownian rough path), there are positive constants c and α independent of r ≥ 1, which satisfy that, for all s ≤ t, Here, β p is a positive constant such that (Any such a positive constant which satisfies the above inequality will do.) Proof. We can choose ω(s, t) = ( W 1 p 1/p−Hld + W 2 p/2 2/p−Hld )(t − s) as a control of W . Then, as is explained in (2.3), M is controlled by ω ′ (s, t) = C ′ (1 + ω(0, 1) γ ′ )ω(s, t), where C ′ , γ ′ are positive constants. Now, by using inequality (3.1) in Lemma 3.2 and choosing a suitable constant c > 0, we can see the lemma holds for 2α = 1 + (γ ′ /p).
Proof. We use induction. The cases k = 1, 2 were already shown. Assume the inequality (3.3) up to k − 1 and let us prove (3.3) for k.
Let us write M j s,t = M(a, X) j s,t , where X is the driving rough path and a ∈ W is the initial condition of the RDE. For a dyadic rational number u ∈ [0, 1], we setw t = w u+t −w u for 0 ≤ t ≤ 1 − u. Sincew and {w s } 0≤s≤u are independent,W , the lift ofw, and {W s,s ′ } 0≤s≤s ′ ≤u are independent. For u < t, M(a, W ) j u,t = M(a + Y (a, W ) 1 0,u ,W ) j 0,t−u .
for L ≥ 3. Hence, we see from the definition of β p that This completes the proof.
Remark 3.5 notice that, in the proof of above lemma, we used the independent increment property of Brownian motion. So, if the driving process is replaced with fractional Brownian motion, this proof fails.
Proposition 3.6 There exists a positive constant C = C r which is independent of s, t such that ∞ k=1 A k:s,t L r ≤ C(t − s) 1/p for all s < t and Id W + J 1 0,t L r ≤ C for all t.
Then, the first estimate holds for C = k {r α C 1/p } k /{β p (k/p)!}. The second estimate is clear from the first one since Id W + J 1 0,t = Id W + ∞ k=1 A k:0,t . for a continious path ψ in the usual sesne, which takes its values in a Banach space W and starts at 0. It is well-known that this norm is stronger than the Hölder norm; there exitsts a positive constant C m,θ such that, for all ψ, ψ 1/p−Hld ≤ C m,θ ψ m,1/p . Now we give the main result of this paper, which states that the first level path of the derivative equation (2.2) is L r -integrable for any r ≥ 1, provided X = W . Proof. From the series representation (2.7), it is clear that J 1 s,t = (Id + J 1 0,t ) ∞ k=1 A k:s,t . We obtain from Proposition 3.6 that J 1 s,t L r = Id + J 1 0,t L 2r · ∞ k=1 A k:s,t L 2r ≤ C √ t − s.