Delay equations driven by rough paths

In this article, we illustrate the flexibility of the algebraic integration formalism introduced by M. Gubinelli (2004), by establishing an existence and uniqueness result for delay equations driven by rough paths. We then apply our results to the case where the driving path is a fractional Brownian motion with Hurst parameter H>1/3.


Introduction
In the last years, great efforts have been made to develop a stochastic calculus for fractional Brownian motion.The first results gave a rigorous theory for the stochastic integration with respect to fractional Brownian motion and established a corresponding Itô formula, see e.g.[1,2,3,6,18].Thereafter, stochastic differential equations driven by fractional Brownian motion have been considered.Here different approaches can be used depending on the dimension of the equation and the Hurst parameter of the driving fractional Brownian motion.In the one-dimensional case [17], existence and uniqueness of the solution can be derived by a regularization procedure introduced in [21].The case of a multi-dimensional driving fractional Brownian motion can be treated by means of fractional calculus tools, see e.g.[19,22] or by means of the Young integral [13], when the Hurst coefficient satisfies H > 1  2 .However, only the rough paths theory [13,12] and its application to fractional Brownian motion [5] allow to solve fractional SDEs in any dimension for a Hurst parameter H > 1  4 .The original rough paths theory developed by T. Lyons relies on deeply involved algebraical and analytical tools.Therefore some alternative methods [8,9] have been developed recently, trying to catch the essential results of [12] with less theoretical apparatus.
Since it is based on some rather simple algebraic considerations and an extension of Young's integral, the method given in [9], which we call algebraic integration in the sequel, has been especially attractive to us.Indeed, we think that the basic properties of fractional differential systems can be studied in a natural and nice way using algebraic integration.(See also [16], where this approach is used to study the law of the solution of a fractional SDE.)In the present article, we will illustrate the flexibility of the algebraic integration formalism by studying fractional equations with delay.More specifically, we will consider the following equation: Here the discrete delays satisfy 0 < r 1 < . . .< r k < ∞, the initial condition ξ is a function from [−r k , 0] to R n , the functions σ : R n,k+1 → R n,d , b : R n,k+1 → R n are regular, and B is a d-dimensional fractional Brownian motion with Hurst parameter H > 1 3 .The stochastic integral in equation ( 1) is a generalized Stratonovich integral, which will be explained in detail in Section 2. Actually, in equations like (1), the drift term t 0 b(X s , X s−r 1 , . . ., X s−r k )ds is usually harmless, but causes some cumbersome notations.Thus, for sake of simplicity, we will rather deal in the sequel with delay equations of the type X t = ξ 0 + t 0 σ(X s , X s−r 1 , . . ., X s−r k )dB s , t ∈ [0, T ], Our main result will be as follows: ), and let B be a d-dimensional fractional Brownian motion with Hurst parameter H > 1  3 .Then equation ( 2) admits a unique solution on [0, T ] in the class of controlled processes (see Definition 2.5.)Stochastic delay equations driven by standard Brownian motion have been studied extensively (see e.g.[15] and [14] for an overview) and are used in many applications.However, delay equations driven by fractional Brownian motion have been only considered so far in [7], where the one-dimensional equation is studied for H > 1  2 .Observe that (3) is a particular case of equation (2).To solve equation (2), one requires two main ingredients in the algebraic integration setting.First of all, a natural class of paths, in which the equation can be solved.Here, this will be the paths whose increments are controlled by the increments of B. Namely, writing (δz) st = z t − z s for the increments of an arbitrary function z, a stochastic differential equation driven by B should be solved in the class of paths, whose increments can be decomposed into z t − z s = ζ s (B t − B s ) + ρ st , for 0 ≤ s < t ≤ T, with ζ belonging to C γ 1 and ρ belonging to C 2γ 2 , for a given γ ∈ ( 1 3 , H). (Here, C µ i denotes a space of µ-Hölder continuous functions of i variables, see Section 2.) This class of functions will be called the class of controlled paths in the sequel.
To solve fractional differential equations without delay, the second main tool would be to define the integral of a controlled path with respect to fractional Brownian motion and to show that the resulting process is still a controlled path.To define the integral of a controlled path, a double iterated integral of fractional Brownian motion, called the Lévy area, will be required.Once the stability of the class of controlled paths under integration is established, the differential equation is solved by an appropriate fixed point argument.
To solve fractional delay equations, we will have to modify this procedure.More specifically, we need a second class of paths, the class of delayed controlled paths, whose increments can be written as where, as above, ζ (i) belongs to C γ 1 for i = 0, . . ., k, and ρ belongs to C 2γ 2 for a given 1 3 < γ < H. (Note that a classical controlled path is a delayed controlled path with ζ (i) = 0 for i = 1, . . ., k.)For such a delayed controlled path we will then define its integral with respect to fractional Brownian motion.We emphasize the fact that the integral of a delayed controlled path is actually a classical controlled path and satisfies a stability property.
To define this integral we have to introduce a delayed Lévy area B 2 (v) of B for v ∈ [−r k , 0].This process, with values in the space of matrices R d,d will also be defined as an iterated integral: for 1 ≤ i, j ≤ d and 0 ≤ s < t ≤ T , we set where the integral on the right hand side is a Russo-Vallois integral [21].Finally, the fractional delay equation ( 2) will be solved by a fixed point argument.
This article is structured as follows: Throughout the remainder of this article, we consider the general delay equation where x is γ-Hölder continuous function with γ > 1 3 and ξ is a 2γ-Hölder continuous function.In Section 2 we recall some basic facts of the algebraic integration and in particular the definition of a classical controlled path, while in Section 3 we introduce the class of delayed controlled paths and the integral of a delayed controlled path with respect to its controlling rough path.Using the stability of the integral, we show the existence of a unique solution of equation ( 4) in the class of classical controlled paths under the assumption of the existence of a delayed Lévy area.Finally, in Section 4 we specialize our results to delay equations driven by a fractional Brownian motion with Hurst parameter H > 1 3 .

Algebraic integration and rough paths equations
Before we consider equation (4), we recall the strategy introduced in [9] in order to solve an equation without delay, i.e., where x is a R d -valued γ-Hölder continuous function with γ > 1 3 .

2.1.
Increments.Here we present the basic algebraic structures, which will allow us to define a pathwise integral with respect to irregular functions.For real numbers 0 ≤ a ≤ b ≤ T < ∞, a vector space V and an integer k ≥ 1 we denote by Such a function will be called a (k − 1)-increment, and we will set ).An important operator for our purposes is given by where ti means that this argument is omitted.A fundamental property of δ is that δδ = 0, where δδ is considered as an operator from Some simple examples of actions of δ are as follows: In particular, the following property holds: Observe that Lemma 2.1 implies in particular that all elements h ∈ C 2 ([a, b]; V ) with δh = 0 can be written as h = δf for some f ∈ C 1 ([a, b]; V ).Thus we have a heuristic interpretation of δ| C 2 ([a,b];V ) : it measures how much a given 1-increment differs from being an exact increment of a function, i.e., a finite difference.
Our further discussion will mainly rely on k-increments with k ≤ 2. For simplicity of the exposition, we will assume that V = R d in what follows, although V could be in fact any Banach space.We measure the size of the increments by Hölder norms, which are defined in the following way: For h ∈ C 3 ([a, b]; V ) we define in the same way Then • µ is a norm on C 3 ([a, b]; V ), see [9], and we define The crucial point in this algebraic approach to the integration of irregular paths is that the operator δ can be inverted under mild smoothness assumptions.This inverse is called Λ.The proof of the following proposition may be found in [9], and in a simpler form in [10].

In other words, for any
This mapping Λ allows to construct a generalised Young integral: for a ≤ s < t ≤ b, where the limit is taken over any partition Π st = {t 0 = s, . . ., t n = t} of [s, t], whose mesh tends to zero.Thus, the 1-increment δf is the indefinite integral of the 1-increment g.
We also need some product rules for the operator δ.Before we consider the technical details, we will make some heuristic considerations about the properties that the solution of equation ( 5) should enjoy.Set σt = σ (y t ), and suppose that y is a solution of (5), which satisfies y ∈ C κ 1 for a given 1 3 < κ < γ.Then the integral form of our equation can be written as Our approach to generalised integrals induces us to work with increments of the form (δy) st = y t − y s instead of (10).It is immediate that one can decompose the increments of ( 10) into We thus have obtained a decomposition of y of the form δy = σδx + ρ.Let us see, still at a heuristic level, which regularity we can expect for σ and ρ: If σ is bounded and continuously differentiable, we have that σ is bounded and where y κ denotes the κ-Hölder norm of y.Hence σ belongs to C κ 1 and is bounded.As far as ρ is concerned, it should inherit both the regularities of δσ and x, provided that the integral t s (σ u − σs )dx u = t s (δσ) su dx u is well defined.Thus, one should expect that ρ ∈ C 2κ 2 .In summary, we have found that a solution δy of equation (10) should be decomposable into This is precisely the structure we will demand for a possible solution of equation ( 5) respectively its integral form (10): Note that in the above definition α corresponds to a given initial condition and ρ can be understood as a regular part.Moreover, observe that a can be negative.Now we can sketch the strategy used in [9], in order to solve equation ( 5): (a) Verify the stability of (b) Define rigorously the integral z u dx u = J (zdx) for a classical controlled path z and compute its decomposition (12).(c) Solve equation ( 5) in the space Q κ,α ([a, b]; R n ) by a fixed point argument.Actually, for the second point we had to impose a priori the following hypothesis on the driving rough path, which is a standard assumption in the rough paths theory: Hypothesis 2.6.The R d -valued γ-Hölder path x admits a Lévy area, i.e. a process Then, using the strategy sketched above, the following result is obtained in [9]: Theorem 2.7.Let x be a process satisfying Hypothesis 2.6 and let σ ∈ C 2 (R n ; R n,d ) be bounded together with its derivatives.Then we have: (1) Equation ( 5) admits a unique solution

The delay equation
In this section, we make a first step towards the solution of the delay equation where x is a R d -valued γ-Hölder continuous function with γ > 1 3 , the function σ ∈ C 3 (R n,k+1 ; R n,d ) is bounded together with its derivatives, ξ is a R n -valued 2γ-Hölder continuous function, and 0 < r 1 < . . .< r k < ∞.For convenience, we set r 0 = 0 and, moreover, we will use the notation 3.1.Delayed controlled paths.As in the previous section, we will first make some heuristic considerations about the properties of a solution: set σt = σ(y t , s(y) t ) and suppose that y is a solution of ( 13) with y ∈ C κ 1 for a given 1 3 < κ < γ.Then we can write the integral form of our equation as Thus, we have again obtained a decomposition of y of the form δy = σδx + ρ.Moreover, it follows (still at a heuristic level) that σ is bounded and satisfies Thus, with the notation of Section 2.1, we have that σ belongs to C γ 1 and is bounded.The term ρ should again inherit both the regularities of δσ and x.Thus, one should have that ρ ∈ C 2κ 2 .In conclusion, the increment δy should be decomposable into This is again the structure we will ask for a possible solution to (13).However, this decomposition does not take into account that equation ( 13) is actually a delay equation.
To define the integral t s σu dx u , we have to enlarge the class of functions we will work with, and hence we will define a delayed controlled path (hereafter DCP in short).
Now we can sketch our strategy to solve the delay equation: (1) Consider the map where we recall that the notation s(z) has been introduced at (14).We will show that and compute its decomposition (12).Let us point out the following important fact: ) By combining the first two points, we will solve equation ( 13) by a fixed point argument on the intervals [0, r 1 ], [r 1 , 2r 1 ], . . . .

3.2.
Action of the map T on controlled paths.The major part of this section will be devoted to the following two stability results: ) and it admits a decomposition of the form where ζ, ζ(i) are the R l,d -valued paths defined by and for i = 1, . . ., k.Moreover, the following estimate holds: where the constant c ϕ,T depends only ϕ and T .
Proof.Fix s, t ∈ [a, b] and set . where For the second remainder term Taylor's formula yields and hence clearly, thanks to some straightforward bounds in the spaces Q, we have The first term can also be bounded easily: it can be checked that Putting together the last two inequalities, we have shown that decomposition (18) holds, that is (ii) Now we have to consider the "density" functions for i = 1, . . ., k.Moreover, for i = 1, . . ., k, we have Hence, the densities satisfy the conditions of Definition 3.1.
We thus have proved that the map T ϕ is quadratically bounded in z and z.Moreover, for fixed z the map ≤ c ϕ,T 1 + C(z (1) , z (2) , z) where C(z (1) , z (2) , and the constant c ϕ,T depends only on ϕ and T .
In the following we will denote constants (which depend only on T and ϕ) by c, regardless of their value.For convenience, we will also use the short notations N [z], N [z (1) ], N [z (2) ] and N [z (1) − z (2) ] instead of the corresponding quantities in (25)-( 26).

Integration of delayed controlled paths (DCP)
. The aim of this section is to define the integral J (m * dx), where m is a delayed controlled path m ∈ D κ,α ([a, b]; R d ).
Here we denote by A * the transposition of a vector or matrix A and by A 1 • A 2 the inner product of two vectors or two matrices A 1 and A 2 .We will also write Note that if the increments of m can be expressed like in (16), m * admits the decomposition where ) and the densities ζ (i) , i = 0, . . ., k satisfy the conditions of Definition 3.1.
To illustrate the structure of the integral of a DCP, we first assume that the paths x, ζ (i) and ρ are smooth, and we express J (m * dx) in terms of the operators δ and Λ.In this case, J (m * dx) is well defined, and we have Now consider the term J (δm * dx): Using the decomposition (41) we obtain with Since, for the moment, we are dealing with smooth paths, the density ζ (i) can be taken out of the integral above, and we have with the d × d matrix x 2 st (v) defined by Inserting the expression of A st into (42) and ( 43) we obtain Let us now consider the Lévy area term for any i = 0, . . ., k.This decomposition of δx 2 (−r i ) into a product of increments is the fundamental algebraic property we will use to extend the above integral to non-smooth paths.Hence, we will need the following assumption: and admits a delayed Lévy area, i.e., for all v ∈ {−r k , . . ., −r 0 }, there exists a path that is ] ut for all s, u, t ∈ [0, T ], i, j ∈ {1, . . ., d}.
In the above formulae, we have set x v for the shifted path x v s = x s+v .
To finish the analysis of the smooth case it remains to find a suitable expression for J (ρ * dx).For this, we write (44) as and we apply δ to both sides of the above equation.For smooth paths m and x we have by Proposition 2.4.Hence, applying these relations to the right hand side of (46), using the decomposition (41) and again Proposition 2.4, we obtain In summary, we have derived the representation for two regular paths m and x.
If m, x, ζ (i) , i = 0, . . ., k and x 2 are smooth enough, we have δ[J (ρ * dx)] ∈ ZC 1+ 3 and thus belongs to the domain of Λ due to Proposition 2.2.(Recall that δδ = 0.) Hence, it follows and inserting this identity into (44), we end up with The expression above can be generalised to the non-smooth case, since J (m * dx) has been expressed only in terms of increments of m and x.Consequently, we will use (47) as the definition for our extended integral.Proposition 3.5.For fixed 1  3 < κ < γ, let x be a path satisfying Hypothesis 3.4.Furthermore, let m ∈ D κ, α([a, b]; R d ) such that the increments of m are given by (16).Define z by z a = α with α ∈ R and (49) Then: (1) J (m * dx) coincides with the usual Riemann integral, whenever m and x are smooth functions.
(2) z is well-defined as an element of (3) The semi-norm of z can be estimated as where with the constant c κ,γ,ϕ,T depending only on κ, γ, ϕ and T .Moreover, for any a ≤ s < t ≤ b, where the limit is taken over all partitions Π st = {s = t 0 , . . ., t N = t} of [s, t], as the mesh of the partition goes to zero.
Proof.(1) The first of our claims is a direct consequence of the derivation of equation ( 47). ( . Now we show that equation (48) defines a classical controlled path.Actually, the term m * δx is trivially of the desired form for an element of Q κ,α .So consider the term h satisfies δh (2) = 0. Indeed, we can write by Proposition 2.4 and because δδ = 0. Applying (45) to the right hand side of the above equation it follows that However, due to Proposition 2.4, it holds Since the increments of m are given by ( 16) we finally obtain that δh (2) = δ(δm * )δx = 0.Moreover, recalling the notation (7), it holds Since γ > κ > 1 3 and δh (2) = 0, we have h (2) ∈ Dom(Λ) and 3κ and we finally obtain (4) By Proposition 2.4 (ii) and the decomposition (16) we have that Thus, applying again Proposition 2.4 (ii), and recalling Hypothesis 3.4 for the Lévy area, we obtain that Hence, equation (48) can also be written as and a direct application of Corollary 2.3 yields (52), which ends our proof.
Recall that the notation A * stands for the transpose of a matrix A. Moreover, in the sequel, we will denote by c norm a constant, which depends only on the chosen norm of R n,d .Then, for a matrix-valued delayed controlled path m ∈ D κ, α([a, b]; R n,d ), the integral J (m dx) will be defined by where . ., n and we have set m = (m (1) , . . ., m (n) ) * .Then we have by (50) that For two paths m (1) , m (2) ) we obtain the following estimate for the difference of z (1) = J (m (1) dx) and z (2) = J (m (2) dx): As above, we have clearly However, since m (55)

Solution to the delay equation
With the preparations of the last section, we can now solve the equation in the class of classical controlled paths.For this, it will be crucial to use mappings of the type for 0 ≤ a ≤ b ≤ T , which are defined by (z, z) → ẑ, where ẑ0 = α and δẑ given by δẑ = J (T σ (z, z)), with T σ defined in Proposition 3.2.¿From now on, we will use the convention that z t = zt = ẑt = ξ t for t ∈ [−r, 0].Note that this convention is consistent with the definition of a classical controlled path, see Definition 2.5: since ξ is 2γ-Hölder continuous, it can be considered as a part of the remainder term ρ.
The first part of the current section will be devoted to the study of the map T .By (54) we have that d ) for i = 1, . . ., n and σ = (σ (1) , . . ., σ (n) ) * , it follows by ( 19) that Combining these two estimates we obtain where the constant c growth depends only on c int , c norm , σ, κ, γ and T .Thus the semi-norm of the mapping Γ is quadratically bounded in terms of the semi-norm of z and z.
Then, by (55) we have Applying Proposition 3.3, i.e. inequality (25), to the right hand side of the above equation we obtain that ≤ c lip 1 + C(z (1) , z (2) , z) with a constant c lip depending only on c int , c norm , σ, κ, γ and T , and moreover C(z (1) , z (2) , Thus, for fixed z the mappings Γ(•, z) are locally Lipschitz continuous with respect to the semi-norm We also need the following Lemma, which can be shown by straightforward calculations: ).Then we have: (1) Equation ( 56) admits a unique solution y in Q κ,ξ 0 ([0, T ]; R n ) for any 1  3 < κ < γ and any T > 0. (2) where y is the unique solution of equation ( 56).This mapping is locally Lipschitz continuous in the following sense: Let x be another driving rough path with corresponding delayed Lévy area x2 (−v), v ∈ {−r k , . . ., −r 0 }, and ξ another initial condition.Moreover denote by ỹ the unique solution of the corresponding delay equation.Then, for every N > 0, there exists a constant K N > 0 such that holds for all tuples (ξ, x, x 2 , x 2 (−r 1 ), . . ., Proof.The proof of Theorem 4.2 is obtained by means of a fixed point argument, based on the map Γ defined above.
Since lr 1 + τ l+1 ≤ (l + 1)r 1 , we still have the estimate Now the existence of a unique solution z (2,l+1) of (56) on the interval I 2,l+1 follows again by the same fixed point argument.
Thus we have shown that there exists a unique path z ∈ Q κ,ξ 0 ([0, T ]; R n ), which is a solution of the equation (56).Moreover, by the above construction we obtain the following bound on the norm of this path: where f : [0, ∞) → (0, ∞) is a continuous non-decreasing function, which depends only on κ, γ, n, d, σ, T and r 1 , . . ., r k .
(i) We first analyse the difference between ρ and ρ.Here we have st + Λ st (e (2) ), define C(ỹ) accordingly for ỹ, and let R be the quantity In the following we will denote constants, which depend only on κ, γ, n, d, σ and T , by c regardless of their value.
for s, t ∈ [a, b].Now, consider the term e (2) .We have e Furthermore, we also have, for any i = 0, . . ., k that Recall that the Hölder norm of a path f is defined by , where c is again an arbitrary constant depending only on κ, γ, n, d, σ and T .Using these notations and combining the previous estimates, we end up with: Hence e (2) belongs to Dom(Λ) and we obtain by Proposition 2.2 that Λ(e (2) Inserting the estimates for e (1) and Λ(e (2) ), i.e. ( 69) and ( 71), into the definition (68) of ρ − ρ gives finally , and due to the subadditivity of the Hölder norms, we get where and D is again defined accordingly.Thus, we obtain now from (76) that there exists a continuous function ḡ : [0, ∞) → [0, ∞), which depends only on κ, γ, σ, n, d, T and r 1 , . . ., r k , such that ∆(0, T ) ≤ ḡ(D + D) R. Hence, the assertion follows.
We deduce  Thus, the delayed Lévy area B 2 0,t−s (v)(i, j) = t−s 0 (B j u+v −B j v )d • B i u for v < 0 behaves as in the case where v = 0.But it is a classical result that B 2 0,t−s (0) is well-defined for H > 1/3 (see, e.g., [20]).Moreover, it follows again by the stationarity (79) and the scaling (78) properties that Both in the cases i = j and i = j, the substitution formula for Russo-Vallois integrals easily yields that δB 2 (v) = δB v ⊗ δB.Furthermore, since B 2 (v) is a process belonging to the second chaos of the fractional Brownian motion B, on which all L p norms are equivalent for p > 1, we get that E|B when i = j.In order to conclude that B 2 (v) ∈ C 2γ 2 (R d×d ) for any 1 3 < γ < H and v ∈ [−r, 0), let us recall the following inequality from [9]: let g ∈ C 2 (V ) for a given Banach space V ; then, for any κ > 0 and p ≥ 1 we have g κ ≤ c U κ+2/p;p (g) + δg κ with U γ;p (g) =  89) into (90), by recalling that δB 2 (v) = δB v ⊗ δB and (83) hold, we obtain that B 2 (v)(i, j) ∈ C 2γ 2 (R d×d ) for any 1 3 < γ < H and i, j = 1, . . ., d.
the usual Hölder spaces C µ 1 ([a, b]; V ) are determined in the following way: for a continuous function g ∈ C 1 ([a, b]; V ) set g µ = δg µ , and we will say that g ∈ C µ 1 ([a, b]; V ) iff g µ is finite.Note that • µ is only a semi-norm on C 1 ([a, b]; V ), but we will work in general on spaces of the type , b]; V ) and note that the same kind of norms can be considered on the spaces ZC 3 ([a; b]; V ), leading to the definition of the spaces ZC µ 3 ([a; b]; V ) and ZC 1+ 3 ([a, b]; V ).
st (f dg) for the integral of a function f with respect to a given function g on the interval [s, t].Moreover, we also set f ∞ = sup x∈R d,l |f (x)| for a function f : R d,l → R m,n .To simplify the notation we will write C γ k instead of C γ k ([a, b]; V ), if [a, b] and V are obvious from the context.