Rough differential equations containing path-dependent bounded variation terms

We consider rough differential equations whose coefficients contain path-dependent bounded variation terms and prove the existence and a priori estimate of solutions. These equations include classical path-dependent SDEs containing running maximum processes and normal reflection terms. We apply these results to determine the topological support of the solution processes.


Introduction
In the framework of Itô's calculus, path-dependent stochastic differential equations(=SDEs) are naturally formulated and the existence and uniqueness hold under suitable standard assumptions on the coefficients.For example, reflected SDEs and SDEs containing running maximum and minimum processes are typical examples.In one dimensional cases, very simple SDEs containing the maximum and minimum processes and reflection term have been studied in detail.In this paper, we consider rough differential equations (=RDEs) whose coefficients contain pathdependent bounded variation terms and prove the existence and a priori estimate of solutions.This class of equations include the classical path-dependent SDEs mentioned above.Although the solutions are not unique in general, the uniqueness holds for smooth rough paths in many cases.Under the uniqueness assumption, we prove a continuity property of solution mappings at smooth rough paths which is useful to determine the topological support of the solution processes.
The structure of this paper is as follows.In Section 2, we introduce a class of RDEs containing bounded variation terms: where X t is a 1/β rough path (1/3 < β ≤ 1/2) and A(Z) t is a continuous bounded variation path which depends on the past path (Z s ) s≤t .After that, we state our main theorem (Theorem 2.7) which proves the existence and a priori estimate of solutions under σ ∈ Lip γ−1 (γ > 1/β) and suitable assumptions on A. Note that the regularity assumption on σ for the existence of solutions is standard in the case of usual RDEs which corresponds to A ≡ 0. The solution Z t is a controlled path of the driving rough path X.Actually, we solve this equation in product Banach spaces consisting of Z and Ψ = A(Z) by applying Schauder's fixed point theorem.
To this end, we introduce Hölder continuous path spaces C θ and Banach spaces C q-var,θ consisting of Ψ based on the control function ω of X.The latter is a set of paths whose qvariation norms (q ≥ 1) are finite and satisfy a certain Hölder continuity defined by ω.We also study basic properties of the functional A. We briefly explain examples but we will discuss the detail in Section 5.
In Section 3, we prove our main theorem.The uniqueness does not hold in general.See Remark 2.8 (6).
In Section 4, we consider usual β-Hölder rough path X with the control function ω(s, t) = |t − s|.We show that the (generally multivalued) solution mapping is continuous at a rough path for which the solution is unique in Proposition 4.2 using a priori estimate of solutions.We use this result to prove support theorems in Section 6.
In Section 5, we give examples.In Subsection 5.1, we consider reflected rough differential equations on a domain D in R n : where Φ t is the reflection term which forces Y t ∈ D. This equation looks different from the equation studied in the main theorem.However, it is well-known that reflected Itô (Stratonovich) SDEs can be transformed to certain path-dependent Itô (Stratonovich) SDEs without reflection term.This is used to prove Freidlin-Wentzell type large deviation principle ( [5]) and the support theorem ( [14]) for reflected diffusions on domains with smooth boundary.We prove the existence theorem (Theorem 5.6) under standard assumptions (A) and (B) on D and σ ∈ Lip γ−1 by transforming the equation (1.2) to the corresponding path-dependent RDE (1.1).This is an extension of the result in [2] in which we proved the existence of solutions of (1.2) under stronger assumptions that D satisfies the condition (H1) and σ ∈ C 3 b .In 1-dimensional cases, perturbed SDEs and perturbed reflected SDEs were studied by many people.See e.g.[7,8,10,11,13,31,36].In Subsection 5.2, we give a short review of these subjects.
In Subsection 5.3, we consider multidimensional and rough path versions of 1-dimensional perturbed SDEs and perturbed reflected SDEs.In the study of the latter one, we need to consider an implicit Skorohod equation as in [2].As for perturbed reflected SDE whose driving process is the standard Brownian motion, we can extend the existence and uniqueness result of the solution due to Doney and Zhang [13] by using our approach.See Remark 5.22.
Path-dependent functional A(x) t which we are mainly concerned with in this paper is a kind of generalization of the maximum process max 0≤s≤t |x s | and the local time term L(x) t .The maximum process max 0≤s≤t |x s | is obtained as the limit of x L p ([0,t]) as p → ∞.Hence it may be natural to study the case where A(x) t = x L p ([0,t]) .In Subsection 5.4, we study such examples.
In Section 6, we prove support theorems for solution processes by using Proposition 4.2 and Wong-Zakai theorems.In this section, except Theorem 6.4, we consider the Brownian rough path W which implies that we consider the usual Stratonovich SDEs driven by the standard Brownian motion.
Section 7 is an appendix.The solution Y t studied in Section 5 is a sum of a controlled path Z t and a continuous bounded variation path Φ t .For a given controlled path Z, the Gubinelli derivative Z ′ is uniquely determined if the first level path X of X is truly rough in the sense of [20].In our case, Φ is certainly bounded variation but does not have good regularity property in Hölder norm.Hence it is natural to ask whether Z ′ is unique or not for Y in our setting.We study this problem by using a certain rough property of the path X in Subsection 7.1.In Subsection 7.2, we make a remark on path-dependent rough differential equations with drift.This consideration is necessary for the study of the reflected diffusions with the drift terms.

Preliminary and Main Theorem
Let us fix a positive number T .Let ω(s, t) (0 ≤ s ≤ t ≤ T ) be a control function.That is, (s, t) → ω(s, t) ∈ R + is a continuous function and ω(s, u) + ω(u, t) ≤ ω(s, t) (0 ≤ s ≤ u ≤ t ≤ T ) holds.We introduce a mixed norm by using ω and p-variation norm.We refer the readers to [21] for the related studies.Let E be a finite dimensional normed linear space.For a continuous path (x t ) (0 ≤ t ≤ T ) on E, we define for [s, t] ⊂ [0, T ], x ∞,[s,t] = max x ∞-var,[s,t] = max x p-var,[s,t] = sup where P = {s = t 0 < • • • < t N = t} is a partition of the interval [s, t] and x u,v = x v − x u .When [s, t] = [0, T ], we may omit denoting [0, T ].For 0 < θ ≤ 1, q ≥ 1, 0 ≤ s ≤ t ≤ T and a continuous path x, we define ) We use the convention that inf is a Hölder continuous path with the exponent θ in usual sense.Hence we may say x is an ω-Hölder continuous path with the exponent θ ((ω, θ)-Hölder continuous path in short).For two parameter function F s,t (0 ≤ s ≤ t ≤ T ), we define F θ,[s,t] similarly.We denote by C θ ([0, T ], E) the set of ω-Hölder continuous paths x with values in E satisfying x θ = x θ,[0,T ] < ∞.We may denote the function space by (C θ ([0, T ], E), ω) to specify the control function.C θ ([0, T ], E) is a Banach space with the norm |x 0 | + x θ .We may just write C θ (E) if there is no confusion.Let C q-var,θ (E) denote the set of E-valued continuous paths of finite q-variation defined on [0, T ] satisfying x q-var,θ := x q-var,θ,[0,T ] < ∞.Note that C q-var,θ (E) is a Banach space with the norm |x 0 |+ x q-var,θ .Obviously, any path x ∈ C q-var,θ (E) satisfy |x s,t | ≤ x q-var,θ ω(s, t) θ .We may write C θ , C q-var,θ for simplicity.
We next introduce the notation for mappings between normed linear spaces.Let E, F be finite dimensional normed linear spaces.For γ = n + θ (n ∈ N ∪ {0}, 0 < θ ≤ 1), Lip γ (E, F ) denotes the set of bounded functions f on E with values in F which are n-times continuously differentiable and whose derivatives up to n-th order are bounded and D n f is a Hölder continuous function with the exponent θ in usual sense.
We use the following lemma.The compact embedding in ( 2) is necessary for the application of the Schauder fixed point theorem.
(2) By (1), we have var,θ ′ < ∞, then by their equicontinuities, there exists a subsequence such that x n k converges to a certain function x ∞ in the uniform norm.By (2.7), we can conclude that the convergence takes place with respect to the norm on C q-var,θ .
Then for q > q 0 , sup (2.10) Taking the limit q → ∞, we obtain the desired estimate.
We use the following quantity, We introduce a set of controlled paths D 2θ X (R n ) of X s,t , where 1/3 < θ ≤ β following [24,20].A pair of ω-Hölder continuous paths (Z, The rough differential equations which we will study contain path dependent bounded variation term A(x) t .We consider the following condition on A. Note that the function space C β in the following statement depends on the control function ω.
(2) (Continuity) There exists 1/3 < β 0 < β such that A can be extended to a continuous mapping from ).We use the same notation A to denote the extended mapping on C β 0 .
(3) There exists a non-decreasing positive continuous function hold.
Remark 2.3.The conditions (1), ( 2) are natural.In many cases, A is defined on continuous path spaces and is continuous with respect to the uniform norm.The condition (3) is strong assumption.This implies that the total variation of A(x) on [s, t] can be estimated by the norm of the path (x u − x s ) on s ≤ u ≤ t.Note that this does not exclude the case where We have the following simple result.
Lemma 2.4.Let ω be a control function and let C β ([0, T ], R n ) be the corresponding Hölder space.
Actually, the condition (3) automatically implies the following stronger estimate.By this result, we may assume that the growth rate of F (u) is at most of order u 1/β , that is, a polynomial order.
(2) Applying Lemma 2.1 (2) in the case where q ′ = 1, θ ′ = β, θ = α, we have ≤ ω(0, T Note that By the assumption on A, we have This completes the proof. Remark 2.6.Of course, we may optimize the estimate (2.15) as follows: where We now introduce our RDEs and state our main theorem.
satisfies the condition in Assumption 2.2.Then the following hold.
(1) There exists a controlled path (Z, (2) All solutions (Z, Z ′ ) of (2.20) satisfy the following estimate: there exist positive constants K and κ 1 , κ 2 , κ 3 which depend only on σ, β, γ, F such that First we make some remarks for this theorem and after that we explain some examples.
(1) From now on, we always set γ > 1/β for 1/3 < β ≤ 1/2 if there is no further comment. ( Let {Ψ t } 0≤t≤T be a continuous bounded variation path on R n .Then we can define the integral t 0 σ(Z s , Ψ s )dX s in a similar way to the usual rough integral.We denote the derivative of σ = σ(ξ, η)

ΞΨ
t i−1 ,t i also converges to the same limit value.We denote the limit by t s σ(Z u , Ψ u )dX u .Hence the sum of the term t s Ψ s,r ⊗ dX r does not have any effect on the integral.However, we need to consider Ξ Ψ instead of ΞΨ to obtain estimates of the integral in Lemma 3.2 which is necessary for the proof of the main theorem.
(3) Let us consider the case σ(ξ, η) = σ(ξ + η), where σ Clearly, the decomposition of Y to controlled path part Z and the bounded variation part Ψ is not unique.We should note that our definition of t 0 σ(Y s )dX s depends on Z ′ and Y .However, under a natural assumption, the Gubinelli derivative Z ′ t is uniquely defined for Y and the integral does not depend on the decomposition (Z, Ψ).We discuss this problem in the appendix.(4) Theorem 2.7 implies that the solution Z t satisfies the following estimate: Here G is a certain polynomial function which depends on σ, β, γ, F .Also θ(> 1) is a positive constant which depends on β and γ (When γ = 3, θ = 3β holds).Clearly, a path Z t which satisfies (2.22) is a solution of (2.20).(5) Let ω be a control function and C i be positive constants.Actually, under the assumption that for all 0 ≤ s ≤ t ≤ T , we can prove similar results to Theorem 2.7 for β-Hölder rough paths X with ω(s, t) = |t − s| by a similar proof of the main theorem.This extension is necessary to treat the examples in Example 2.9 (3) and ( 4).However, we need to change the upper bound function in (2.21).The reason is as follows.The β-Hölder rough path X can be regarded as a (ω, β)-Hölder rough path, where ω(s, t) = ω(s, t) + |t − s|.We can do the same proof as in the main theorem in this setting.The control function ω in (2.21) should be changed to this ω and accordingly |||X||| β also should be changed to the corresponding quantity.Also we should replace the term the uniqueness of the solutions hold under the assumption σ ∈ Lip γ .However, even if A ≡ 0, the uniqueness does not hold in general under σ ∈ Lip γ−1 .See Davie [9].Gassiat [23] gave an example which showed that the uniqueness does not hold for reflected RDE even if the coefficient is smooth and the domain is just a half space.Contrary to this, in one dimensional case (note that the driving noise is multidimensional one), the uniqueness of the solutions of reflected RDEs were proved by Deya-Gubinelli-Hofmanová-Tindel in [12].It may be interesting problem to find natural class of solutions for which the uniqueness hold and a non-trivial class of reflected RDEs or more generally path-dependent RDEs for which the uniqueness hold in an appropriate sense.See also Subsection 5.4 for some examples for which the uniqueness hold.
The situation is different if β > 1/2.Ferrante and Rovira [19] proved the existence of solutions of reflected (Young) ODE on half space driven by fractional Brownian motion with the Hurst parameter H > 1/2.Falkowski and S lomiński [18] proved the Lipschitz continuity of the Skorohod mapping on a half space in the Hölder space and proved the uniqueness in that case.
We briefly explain examples.We refer the reader for the detail to Section 5.
Example 2.9.(1) Let D be a domain in R n satisfying conditions (A) and (B).Consider the Skorohod equation y t = x t + φ t , where x is a continuous path whose starting point is in D. Also y t ∈ D (0 ≤ t ≤ T ) and φ t is the bounded variation term.The mapping L : x → φ satisfies Assumption 2.2.Using this result, we can apply the main theorem to reflected rough differential equations. ( (2.23) This satisfies Assumption 2.2.Actually this satisfies the stronger conditions (Lip) ρ and (BV) ρ for certain ρ in Definition 5.12.See Proposition 5.13 for the proof.Note that even if we replace each max 0≤s≤t f i (x s ) by finite products of maximum functions and minimum functions of f (x s ), Assumption 2.2 holds.
(3) Let c 1 , . . ., c n be β-Hölder continuous paths on R n in usual sense.Let f be a Lipschitz map from R n to R n .Let us consider a variant of the example (2) as follows: This does not satisfy Assumption 2.2 (3).However it holds that for some positive constant C.
(4) We consider the case ω(s, t) = |t − s|, that is, usual β-Hölder rough path.Path-dependent functional A(x) t which we are mainly concerned with in this paper is a kind of generalization of the maximum process max 0≤s≤t x s and the local time term L(x) t .The maximum process max 0≤s≤t |x s | is obtained as the limit of x L p ([0,t]) as p → ∞.Hence it may be natural to study the case where A(x) t = x L p ([0,t]) .Theorem 2.7 cannot be applied to this directly.We will study this example in Subsection 5.4.
(5) Let W t be the 1-dimensional standard Brownian motion starting at 0. Let us consider the following equations, Here a denotes a real number.The equation (2.25) contains the local time term Φ t at 0. These processes have been studied e.g. in [7,8,10,11,13,31,36].We see that a multidimensional version of these equations can be transformed to the equation of the form (2.20) in Section 5.2.We also give some brief review of 1-dimensional cases there.

Proof of Theorem 2.7
In the calculation below, we assume γ ≤ 3 as well as γ > 1/β.
If we write A(Z) t = Ψ t , then the equation (2.20) reads We solve this equation by using Schauder's fixed point theorem.First, we give an estimate of the integral t s Ψ s,r ⊗ dx r (0 ≤ s < t ≤ T ), where x ∈ C θ , Ψ ∈ C q-var,θ ′ and ⊗ denotes the tensor product.To this end, we introduce some notations.Let 0 ≤ s ≤ t ≤ T and consider a mapping We use the following estimate.
Let p be a positive number such that θp > 1.Let q be a positive number such that 1/p + 1/q ≥ 1 and Ψ ∈ C q-var,θ ′ (R n ).For any 0 ≤ s < t ≤ T , the integral t s Ψ s,r ⊗ dx r converges in the sense of Young integral and it holds that where Proof.
The assumption implies x is finite 1/θ-variation.Moreover θ + 1/q > 1 holds.Hence the Young integral of t s Ψ s,r ⊗ dx r converges and the following estimate holds: which completes the proof.
By using this lemma, we will give estimates for the integral Until the end of this section, we choose and fix p > 0 such that 1/β < p < γ.For this p, we assume q, α, α satisfy the following condition.
As we explained, we consider By a simple calculation, we have for s < u < t, Thus, under the assumption on Z, Ψ, applying Lemma 3.1 to the case θ = β, θ ′ = α and where C = 3 γ−2 (2 + C β,q ).Therefore, there exists a positive constant C which depends only on γ, β, p such that where (3.10) , by the Sewing lemma (see e.g.[29,22,20]), the following limit exists, We may denote I ((Z, Z ′ ), Ψ) by simply if there are no confusion.This integral satisfies the additivity property The pair (I(Z, Ψ), σ(Y t )) is actually a controlled path of X.In fact, we have the following estimates.
Proof.(1) This follows from the explicit form of (3.5) and Lemma 3.1.
(3) This follows from (2) and Lemma 3.1.(4) This follows from the definition of Y t .
We consider the product Banach space D 2θ 1 X ×C q-var,θ 2 , where 1/3 < θ 1 ≤ 1/2 and 0 < θ 2 ≤ 1.The norm is defined by Let ξ be the starting point of Z and let The solution of RDE could be obtained as a fixed point of the mapping, We prove a continuity property of M.
Proof of Lemma 3.4.First note that X follows from Lemma 3.2.By Assumption 2.2, we have Thus we have proved (3.26).We estimate We argue in a similar way to the sewing lemma for the estimate of the first term.Let By (3.15), Hence this term is small in the ω-Hölder space C 2α on a bounded set of W T,α, α,q,ξ,η if N is large.
We fix a partition so that this term is small.Although the partition number may be big, is a finite sum, and by the explicit form of δΞ as in (3.6), we see that this difference is small in C 2α if ((Z, Z ′ ), Ψ) and (( Z, Z′ ), Ψ) are sufficiently close in W T,α, α,q,ξ,η .The estimate of the second and the third terms are similar to the above and we obtain the continuity of the mapping We next prove the continuity of the mapping Since we choose β 0 ≤ α, it suffices to apply Lemma 2.5 (2) to the case where x = ξ + I(Z, Ψ) and x ′ = ξ + I( Z, Ψ) because of Lemma 3.2 (2) and the continuity (3.33).By using the above lemmas, we prove the existence of solutions on small interval [0, T ′ ].Since the interval can be chosen independent of the initial condition, we obtain the global existence of solutions and the estimate for solutions.We consider balls with radius 1 centered at Lemma 3.6 (Invariance and compactness).Assume (3.24) and let α < α < α < α < β.Also we choose q ′ > 1 such that α α q < q ′ < q.
(1) For sufficiently small T ′ , we have Moreover T ′ does not depend on ξ.
We are in a position to prove our main theorem.
Proof of Theorem 2.7.(1) Let us take α, α, p, q, α, α as in Lemma 3.6.By Lemma 3.4 and Lemma 3.6, applying Schauder's fixed point theorem, we obtain a fixed point for small interval [0, , where K is a certain positive constant.That is, there exists a solution on [0, T ′ ].We now consider the equation on [T ′ , T ].We can rewrite the equation as where Note that we already defined Ãy,T ′ (x) t (0 in Lemma 2.4 (4).Thanks to Lemma 2.4, we can do the same argument as [0, T ′ ] for small interval.By iterating this procedure finite time, say, N -times, we obtain a controlled path (Z t , Z ′ t ) (0 ≤ t ≤ T ).This is a solution to (2.20).Clearly, We need to show (Z, Ψ) ∈ W T,β,β,1,ξ,η and its estimate with respect to the norm • β .We give the estimate of the solution on [0, T ′ ].The solution (Z, Z ′ ) which we obtained satisfies (3.16) and (3.1), we have Second, by (2.14) and (3.43), we have Therefore Z and A(Z) are (ω, β)-Hölder continuous paths.Hence, we have Moreover, we can apply Lemma 3.2 to Z and Φ = A(Z) in the case where α = α = β and q = 1.Thus, by substituting the estimates (3.43) and (3.44) for (3.19), we obtain for 0 . These local estimates hold on other small intervals.By the estimate (3.41), we obtain the desired estimate.
we obtain T ] < ∞, taking the limit β ↑ β, this estimate hold for the norms • β and • 2β as well.The estimates of Z and A(Z) follow from this estimate and the estimates similar to (3.45) and (3.47).This completes the proof.

A continuity property of the solution mapping
In this section, we consider the case where ω(s, t) = |t − s|.That is, we consider usual Hölder rough paths.Also let us denote the set of β-Hölder geometric rough paths (1/3 < β ≤ 1/2) by ) which is the closure of the set of smooth rough paths in the topology of C β (R d ).In this paper, smooth rough path means the rough path h defined by a Lipschitz path h ∈ C 1 and its iterated integral h2 s,t = t s (h u − h s ) ⊗ dh u .We identify h and the Lipschitz path h.Also we denote the set of smooth rough paths by C Lip (R d ).
Let Z(h) be a solution to (2.20) for X = h.Then Z(h) is a solution to the usual integral equation As already explained, we cannot expect the uniqueness of the solution of the RDEs in our setting driven by general rough path X.However, the uniqueness hold in many cases when the driving rough path is a smooth rough path and σ is sufficiently smooth.If the solution to the ODE (4.1) is unique, then Z(h) is uniquely defined and (Z(h), R Z(h) , A(Z(h))) satisfies the same estimate as in Theorem 2.7.We use the notation Z(h) t instead of Z(h) t in this case.We denote the set of solutions (Z, Z ′ ) of our RDE (2.20) by Sol(X).We prove a certain continuity property of multivalued mapping X → Z(X) ∈ Sol(X) at the rough path X for which the solution is unique.Thus, this multivalued map is continuous in such a sense at any smooth rough path if the uniqueness holds on the set of smooth rough paths.
We write Clearly, these spaces are Fréchet spaces with the naturally defined semi-norms.Also note that Z(X) Proof.By the estimate in Theorem 2.7 (2), we can choose {N k } such that Z(X N k ), A(Z(X N k )) converges in C β− and C 1+-var,β− respectively.This implies lim k→∞ t s A(Z(X N k )) s,r dX N k (r) = t s A(Z(X)) s,r dX r which shows the limit Z satisfies the inequality (2.22).This proves (Z, σ(Z, A(Z))) ∈ Sol(X).We have for all (s, t).Combining the uniform estimates of (ω, 2β)-Hölder estimates of them, this completes the proof.
The following proposition follows from the above lemma Proposition 4.2.We consider the equation (2.20) and assume the same assumption on A and σ in Theorem 2.7.Assume the solution of (2.20) is unique for the rough path X 0 ∈ C β (R d ).
Then the multivalued mapping X(∈ C β (R d )) → Sol(X) is continuous at X 0 in the following sense.For any ε > 0, there exists δ > 0 such that for any X satisfying |||X − X 0 ||| β ≤ δ and any Z(X) ∈ Sol(X), it holds that ).It holds that for any sequence {h N } ⊂ C Lip (R d ) satisfying lim N → |||h N − X||| = 0, any accumulation points of {Z(h N )} belong to Sol(X).The set Sol ∞ (X) which consists of such all accumulation points is a subset of Sol(X) and may be a natural class of solutions.However Sol ∞ (X) = Sol(X) may hold.
By a similar argument to the proof of Theorem 4.9 in [2], we can prove the existence of universally measurable selection mapping of solutions as follows.
Proposition 4.4.We consider the equation (3.1) and (3.2) and assume the same assumption on A and σ in Theorem 2.7.Then there exists a universally measurable mapping which satisfies the following. ( is a solution in Theorem 2.7 and satisfies the estimate in (5.13).
(2) There exists a sequence of Lipschitz paths h Proof.Below, we omit writing ξ.We consider the product space, and its subset Let Ē0 be the closure of E 0 in E. Then Ē0 is a separable closed subset of E. The separability follows from the continuity of the mapping h → ((Z(h), σ(Y (h))), Ψ(h)).Note that Sol ∞ (X) coincides with the projection of the subset of Ē0 whose first component is X.Hence by the measurable selection theorem (See 13.2.7.Theorem in [15]), there exists a universally measurable mapping I : . This mapping satisfies the required properties in ( 1) and ( 2).Remark 4.5.It is not clear that we could obtain the adapted measurable solution mapping I.

Reflected rough differential equations
Let D be a connected domain in R n .As in [34,27], we consider the following conditions (A), (B) on the boundary.See also [35].
Definition 5.1.We write B(z, r) = {y ∈ R n | |y − z| < r}, where z ∈ R n , r > 0. The set N x of inward unit normal vectors at the boundary point x ∈ ∂D is defined by

.2)
(A) There exists a constant r 0 > 0 such that (B) There exist constants δ > 0 and 0 < δ ′ ≤ 1 satisfying: for any x ∈ ∂D there exists a unit vector l x such that Let us recall the Skorohod equation.The Skorohod equation associated with a continuous path x ∈ C([0, ∞), R n ) with x 0 ∈ D is given by Under the assumptions (A) and (B) on D, the Skorohod equation is uniquely solved.This is due to Saisho [34].We write Γ(x) t = y t and L(x) t = φ t .By the uniqueness, we have the following flow property.
We obtain the following estimate of L(x).
Lemma 5.3.Assume conditions (A) and (B) hold.Let x t be a continuous path of finite qvariation (q ≥ 1).Then we have the following estimate.
where C is a positive constant which depends on the constants δ, δ ′ , r 0 in conditions (A) and (B).
Proof.We proved the following estimate in [2,4] following the argument in [34]. where (5.9) By combining this and Lemma 2.5, we complete the proof.
Definition 5.5.We call Y t is a solution of (5.11) if and only if the following holds: (iii) Z satisfies Note that if Y is a solution in the above sense, Z is uniquely determined by Y and X since See also Remark 5.7 (1).By applying Theorem 2.7, we obtain the following result.(5.11).Moreover the following estimate holds, where K, κ i are constants which depend only on σ, β, γ, δ, δ ′ , r 0 .
Proof.By applying Theorem 2.7, we have at least one solution Z and the estimate of (5.12).Let Y t = Z t + L(Z) t and Φ t = L(Z) t .Then this pair is a solution to the original equation.
(2) In [2], we consider the following condition (H1) on D : (i) The condition (A) holds, (ii) There exists a positive constant C such that for any x, it holds that This condition holds if D is convex and there exists a unit vector Under (H1) and σ ∈ C 3 b , we proved the existence of solutions of reflected RDEs driven by 1/β rough paths and gave estimates for the solutions in Theorem 4.5 in [2].We used Euler approximation of the solution modifying Davie's proof in [9].In the proof, we need to solve the following implicit Skorohod equation in each step, where 0 < T ′ < T , ) and Φ t is a continuous bounded variation path.Also η t , x t are finite 1/β-variation paths which are defined by X and X.If we replace t 0 Φ r ⊗ dx r in (5.15) and (5.16) by t 0 f (Φ r ) ⊗ dx r , where f is a bounded Lipschitz map between R n , then we can solve the equation under general condition (A) and (B).To avoid the explosion problem, that is, to handle the linear growth term of Φ t , we put stronger assumption (H1)(ii) on D in [2].Also we used Lyon's continuity theorem of rough integrals in the proof and so we need to assume σ ∈ C 3 b .In this paper, we adopt different approach to the problem and obtain an extension of the previous result in the sense that the assumption on σ and D can be relaxed.
In Section 4, we prove a continuity property of solution mappings at Lipschitz paths under the uniqueness of the solutions.For reflected RDEs, we can give more explicit estimate of the continuity of the solution mapping Y at the Lipschitz paths.As before we consider a domain D ⊂ R n which satisfies the conditions (A) and (B).Let h be a Lipschitz path on R d starting at 0. If σ is Lipschitz continuous, there exists a unique solution (Y (h, ξ) t , Φ(h, ξ) t ) to the reflected ODE in usual sense (see Proposition 4.1 in [4] for example), (5.17) We may omit denoting h, ξ.Moreover, Z(h ) and Φ(h) t are a unique pair of solution to the equation in Theorem 5.6 for the smooth rough path h s,t = (h s,t , h2 s,t ) defined by h.Hence the solution (Z(h), R Z(h) , Φ(h)) satisfies the estimate (5.13) with the same constant C 1 , C 2 .
From now on, we will give an explicit estimate for Y t (ξ, X) − Y t (η, h).Let X be a general (not necessarily geometric) β-Hölder rough path.Let X −h s,t be the translated rough path of X by h.That is, the 1st level path and the second level path are given by, Hence These imply that if |||h − X||| β ≤ 1, then By the definition of controlled paths, we immediately obtain the following.
) with Φ 0 = 0 and q, α, α satisfy the assumptions in Lemma 3.1.By the above lemma, we can define the integral t s σ(Y u )dX −h u and the estimates in Lemma 3.2 hold for this integral.Here Y u = Z u + Φ u .Moreover, Ξ s,t in (3.5) which is defined by where Since | Ξs,t | ≤ C(t − s) 1+ α, the sum of these terms converges to 0. Thus we obtain We now consider the following condition on the boundary.Usually, the function f is assume to be C 2 b in the condition (C).See [27,34].Here, we assume f ∈ Lip γ to make use of estimates in Lemma 3.2.
Under additional condition (C), we can prove the following explicit modulus of continuity.

Perturbed reflected SDEs: a short review
Let us recall basic results for the following equation driven by a continuous path x t on R, (5.32)When x t is a sample path of a standard Brownian motion, the solutions to (5.31) and (5.32) are called (doubly) perturbed Brownian motion and perturbed reflected Brownian motion respectively.
First we consider the equation (5.31).Clearly, if either a ≥ 1 or b ≥ 1, then there are no solutions to this equation for certain x.So we consider the case where a < 1 and b < 1. Suppose b = 0. Then we have explicitly, Y t = x t + a 1−a sup 0≤s≤t x s .By [7], when |  [10].Consider the case where x t is a sample path of 1-dimensional Brownian motion W t with W 0 = 0.For any 0 ≤ a < 1, 0 ≤ b < 1, it is proved in [31] that the pathwise uniqueness holds and the solution is adapted to the Brownian filtration.Finally, for any a < 1, b < 1, the same results is proved in [8].
We consider the equation (5.32).By a fixed point argument, the unique existence is proved in [25] the case (1) a < 1/2 and (2) a < 1 with x 0 > 0. Next, the pathwise uniqueness is proved by [8] for a < 1 when x t is the Brownian path W t with W 0 = 0.The unique existence for a < 1 is extended by [13] for any continuous path x t .
We next explain results for the variable coefficient version driven by a standard 1-dimensional Brownian motion W t , where σ is a Lipschitz continuous function on R and the integral is the Itô integral.The unique existence of the solution to (5.33) is proved for a < 1 by [13].The same authors prove the unique existence of the solution to (5.34) for two cases where (1) a < 1 and ξ > 0 and (2) 0 ≤ a < 1/2 and ξ = 0.Under the same assumption on a, the absolutely continuity of the law of Y t with respect to the Lebesgue measure was studied in [36].

Perturbed reflected rough differential equations
We consider the multidimensional versions of (5.33) and (5.34) driven by rough paths.Our objectives are the following two equations.
where e n = t (0, . . ., 0, 1) and σ ∈ Lip γ−1 (R n , L(R d , R n )).We assume that C is a mapping from C([0, T ], R n ) to the subspace of continuous and bounded variation paths on R n and {C(x) s } 0≤s≤t is measurable with respect to σ({x s } 0≤s≤t ) for all 0 ≤ t ≤ T .The first equation (5.35)   (i) There exists a ) is a solution of (5.36) if the following holds: (i) (Y t , Φ t ) satisfies (5.37) and (5.38).
(ii) There exists a We solve these equations by transforming them to the equations in Theorem 2.7.To this end, we introduce the following conditions.Definition 5.12.For a mapping C : C([0, T ], R n ) → C([0, T ], R m ), we consider the following conditions, where ρ denotes a positive number.
Example 5.17.(1) We consider the following C: where Y j t and C i (Y ) t are the j-th coordinate and i-th coordinate of Y t and C(Y ) t respectively.By Proposition 5.13 and Lemma 5.16, we see that this C satisfies the assumption in Theorem 5.15 for sufficiently small a i j , b i j .In this paper, we do not consider the subtle case as in the previous Subsection, e.g., |ab/(1 − a)(1 − b)| ≤ 1 or a < 1, b < 1, etc.We just mention the following simple result.
Let a i < 1 (1 ≤ i ≤ n) and consider C defined by C i (x) t = a i max 0≤s≤t x i (s), where 1−a max 0≤s≤t x s .Therefore, we have explicitly x n s .
Hence, this example satisfies the assumption in Theorem 5.15. ( and (BV) i ρ i .Hence, if i ρ i < 1, then the assumption in Theorem 5.15 holds.This follows from Proposition 5.13 and Lemma 5.16.
We now consider (5.36) on D = {(x 1 , . . ., x n ) | x n ≥ 0}.For the moment, we suppose C satisfies (Condition C) and ξ is chosen so that the solution η of η = ξ + C 0 (η) satisfies η ∈ D as we noted before.Let Y t be a solution of (5.36) and suppose (5.43)By (5.39), we get an equation for Φ t , where Z n s and Cn is the n-th coordinate of Z s and C respectively.This is a nonlinear implicit Skorohod equation.This kind of equation appeared in the study of the Euler approximation of the solutions for reflected RDEs in [2].
Fix x ∈ C([0, T ], R n ; x 0 = ξ) and consider a mapping on C([0, T ], R; φ 0 = 0): where x n is the n-th coordinate of x.Now suppose that x → Cn (x) is a Lipschitz map belonging to (Lip) κ .Then we have for any φ, φ Hence, if κ < 1, that is, Cn is strict contraction, then M x is a contraction mapping for all x ∈ C([0, T ], R n ; x 0 = ξ).Let us denote the fixed point by L(x).Then we have Φ = L(Z).Thus, under the assumption that Cn : we obtain a mapping x(∈ C([0, T ], R n ; x 0 = ξ)) → L(x) ∈ C([0, T ], R; φ 0 = 0) and the equation for Z: (5.47) We have the following estimate of L.
Let Ã(x) t = C(x + L(x)e n ) t + L(x) t e n .Then the following hold. ( we have (2) We have Thus, we obtain L (3) By using ( 1) and ( 2), we have which implies the desired result.
(4) We have The following lemma follows from Lemma 5.16.
(2) Let (Y, Φ) be a solution to (5.36) and Z be a controlled path appearing in Definition 5.11 (2).Then Z is a solution to (5.47).Moreover, Z is uniquely determined by Y and X.
(3) The transformations defined in (1) and (2) are inverse mapping each other and the uniqueness of the solution of (5.36) and (5.47) is equivalent.(2) We consider the example in Example 5.17 (2).Suppose i ρ i < 1/2.Then the assumption on C in Lemma 5.18 holds.This follows from Proposition 5.13, Lemma 5.16 and Lemma 5.19.
(3) Let a ∈ R and we consider Lipschitz functions f i (1 ≤ i ≤ n) in Example 5.17 x n s .
(5.50) Suppose ξ is chosen so that η ∈ D. For example, if a < 1, f n (η 1 , . . ., η n−1 , 0) ≥ 0 for all η and ∂fn ∂ηn ∞ is sufficiently small, ξ ∈ D is sufficient for η ∈ D. We prove that if a < 1/2 and n i=1 ρ i is sufficiently small, C satisfies the assumption in Lemma 5. 18.Let By this expression, for any ε > 0, if i ρ i is sufficiently small, we have, for any x, x ′ ∈ C([0, T ], R n ), This shows that if a < 1/2 and i ρ i sufficiently small, the assumption of Lemma 5.18 is satisfied.where µ is the 1 dimensional standard normal distribution.This implies that if pH < 1 for sufficiently small δ there exists a subsequence N k ↑ ∞ such that lim N k →∞ I N k = +∞ for almost all W . Proposition 7.1.Let X ∈ C β (R d ) (1/3 < β ≤ 1/2).Let us choose p such that 1/(2β) < p < 1/β.Assume that the first level path X t satisfies C(δ, p, ξ) for any ξ ∈ K and a fixed positive δ, where K is a countable dense subset of S d−1 .Let Y t be a continuous path on R n .Suppose Y t = Z t + Ψ t = Zt + Ψt , where Z, Z ∈ D 2β X (R n ) and Ψ, Ψ ′ are continuous bounded variation paths.Then Z ′ t = Z′ t (0 ≤ t ≤ T ) holds.
Proof.Suppose that there exists s 0 such that Z ′ s 0 − Z′ s 0 = 0. Then there exist 0 < s 1 < s 0 < s 2 < T and unit vectors ξ 1 , ξ 2 ∈ K such that Therefore, for i ∈ P N (X, ξ 2 , δ), Also we note that Thus we obtain (7.9) Since p > 1/(2β), the right hand side of the above inequality is bounded.This contradicts the assumption on X.

ab ( 1
−a)(1−b) | < 1, a fixed point argument works and the unique existence holds for any continuous path x t with x 0 = 0.The unique existence extends to | ab (1−a)(1−b) | = 1 by

( 3 )
The transformations defined in (1) and (2) are inverse mapping each other and the uniqueness of the solution of (5.35) and (5.40) is equivalent.Proof.(1) The existence and the estimate of the solution follows from Theorem 2.7.By Y t = Z t + C(Z) t and by the definition of C, we have C(Z) = C(Y ).Hence Z ′ t = σ(Z t + C(Y ) t ) and Y t is a solution to (5.35).(2) By the definition of C, C(Z) t = C(Y ) t holds.Hence Z is a solution to (5.40).Also the uniqueness follows from the assumption on C. (3) These follows from the assumption on C. We give sufficient conditions on C under which C satisfies (Condition C).Lemma 5.16.Let C be a continuous mapping between C

Proof. ( 1 )
By Lemma 5.18 and Theorem 2.7, there exists a solution Z to (5.47) and has the estimate given in Theorem 2.7.By the definition of L and C, we have L(Z) = L(Z + C(Z + L(Z)e n )) and C(Z + L(Z)) = C(Y ) which shows (5.49).(2) The argument by which we derived the equation (5.47) shows the former half part.Z is uniquely defined by Y and X only because Z t = ξ + t 0 σ(Y s )dX s , Y is a sum of Z and a continuous bounded variation path and Z ′ t = σ(Y t ).(3) The invertibility of the mapping follows from the definition.The latter half statement follows from this property of the mapping.

Example 5 .
21. (1) We consider C in (5.42).If a i j , b i j are sufficiently small, then the assumption on C in Lemma 5.18 holds by Proposition 5.13, Lemma 5.16 and Lemma 5.19.

C
f (x) t = max 0≤s≤t f 1 (x s ), . . ., max 0≤s≤t f n (x s ) , C fn (x) t = max 0≤s≤t f n (x s ).The equation y = x + C(y) is equivalent to y = x + C f (y) + a 1 − a max 0≤s≤t (C fn (y) s + x n s )e n =: Φ x (y)If i ρ i is sufficiently small, then the mapping y → Φ x (y) is a strict contraction mapping for all x.Thus, y = x + C(y) is uniquely solved and C(x) = y − x is defined.Note thatC(x) = C f (x + C(x)) + a 1 − a max0≤s≤t (C fn (x + C(x)) s + x n s )e n .

Remark 5 . 22 (R
Remark on the Itô and Stratonovich SDEs).The equations,(5.35)and(5.36)are formulated by using rough integrals.We now consider the equations replacing the rough integrals by Itô and Stratonovich integrals against the standard Brownian motion W t .The solutions are semimartingales and the equations are well-defined.We need to assume σ is Lipschitz continuous and σ ∈ C 2 b for the Itô and Stratonovich integrals respectively.Under the same assumptions on C in Theorem 5.15 and Theorem 5.20, the existence and the pathwise uniqueness hold for the stochastic integral's version of (5.40) and (5.47) by the Lipschitz continuity of their coefficients which implies the uniqueness of the solutions of the stochastic integral's version of(5.35)  and By Remark D.3.2 in[30], for all 1 ≤ k ≤ d, we have lim|x| p dµ(x) in L 2 ,