Valérie Le Blanc
email: leblanc@math.univ-lyon1.fr.

Persistence of generalized roll-waves under viscous perturbation

Keywords: vanishing viscosity, roll-waves, Green's function

The purpose of this article is to study the persistence of solution of a hyperbolic system under small viscous perturbation. Here, the solution of the hyperbolic system is supposed to be periodic: it is a periodic perturbation of a roll-wave. So, it has an infinity of shocks. The proof of the persistence is based on an expansion of the viscous solution and estimates on Green's functions.

Introduction

In this paper, we consider a one-dimensional system

u ε t + f (u ε ) x = g(u ε ) + εu ε xx (1)
with a smooth flux f : R n → R n and a smooth function g : R n → R n . We assume that the corresponding system without viscosity

u t + f (u) x = g(u) (2)
is strictly hyperbolic. We consider a piecewise smooth function u which is a distributional solution of (2) on the domain R × [0; T * ]. We assume that u is periodic in the x variable, with a period L and that u has m noninteracting Lax shocks per period.

We show that u is a strong limit of solutions u ε of (1) as ε → 0. This work is of course motivated by the conjecture that the admissible solutions of (2) are strong limits of solutions of [START_REF] Diperna | Convergence of the viscosity method for isentropic gas dynamics[END_REF] with the same initial data.

In the case of scalar conservation laws, the proof of this conjecture uses the maximum principle [START_REF] Volpert | Spaces BV and quasilinear equations[END_REF], and in the case of special 2×2 systems, R. J. DiPerna proved it by a compensated compactness argument [START_REF] Diperna | Convergence of the viscosity method for isentropic gas dynamics[END_REF]. For the general case of shocks, there is a first paper of J. Goodman and Z. P. Xin which proves this conjecture for small amplitude Lax shocks [START_REF] Goodman | Viscous limits for piecewise smooth solutions to systems of conservation laws[END_REF]. This conjecture is also proved for a single noncharacteristic Lax shock or overcompressive shock by F. Rousset [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF]. Here, we only consider Lax shocks but we have an infinity of shocks.

An other motivation of this work states in the study of roll-waves, in fluid mechanics or in general hyperbolic systems with source terms. Indeed, P. Noble proved the existence of roll-waves for this kind of system under assumptions on the source term [START_REF] Noble | Roll-waves in general hyperbolic systems with source terms[END_REF]. Specifically, in the case of inviscid Saint Venant equations

   h t + (hu) x = 0, (hu) t + (g cos θ h 2 2 + hu 2 ) x = gh sin θ -c f u 2 ,
one can prove that there exist roll-waves which are persistent under small perturbation [START_REF] Noble | Persistence of roll waves for the Saint Venant equations[END_REF]. So, there exist solutions of inviscid Saint Venant equations, near roll-waves.

Here, the idea is to prove that there exists a family of solutions of the viscous Saint Venant system

   h t + (hu) x = 0, (hu) t + (g cos θ h 2 2 + hu 2 ) x = gh sin θ -c f u 2 + ε(hu x ) x , (3)
which tends to a solution of inviscid system as ε goes to 0. We prove this result in the case of full viscosity.

We can now give the full set of assumptions and formulate our main result. First, we suppose that (H1) system ( 2) is strictly hyperbolic.

That means that there exist smooth matrices P (u), D(u) such that df (u) = P (u)D(u)P (u) -1 where D(u) = diag(λ 1 (u), . . . , λ n (u)) is a diagonal matrix and λ i = λ j for all i = j.

(H2) u is a distributional solution of (2) on [0; T * ]. Moreover, we suppose that u is piecewise smooth, periodic, and has m noninteracting and non-characteristic Lax shocks per period. That means that u is smooth except at the points (x, t) of smooth curves x = X j (t) + iL, j = 1, . . . , m, i ∈ Z and that for all j, k, t, |X j (t) -X k (t)| > 2r > 0 (see Figure 1). Moreover, following limits are finite:

x u ct L m + ct 2L m + ct 3L m + ct 4L m + ct X 1 (t) X 2 (t) X 3 (t) X 4 (t) X 1 (t) + L
∂ k x u j± (t) := ∂ k x u(X j (t) ± 0, t) = lim x→X j (t) ± ∂ k x u(x, t).

Since the shocks are non-characteristic k-Lax shocks, we have:

λ 1 (u j-) ≤ • • • ≤ λ k-1 (u j-) < X ′ j (t) < λ k (u j-) ≤ • • • ≤ λ n (u j-), λ 1 (u j+ ) ≤ • • • ≤ λ k (u j+ ) < X ′ j (t) < λ k+1 (u j+ ) ≤ • • • ≤ λ n (u j+
). This assumption ensures the existence of at least one sonic point between two shocks.

We refer to [START_REF] Noble | Persistence of roll waves for the Saint Venant equations[END_REF] for the existence of such a solution in the case of Saint Venant equations. This result can be extended to general hyperbolic systems.

These assumptions imply that there exists a viscous profile for each shock. More precisely, for all j, there exists V j such that We assume that the viscous shock profiles are linearly stable. This assumption is equivalent to an Evans function criterion [START_REF] Zumbrun | Pointwise semigroup methods and stability of viscous shock waves[END_REF].

(H3) ∀τ ∈ [0; T * ], j = 1, . . . , m, L j τ is such that D j τ (λ) = 0 ∀λ, ℜλ ≥ 0, λ = 0, and D j τ ′ (0) = 0, where D j τ is the Evans function of L j τ .

We can now state our main theorem:

Theorem 1. Under assumptions (H1)-(H2)-(H3), for all ε > 0, there exists a unique solution u ε of (1) on [0; T * ] such that

u ε (t = 0, x) = u(t = 0, x). ( 5
)
Moreover, we have the convergences

u ε -u L ∞ ([0;T * ],L 1 (0;L)) → 0, as ε → 0.
And for any η ∈ (0, 1),

sup 0≤t≤T * ,|x-X j (t)|≥ε η |u ε (x, t) -u(x, t)| → 0, as ε → 0.
The proof of this theorem is done in three steps: construction of an approximate solution (which gives us an expansion of u ε in ε), estimates on the semigroup generated by linearized operator around this approximate solution and a Banach fixed point argument to deal with the full problem.

The paper is organized as follows. In Section 2, we build an approximate solution u ε app of the full problem (1) close to u, solution of (2) up to order 2 with respect to ε. This is done separating slow parts where u ε app is close to u and shock parts where u ε app | X j ±ε γ is close to V j . More precisely, one expands u ε in the slow part as

u ε (x, t) = u(x, t) + εu 1 (x, t) + ε 2 u 2 (x, t) + o(ε 2 )
where u is the solution of (2) and u i are solutions of the linearized equation of (2) around u, which is well-posed thanks to assumption (H3). In shock parts, the expansion at shock j is

u ε (x, t) = V j (ξ j (x, t, ε), t) + εV j 1 (ξ j (x, t, ε), t) + ε 2 V j 2 (ξ j (x, t, ε), t) + o(ε 2 )
where the stretched variable is ξ j (x, t, ε) =

x-X j (t) ε + δ j (t), V j is solution of viscous equation ( 4) and V j i , i = 1, 2, are solutions of the linearized equation of (4) around V j . Moreover, the functions u i , V j i are related by matching conditions which ensure regularity on the approximate solution u ε app , built by convex combination of the expansions:

u ε app = m j=1 µ x -X j (t) ε γ I jε (x, t) + 1 - m j=1 µ x -X j (t) ε γ O ε (x, t).
where

µ(x) = 0 if |x| > 2, 1 if |x| < 1, I jε (x, t) = V j (ξ j (x, t, ε), t) + εV j 1 (ξ j (x, t, ε), t) + ε 2 V j 2 (ξ j (x, t, ε), t), O ε (x, t) = u(x, t) + εu 1 (x, t) + ε 2 u 2 (x, t).

With this construction, we prove the theorem:

Theorem 2. There exists an approximate solution u ε app of (1) defined on [0; T * ]. If ϕ is a smooth change of variable which fixes the shocks (∀t, i, j, ϕ((j -1) L m + iL, t) = X j (t) + iL), and ũε app (z, t) = u ε app (ϕ(z, t), t), then ũε app verifies the equation

(ũ ε app ) t + f (ũ ε app ) x -ε(ũ ε app ) xx -g(ũ ε app ) = qε (x, t)
with the following estimates on qε

qε L ∞ , qε t L ∞ , qε tt L ∞ ≤ Cε 2γ , (6) qε L 1 (0;L) , qε t L 1 (0;L) , qε tt L 1 (0;L) ≤ Cε 3γ , (7) qε z L ∞ , qε zt L ∞ ≤ Cε γ , qε z L 1 (0;L) , qε zt L 1 (0;L) ≤ Cε 2γ , (8) qε zz L ∞ ≤ C, qε zz L 1 (0;L) ≤ Cε γ . (9)
Here, u ε app is constructed as a perturbation of u going to order 2, which allows us to have estimates on qε zz in L 1 in ε γ . This property will be useful to prove the convergence of u ε appu ε to 0. In Section 3, we linearize (1) in the neighbourhood of the approximate solution u ε app and we compute estimates on the Green's function. To do so, we use the method of iterative construction of the Green's function, which was first used by E. Grenier and F. Rousset in [START_REF] Grenier | Stability of one-dimensional boundary layers by using Green's functions[END_REF]. So, we consider approximations of the Green's functions in neighbourhood of the shocks (given by K. Zumbrun and P. Howard in [START_REF] Zumbrun | Pointwise semigroup methods and stability of viscous shock waves[END_REF]) and we build our own approximation far away from the shock, using the characteristic curves.

The last section is dedicated to the proof of theorem Theorem 3. Under assumptions (H1)-(H2)-(H3), for all ε, there exists u ε solution of (1)-( 5) on (0; T * ). And this u ε verifies the convergences:

u ε -u ε app L ∞ ((0;T * )×R) → 0, u ε -u ε app L ∞ ((0;T * ),L 1 (R)) → 0 when ε goes to zero.
This is done using standard arguments for parabolic problems. Indeed, we combine estimates on qε , and estimate on the Green's function to obtain estimates on u ε -u ε app , and its derivatives, depending on ε and uniform in time for ε small enough. Then, using the convergence of u ε app to u, we immediately deduce Theorem 1.

Construction of the approximate solution

The purpose of this section is to prove Theorem 2 on the existence of the approximate solution u ε app of (1). In a first step, we compute formally this approximate solution using outer and inner expansions of order 2. Indeed, in slow part, where ∇u is bounded, the solution u ε of (1) may be approximated by truncation of the formal series

u ε (x, t) ∼ O ε (x, t) = u(x, t) + εu 1 (x, t) + ε 2 u 2 (x, t)
where u is the solution of (2) we want to approach. Similarly, near the shocks j, we search for u ε app with the inner expansion

I jε (x, t) = V j (ξ j (x, t, ε), t) + εV j 1 (ξ j (x, t, ε), t) + ε 2 V j 2 (ξ j (x, t, ε), t)
where ξ j (x, t, ε) =

x-X j (t) ε + δ j 0 (t) + εδ j 1 (t) is the stretched variable and V j is the viscous shock profile, solution of (4). We match this expansion by continuity of u ε app and its spatial derivatives.

In this section, we formally substitute these expansions in (1) to find equations satisfied by u i and V j i , i = 1, 2, j = 1, . . . , m, and matching conditions. Then, we prove the existence of the u i and V j i on (0; T * ). Furthermore, we give rigorous estimates on the error terms. We can remark here that we search for an approximation of order 2, this will be useful to obtain good estimates on the second derivatives of the error term.

Formal calculation and derivation of the equations

Substituting O ε into (1) and identifying the power of ε in the expressions, we get for x = X j (t):

O(ε 0 ) : u t + (f (u)) x -g(u) = 0, O(ε 1 ) : u 1,t + (df (u) • u 1 ) x -dg(u) • u 1 = u xx , O(ε 2 ) : u 2,t + (df (u) • u 2 ) x -dg(u) • u 2 = u 1xx - 1 2 (d 2 f (u) • (u 1 , u 1 )) x + 1 2 d 2 g(u) • (u 1 , u 1 ).
A similar calculation for I jε yields the set of equations:

O(ε -1 ) : V j ξξ -(f (V j ) -X ′ j V j ) ξ = 0, O(ε 0 ) : V j 1ξξ -((df (V j ) -X ′ j ) • V j 1 ) ξ = V j t + V j ξ δ j 0t -g(V j ), O(ε 1 ) : V j 2ξξ -((df (V j ) -X ′ j )•V j 2 ) ξ = V j 1t + V j 1ξ δ j 0t + V j ξ δ j 1t + 1 2 (d 2 f (V j )•(V j 1 , V j 1 )) ξ -dg(V j ) • V j 1 .
We remark that the equations for u i are hyperbolic equations: the first one is (2), so nonlinear, and the others are the linearization of (2) around u. Similarly, the equations for the shock profiles are ordinary equations: nonlinear for V j , we recognize (4), and its linearization around V j for V j 1 and V j 2 . To maximize the order of the approximation, we couple these equations with boundary conditions, connecting u i and V j i . First, we note

∂ k x u j± i (t) := ∂ k x u i (X j (t) ± 0, t) = lim x→X j (t) ± ∂ k x u i (x, t), i = 1, 2.
Then, we rewrite O ε and I ε with the variable ξ, in a vicinity of shock j, and we ask the two functions to coincide as ε goes to 0. Therefore, we make Taylor expansion of order 2 with respect to ε. For example, for ξ > 0, large enough,

O ε (X j (t) + ε(ξ -δ j 0 (t) -εδ j 1 (t)), t) = u j+ (t) + ε u j+ 1 (t) + u j+ x (ξ -δ j 0 ) + ε 2 2 u j+ 2 (t) + 2u j+ 1x (t)(ξ -δ j 0 ) -2u j+ x δ j 1 + u j+ xx (t)(ξ -δ j 0 ) 2 + o(ε 2
) and

I ε (X j (t) + ε(ξ -δ j 0 (t) -εδ j 1 (t)), t) = V j (ξ, t) + εV j 1 (ξ, t) + ε 2 V j 2 (ξ, t) + o(ε 2 )
Identifying the terms of same order on ε, we get as ξ → ±∞:

V j (±∞, t) = u j± (t), (10)
V j 1 (ξ, t) = u j± 1 (t) + u j± x (t)(ξ -δ j 0 (t)) + o(1), (11)
V j 2 (ξ, t) = u j± 2 (t) + u j± 1x (t)(ξ -δ j 0 (t)) + 1 2 u j± xx (t)(ξ -δ j 0 (t)) 2 -u j± x (t)δ j 1 (t) + o(1). ( 12
)
For more details on the computation of these conditions, we refer to [START_REF] Paul | Dynamics of internal layers and diffusive interfaces[END_REF].

Existence of solutions of the outer and inner problems

In this section, we show that the solutions u i and V j i of the previous equations exist under assumption (H3) on the spectral stability of the viscous shock profile. We first remark that the leading-order outer function u is exactly the solution of (2) which we want to approximate. Therefore, we first prove the existence of the V j , and then we prove the existence of u 1 and all the V j 1 . Simultaneously, we prove the existence of the δ j 0 . Similarly, we prove the existence of u 2 , V j 2 and δ j 1 .

Construction at order 0

In this section, we deal with the existence of u, V j which satisfy equations ( 2)-( 4) and matching condition [START_REF] Volpert | Spaces BV and quasilinear equations[END_REF]. The existence of u is exactly assumption (H2). Since u is a distributional solution, u verifies Rankine-Hugoniot conditions at each shock j:

f (u j+ ) -f (u j-) = X ′ j (t)(u j+ -u j-
). This property ensures the existence of the viscous shock profile V j which verifies

V j ξξ -(f (V j ) -X ′ j (t)V j ) ξ = 0 (13)
and the asymptotic conditions:

V j (±∞, t) = u j± (t).
For the existence for all t of such a profile and its properties, we refer to N. Kopell and L. N. Howard [START_REF] Kopell | Bifurcations and trajectories joining critical points[END_REF]. In this section, we just recall the convergence rate of the profile and its derivatives as ξ → ±∞. Since u j+ and u j-are hyperbolic rest points for the ordinary differential equation ( 13), we have for some ω > 0 and for any α ∈ N,

|∂ α t V j (ξ, t) -∂ α t u j± (t)| ≤ e -ω|ξ| , ∀ξ ∈ R, (14) |∂ α ξ V j (ξ, t)| ≤ e -ω|ξ| , ∀ξ ∈ R. (15)
2.2.2 Construction at order 1: existence of V j 1 and u 1 In this section, we prove the existence of u 1 , V j 1 , δ j 0 on (0; T * ) such that

u 1,t + (df (u) • u 1 ) x -dg(u) • u 1 = u xx , (16) V j 1ξξ -((df (V j ) -X ′ j ) • V j 1 ) ξ = V j t + V j ξ δ j 0t -g(V j ), (17)
V j 1 (ξ, t) = u j± 1 (t) + u j± x (t)(ξ -δ j 0 (t)) + o(1), ξ → ±∞. (18)
We first remark that these equations are linear. As in [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF], it is convenient to deal with bounded solutions. Therefore, we write

U j 1 = V j 1 -D j 1
where D j 1 is a smooth function such that:

D j 1 = ξu j- x (t) if ξ < -1, ξu j+ x (t) if ξ > 1.
Consequently, U j 1 solves:

U j 1ξξ -((df (V j ) -X ′ j ) • U j 1 ) ξ = δ j 0t V j ξ + h j (ξ, t), (19)
U j 1 (±∞, t) = u j± 1 (t) -δ j 0 (t)u j± x (t) (20) with h j (ξ, t) = -D j 1ξξ + V j t + ((df (V j ) -X ′ j )D j 1 ) ξ -g(V j
). From estimates ( 14), (15), we deduce that h satisfies:

h j (ξ, t) = d dt u j± (t) + (df (u j± ) -X ′ j )u j± x (t) -g(u j± ) + O(e -α|ξ| ), α > 0.
And, since u is a smooth solution of (2), we have h ∈ L 1 (R) and:

h j (ξ, t) = O(e -α|ξ| ).
Integrating ( 19) with respect to ξ yields

U j 1ξ -(df (V j ) -X ′ j )U j 1 = δ j 0t V j + ξ 0 h j (η, t) dη + C j (t) (21)
where C j (t) is a constant, only depending on t.

Let us solve the problem ( 16)-( 21) with matching condition (20). Following [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF], we construct the solution of this system in two steps. First, for all j, we fix t and δ j 0 , we find U j 1 solution of (21) with finite limits at ±∞. Since these limits are explicit and only depends on t, δ j 0 , and C j , we use the matching condition (20) to rewrite (16) as a hyperbolic boundary value problem where u 1 and δ j 0 are the only unknowns. After solving this system, we use the previous construction to obtain U j 1 solution of ( 21) with matching conditions (20).

So, we fix t and δ j 0 for all j. With exactly the same arguments as in [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF], we prove the existence of U j 1 for all j. Hence, using assumption (H3) on the viscous shock profile and theory of Fredholm operators, we show that U j 1 exists and the limits satisfy:

lim ξ→±∞ U j 1 (ξ, t) = -(df (u j± ) -X ′ j ) -1 (δ j 0t u j± + H j± + C j )
where H j± = ±∞ 0 h(η, t) dη. We now use matching conditions (20) to eliminate C j in these relations. Indeed, we have

(df (u j+ ) -X ′ j )(u j+ 1 -δ j 0 u j+ x ) = -(δ j 0t u j+ + H j+ + C j ), (df (u j-) -X ′ j )(u j- 1 -δ j 0 u j- x ) = -(δ j 0t u j-+ H j-+ C j ),

and their difference is

A j+ u j+ 1 -A j-u j- 1 + δ j 0t (u j+ -u j-) = δ j 0 (A j+ u j+ x -A j-u j- x ) -(H j+ -H j-) (22) where A j± = df (u j± ) -X ′ j (t)
. Now, we have to solve (16), (22). In order to find a solution of this system, we rewrite it by fixing the shocks. Since the shocks do not interact, we can define a change of variable Z (see Figure 2) which is bijective, continuous in (x, t), piecewise linear in x and piecewise smooth:

Z(t, x) = x -X j (t) X j+1 (t) -X j (t) L m + (j -1) L m if x ∈ [X j (t); X j+1 (t)], i = 1, . . . , m. X 1 X 2 X 3 X 4 X 1 + L L m 2L m 3L m L Figure 2:
Example of the change of variable Z in the case m = 4, for some t.

We also define v 1 by

u 1 (x, t) = v 1 (Z(t, x), t).
It follows from these definitions that v 1 solves

v 1t + (Z x df (u) + Z t )(Z -1 (t, z), t)v 1z + h(z, t)v 1 -h(z, t) = 0 ( 23
)
where z → x = Z -1 (t, z) is the inverse of x → z = Z(t, x), and, h and h only depend on Z, u, u x , and u xx . We now use the fact that df (u) is diagonalizable, df (u) = P (u) -1 D(u)P (u) so

(Z x df (u) + Z t )(Z -1 (t, z), t) = P (z, t) -1 D(z, t) P (z, t).
Since zeroth order terms do not play any role in the wellposedness issue, we consider the simplified system

w 1t + D(z, t)w 1x = k(z, t), ( 24
) A j+ (P j+ ) -1 w j+ 1 -A j-(P j-) -1 w j- 1 + δ j 0t (u j+ -u j-) = l j (t) (25)
with k and l j known functions. Therefore, we have to solve this system on [0; L] under periodic boundary conditions. Since 0 and L correspond to the same shock, the periodic boundary conditions are in fact the shock conditions (25) for j = 1. Equation ( 24) is a linear transport equation on w 1i , i = 1, . . . , n. Since, generically, the existence of u smooth on [0; T * ] ensures that the characteristics do not intersect on [0; T * ], they can be used to build w 1,i smooth between shocks, using the initial condition. So, it suffices to verify that the conditions (25) at the shocks are well-posed. We must therefore count the incoming and outgoing information at the shock. As we can see in Figure 3, for i < k, the incoming characteristics come from the right, so we obtain the value of w j+ 1i . For i > k, the incoming characteristics come from the left, so we get w j- 1i . And for i = k, the sign of the eigenvalue change between two shocks: negative on the right of a shock and positive on the left, so, using again characteristic construction, w 1k is defined on the whole interval delimited by the shocks: we obtain w j+ 1k and w j- 1k .By this method we have built w 1i on the right or left side of each shock. We now use the boundary conditions ( 25) to obtain all the components of w j 1 , and δ j 0t .

x t

0 L/2 L i > k i = k i < k i = k s 1 s 2
Figure 3: Characteristic curves between two shocks, example with m = 2. We also plot s 1 and s 2 which are sonic points for k-th eigenvalue, that means λ k (s j ) = 0.

Indeed, if we note by r i the i-th eigenvector of df (u), and w 1 = P i a i r i , then a j+ i is known for i ≤ k and a j- i for i ≥ k by our construction and we rewrite (25) as

n i=1 A j+ a j+ i r j+ i - n i=1 A j-a j- i r j- i + δ j 0t (u j+ -u j-) = l j (t) or equivalently n i=1 (λ j+ i -X ′ j )a j+ i r j+ i - n i=1 (λ j- i -X ′ j )a j- i r j- i + δ j 0t (u j+ -u j-) = l j (t).
This yields the linear system on the unknowns a j+ i for i > k, a j- i for i < k, and

δ j 0t i>k (λ j+ i -X ′ j )a j+ i r j+ i - i<k (λ j- i -X ′ j )a j- i r j- i + δ j 0t (u j+ -u j-) = l j (t) - i≤k (λ j+ i -X ′ j )a j+ i r j+ i + i≥k (λ j- i -X ′ j )a j- i r j- i .
These equations have unique solutions if and only if the system obtained is invertible for all t, that is the Majda-Liu condition:

∀j = 1, . . . , m, det(r j- 1 , . . . , r j- k-1 , u j+ -u j-, r j+ k+1 , . . . , r j+ n ) = 0.
Using [START_REF] Zumbrun | Viscous and inviscid stability of multidimensional planar shock fronts[END_REF], our assumption (H3) implies Majda-Liu condition. To finish the construction of the approximate solution, we use again the characteristics. By this way, we have built a solution on the whole space R. To ensure the regularity of the solution far away from the shocks, we only need suitable compatibility conditions on the initial data. Finally, we have proved the existence of u 1 and δ j 0t for all j and for 0 ≤ t ≤ T * . The previous construction give us V j 1 for all j. We can then apply the same method to obtain the existence of V j 2 , δ j 1t and u 2 , since the linear system has the same terms of maximal order. Remark 1. Since the construction of viscous shock profile only depends on the shock, we can use the previous construction even if u is not periodic. However, we will see in the following that the periodicity of u allows us first to obtain bounds on u 1 and secondly to build the Green's function in Section 3.

Construction of the approximate solution

We complete the construction of an approximate solution of equation [START_REF] Diperna | Convergence of the viscosity method for isentropic gas dynamics[END_REF]. First, we define a smooth function µ such that:

µ(x) = 0 if |x| > 2, 1 if |x| < 1.
Then, the approximate solution u ε app is defined as

u ε app = m j=1 µ x -X j (t) ε γ I jε (x, t) + 1 - m j=1 µ x -X j (t) ε γ O ε (x, t)
and u ε app verifies

(u ε app ) t + f (u ε app ) x -ε(u ε app ) xx -g(u ε app ) = q ε
where q ε (x, t) = 3 i=1 q ε i (x, t) is an error term given by

q ε 1 (x, t) =(1 -µ j ) f (O ε ) -f (u) -ε df (u) • u 1 -ε 2 df (u)•u 2 - ε 2 2 d 2 f (u)•(u 1 , u 1 ) x -g(O ε ) -g(u) -ε dg(u) • u 1 -ε 2 dg(u) • u 2 - ε 2 2 d 2 g(u) • (u 1 , u 1 ) -ε 3 u 2xx , q ε 2 (x, t) =µ j [(f (I jε )-f (V j )-ε df (V j ) • V j 1 -ε 2 df (V j )•V j 2 - ε 2 2 d 2 f (V j )•(V j 1 , V j 1 )) x -(g(I jε ) -g(V j ) -ε dg(V j ) • V j 1 ) + ε 2 (δ 1t V j 1ξ + V j 2t + δ ′ V j 2ξ )], q ε 3 (x, t) =µ j t (I jε -O ε ) -εµ j xx (I jε -O ε ) -2εµ j x (I jε -O ε ) x + µ j x (f (I jε ) -f (O ε )) + f (µ j I jε + (1 -µ j )O ε ) x -(µ j f (I jε ) + (1 -µ j )f (O ε )) x -g(µ j I jε + (1 -µ j )O ε ) -(µ j g(I jε ) + (1 -µ j )g(O ε )),
and

µ j = µ x-X j (t) ε γ
. We now want to prove that u ε app is a good approximation of u ε , that means u εu ε app → 0 when ε → 0. Therefore, we define w ε = u εu ε app which solves:

w t + (df (u ε app ) • w) x -εw xx -dg(u ε app ) • w = -q ε + Q 1 (u ε app , w) -Q 2 (u ε app , w) x (26) with Q 1 (u ε app , w) = g(w + u ε app ) -g(u ε app ) -dg(u ε app ) • w and Q 2 (u ε app , w) = f (w + u ε app ) -f (u ε app ) -df (u ε app )
• w which are at least quadratic terms in w.

Estimates on the error term

To end with the proof of Theorem 2, it remains to compute the estimates on the error term q ε . As in [START_REF] Goodman | Viscous limits for piecewise smooth solutions to systems of conservation laws[END_REF], we can estimate the support of functions

q ε i : supp(q ε 1 ) ⊂ {(x, t) : |x -X j (t)| ≥ ε γ }, supp(q ε 2 ) ⊂ {(x, t) : |x -X j (t)| ≤ 2ε γ }, supp(q ε 3 ) ⊂ {(x, t) : ε γ ≤ |x -X j (t)| ≤ 2ε γ }.
To obtain estimates on q ε i and their derivatives, we first need to fix the shocks by a smooth change of variable. This manipulation cannot be avoided for the estimates on q ε it , q ε itt . So, we define

ϕ(z, t) = z + m j=1 α j (z, t) X j (t) -(j -1) L m -εδ j (t) (27)
where α j (•, t) are smooth functions, such that j α j ≡ 1, ϕ is increasing, ϕ z > 0 and α j (•, t) ≡ 1 on a neighbourhood [(j -1) L m -r; (j -1) L m +r] of (j -1) L m . We recall that in assumption (H2) we have supposed that |X j+1 -X j | > 2r, which ensures the existence of such a ϕ.

With the notations w(z, t) = w(ϕ(z, t), t), ũε app (z, t) = u ε app (ϕ(z, t), t), qε (z, t) = q ε (ϕ(z, t), t), equation (26) becomes:

wt + 1 ϕ z (df (ũ ε app ) • w) z - ϕ t ϕ z + εϕ zz ϕ 3 z wz -ε 1 ϕ 2 z wz z -dg(ũ ε app ) • w = -q ε + Q 1 (ũ ε app , w) - 1 ϕ z Q 2 (ũ ε app , w) z . ( 28
)
We now prove estimates ( 6)-( 7)-( 8)-( 9). Using the fact that f and g are smooth and u i is piecewise smooth, with discontinuities only at x = X j (t), we have the following estimates for qε 1 : qε 1 L ∞ ≤ Cε 3 where C is a constant which does not depend on ε. Integrating this inequality, we get: qε 1 L 1 (0;L) ≤ Cε 3 . Moreover, if ε is small enough (such that 2ε γ < r),

qε 1t (z, t) = (1 -μj )[. . . ] t + ε 1-γ δ j t µ ′ ϕ(z, t) -X j (t) ε γ [. . . ]
where the [. . . ] is dominated by ε 3 . Therefore we have

qε 1t L ∞ ≤ Cε 3 , qε 1t L 1 (0;L) ≤ Cε 3 .
Similarly, we prove the same estimates for qε 1tt . Nevertheless we cannot have such estimates for qε 1z . Indeed, we have

qε 1z (z, t) = (1 -μj )[. . . ] z -ε -γ µ ′ ϕ(z, t) -X j (t) ε γ [. . . ] so we just have qε 1z L ∞ ≤ Cε 3-γ , qε 1z L 1 (0;L) ≤ Cε 3 . Similarly, we prove qε 1zt L ∞ ≤ Cε 3-γ , qε 1zt L 1 (0;L) ≤ Cε 3 , qε 1zz L ∞ ≤ Cε 3-2γ , qε 1zz L 1 (0;L) ≤ Cε 3-γ .
Then, we compute estimates for qε 2 :

qε 2 L ∞ , qε 2t L ∞ , qε 2tt L ∞ ≤ Cε 2 , qε 2 L 1 (0;L) , qε 2t L 1 (0;L) , qε 2tt L 1 (0;L) ≤ Cε 2+γ , qε 2z L ∞ , qε 2zt L ∞ ≤ Cε 2-γ , qε 2z L 1 (0;L) , qε 2zt L 1 (0;L) ≤ Cε 2 . qε 2zz L ∞ ≤ Cε 2-2γ , qε 2zz L 1 (0;L) ≤ Cε 2-2γ
. Eventually, we use matching conditions ( 10)-( 11)-( 12) to prove the estimates on qε 3 . Indeed, the properties of viscous shock profiles provide that terms o(1) can be replaced by e -α|ξ| in the matching conditions with α a positive number. So, we have for z -(j -1) L m < r:

I jε (ϕ(z, t), t) = V j z-(j-1) L m ε , t + εV j 1 z-(j-1) L m ε , t + ε 2 V j 2 z-(j-1) L m ε , t = u j± (t) + ε u j± 1 (t) + u j± x (t) z-(j-1) L m -εδ j 0 ε + ε 2 u j± 2 (t) +u j± 1x (t) z-(j-1) L m -εδ j 0 ε + 1 2 u j± xx (t) z-(j-1) L m -εδ j 0 ε 2 -u j± x (t)δ j 1 (t) +O e -α |z-(j-1) L m | ε
and, using Taylor expansion for u i (X j + (z + (j -1) L mεδ j 0 )),

O ε (ϕ(z, t), t) = u(z + X j -(j -1) L m -εδ j , t) +εu 1 (z + X j -(j -1) L m -εδ j , t) +ε 2 u 2 (z + X j -(j -1) L m -εδ j , t) = +u j± (t) + u j± x (t)(z -(j -1) L m + εδ j 0 + ε 2 δ j 1 ) + 1 2 u j± xx (t)(z -(j -1) L m -εδ j 0 ) 2 + εu j± 1 +εu j± 1x (z -(j -1) L m -εδ j 0 ) + ε 2 u j± 2 +O(ε 3 + (z -(j -1) L m -εδ j 0 ) 3 ). Since supp(q ε 3 ) ⊂ (z, t) : ε γ ≤ z -(j -1) L m -εδ j ≤ 2ε γ ,
we have

O ε 3 + z -(j -1) L m -εδ j 0 3 = O(ε 3γ )
and we obtain the estimates:

qε 3 L ∞ , qε 3t L ∞ , qε 3tt L ∞ ≤ Cε 2γ , qε 3 L 1 (0;L) , qε 3t L 1 (0;L) , qε 3tt L 1 (0;L) ≤ Cε 3γ , qε 3z L ∞ , qε 3zt L ∞ ≤ Cε γ , qε 3z L 1 (0;L) , qε 3zt L 1 (0;L) ≤ Cε 2γ , qε 3zz L ∞ ≤ C, qε 3zz L 1 (0;L) ≤ Cε γ .
This ends the proof of Theorem 2.

Estimates on the Green's function

We now consider the linear operator

L ε w = wt + 1 ϕ z df (ũ ε app ) -ϕ t + ε ϕ zz ϕ 2 z • wz -ε 1 ϕ 2 z wzz + 1 ϕ z df (ũ ε app ) -ϕ t + ε ϕ zz ϕ 2 z z -dg(ũ ε app ) • w.
The aim of this section is to prove the following theorem:

Theorem 4. There exists a Green's function G ε (t, τ, z, y) of the linear operator

L ε defined for 0 ≤ τ, t ≤ T * , z, y ∈ R such that G ε (t, τ, z, y) = 0 if τ > t and sup y,τ ≤T * T * 0 R |G ε (t, τ, z, y)| dz dt+ √ ε sup y,τ ≤T * T * 0 R |∂ z G ε (t, τ, z, y)| dz dt ≤ C (29)
where C is positive and does not depend on ε.

To find estimates on this Green's function, we use approximations of the Green's function both near the shock and far away from the shocks.

First, we recall the method of iterative construction of the Green's function of E. Grenier and F. Rousset [START_REF] Grenier | Stability of one-dimensional boundary layers by using Green's functions[END_REF]. Then, to approximate the Green's function near the shocks, we recall the result of K. Zumbrun and P. Howard [START_REF] Zumbrun | Pointwise semigroup methods and stability of viscous shock waves[END_REF] about Green's function for pure viscous profile problem. Far away from the shocks, we use characteristic curves to build some approximate Green's functions. Finally, we combine all these Green's functions to obtain an approximate Green's function of L ε and we find bounds on the error terms.

Method

Here, we recall the method used by E. Grenier and F. Rousset in [START_REF] Grenier | Stability of one-dimensional boundary layers by using Green's functions[END_REF]. We want to construct an approximate Green's function G ε app of L ε in the form

G ε app (t, τ, z, y) = N k=1 S k (t, τ, z, y)Π k (τ, y),
where S k are Green's kernels which satisfy (29) and

Π k ∈ C ∞ ([0, T * ] × R, L(R n )) are such that Π k (t, x)v ≤ C v , ∀x ≥ 0, t ∈ [0; T * ], v ∈ R n and N k=1 Π k = Id .
We next define the error R k (•, τ, •, y) = L ε S k (τ, y) for k = 1, . . . , N, and the matrix of errors:

M(T 1 , T 2 ) = (σ kl (T 1 , T 2 )) 1≤k,l≤N with σ kl (T 1 , T 2 ) = sup T 1 ≤τ ≤T 2 ,y∈supp Π l T 2 T 1 R |Π k (t, z)R l (t, τ, z, y)| dz dt.
Thanks to Theorem 2.2 of [START_REF] Grenier | Stability of one-dimensional boundary layers by using Green's functions[END_REF], we just have to prove that there exists

ε 2 > 0 such that 0 < T 2 -T 1 < ε 2 implies lim p→∞ M p (T 1 , T 2 ) = 0.
Remark 2. Since we consider the error in a matrix, we need to consider a finite number of Green's kernel S k . We will see in the following that the number of these kernels is proportional to the number of shocks per period. So, this method does not allow us to treat the case of a non-periodic perturbation.

Near the shocks

Near the shock j, we can approximate

1 ϕ z df (ũ ε app ) -ϕ t + ε ϕ zz ϕ 2 z by df V j z -(j -1) L m ε , τ -X ′ j
and we forget zeroth order term. Therefore, we search the Green's functions for the linear operators

L εj τ w = ∂ t w + df V j z -(j -1) L m ε , τ -X ′ j (τ ) w z -εw zz
which depend on j and τ < T * . As in [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF], we remark that these Green's functions G Sj τ (t, z, y) verify

G Sj τ (t, z, y) = 1 ε G HZj τ t ε , z -(j -1) L m ε , y -(j -1) L m ε
where G HZj τ is the Green's function related to the operator

L j τ w = ∂ t w + (df (V j (z, τ )) -X ′ j (τ ))w z -w zz .
In [START_REF] Zumbrun | Pointwise semigroup methods and stability of viscous shock waves[END_REF], K. Zumbrun and P. Howard obtained estimates on the Green's functions which will be useful to obtain estimates for our operators. Let us denote by ãj± i (τ ), and r j± i (τ ) the eigenvalues and the associated eigenvectors of df (u j± (τ )) -X ′ j (τ ). Proposition 1. Under hypothesis (H3), we have

G Sj τ t, z + (j -1) L m , y + (j -1) L m = i,ã j+ i (τ )>0 O   exp - (z-ã j+ i (τ )t) 2 M εt √ εt   r j+ i (τ )χ z≥0 + i,ã j- i (τ )<0 O   exp - (z-ã j- i (τ )t) 2 M εt √ εt   r j- i (τ )χ z≤0 + O   exp -(z-y) 2 M εt √ εt e -σ t ε   (30) ∂ z G Sj τ t, z + (j -1) L m , y + (j -1) L m = i,ã j+ i (τ )>0 O   exp - (z-ã j+ i (τ )t) 2 M εt εt   r j+ i (τ )χ z≥0 + i,ã j- i (τ )<0 O   exp - (z-ã j- i (τ )t) 2 M εt εt   r j- i (τ )χ z≤0 + O   exp -(z-y) 2 M εt εt e -σ t ε   , ( 31
)
where M and σ are positive constants, and χ designs characteristic function. Moreover, O's are at least linear forms, locally bounded in y and uniformly bounded in (t, z).

Far away from the shocks

As in the previous section, we do not search Green's function for L ε but for the approximate operator Lε defined by

Lε w = w t + 1 ϕ z df (u(ϕ(z, t), t)) -ϕ t + ε ϕ zz ϕ 2 z • w z - ε ϕ 2 z w zz .

Far away from the shocks

We remark that, as in the previous section, we forget the terms in w.

Recall that df (u(x, t)) is diagonalizable for all x, t, so we can write df (u(ϕ(z, t), t)) = P (u(ϕ(z, t), t))D(u(ϕ(z, t), t))P (u(ϕ(z, t), t)) -1

with D(u(ϕ(z, t), t)) = diag(λ i (u(ϕ(z, t), t))).

To obtain approximation of the Green's function between two shocks, we define j approximate problems on R, with continuous solutions. First, we set

λ j i (ϕ(z, t), t) =    λ i (u(X + j (t), t)) if z ∈ -∞; (j -1) L m + εδ j , λ i (u(ϕ(z, t), t)) if z ∈ (j -1) L m + εδ j ; j L m + εδ j+1 , λ i (u(X - j+1 (t), t)) if z ∈ j L m + εδ j+1
; +∞ . Then, we want to find approximate Green's functions for the scalar operators

L j i w = w t + 1 ϕ z λ j i -ϕ t + ε ϕ zz ϕ 2 z w z - ε ϕ 2 z w zz .
To do so, we define characteristic curves χ j i (t, τ, y) by

∂ t χ j i (t, τ, y) = λ j i (χ j i (t, τ, y), t), t ≥ τ, χ j
i (τ, τ, y) = y, and the approximate Green's functions

G j i (t, τ, z, y) = ϕ z (y, τ ) 4πε(t -τ ) exp - (ϕ(z, t) -χ j i (t, τ, ϕ(y, τ ))) 2 4ε(t -τ )
.

We easily compute the error committed here y). Before we build the whole Green's function, we introduce some notations. First, we write G j = diag(G j i ). In the sequel, we need to distinguish at each shock the outgoing waves to the incoming waves. We define D -in = diag(0, . . . , 0, 1, . . . , 1), with k -1 unit coefficients, D -out = diag(1, . . . , 1, 0, . . . , 0), with k -1 null coefficients, D +in = diag(1, . . . , 1, 0, . . . , 0), with k unit coefficients, D +out = diag(0, . . . , 0, 1, . . . , 1), with k null coefficients, so that D ±in + D ±out = Id.

L j i G j i = (λ j i (ϕ(z, t), t) -λ j i (χ j i (t, τ, ϕ(y, τ )), t))G j iz (t, τ, z,
Finally, we define the projections

P ±in (t, z) = P (t, z)D ±in P (t, z) -1 , P ±out (t, z) = P (t, z)D ±out P (t, z) -1 .
Since the shocks are non-characteristic Lax shocks, we have the following inequality on a neighbourhood of each shock j = 1, . . . , m:

|λ i (ũ(z, t)) -X j (t)| > C > 0 if (j -1) L m -4η < z < (j -1) L m + 4η, i = 1, . . . , n.
We can assume that η is such that 4η < r so that α j ≡ 1 in (27) on the previous neighbourhood. Furthermore, we need some cut-off smooth functions

K + (z) = 0 if z ≤ 1, 1 if z ≥ 2 and K -(z) = 1 if z ≤ -2, 0 if z ≥ -1.
We also assume the cut-off function µ already used to read as µ = (1-K + )(1-K -).

We can now build an approximate Green's function in the form

G ε app (t, τ, z, y) = m j=1 7 k=0 S j k (t, τ, z, y)Π j k (τ, y)
where the Green's kernels are periodic with period (0, 0, L, L):

S j k (t, τ, z, y) = l∈Z Sj k (t, τ, z + lL, y + lL) with Sj 0 (t, τ, z, y) = µ z -(j -1) L m 2η µ z -(j -1) L m M 3 ε G Sj τ (t -τ, z , y), Sj 1,2 (t, τ, z, y) = µ .
. . 2η K + z -(j -1) L m M 1 ε P (t, z)D +out G j (t, τ, z, y)P (τ, y) -1 , Sj 3 (t, τ, z, y) = µ . . . 2η K + z -(j -1) L m M 1 ε P (t, z)D +in G j (t, τ, z, y)P (τ, y) -1 , Sj 4 (t, τ, z, y) = K + 4(z -(j -1) L m ) η + K -4(z -j L m ) η -1 P (t, z)G j P (τ, y) -1 , Sj 5 (t, τ, z, y) = µ z -j L m 2η K -z -j L m M 1 ε P (t, z)D -in G j (t, τ, z, y)P (τ, y) -1 , Sj 6,7 (t, τ, z, y) = µ . . . 2η K -z -j L m M 1 ε P (t, z)D -out G j (t, τ, z, y)P (τ, y) -1 ,
and the projectors are also periodic:

Π j k (τ, y) = l∈Z Πj k (τ, y + lL) with Πj 0 (τ, y) = µ y -(j -1) L m η µ y -(j -1) L m M 2 ε , Πj 1 (τ, y) = µ . . . η K + y -(j -1) L m M 2 ε 1 -K + 2(y -(j -1) L m ) M 3 ε P +out (τ, y), Πj 2 (τ, y) = µ . . . η K + 2(y -(j -1) L m ) M 3 ε P +out (τ, y), Πj 3 (τ, y) = µ . . . η K + y -(j -1) L m M 2 ε P +in (τ, y), Πj 4 (τ, y) = K + y -(j -1) L m η + K -y -j L m η -1, Πj 5 (τ, y) = µ y -j L m η K -y -j L m M 2 ε P -in (τ, y), Πj 6 (τ, y) = µ . . . η K -2(y -j L m ) M 3 ε P -out (τ, y), Πj 7 (τ, y) = µ . . . η K -y -j L m M 2 ε 1 -K -2(y -j L m ) M 3 ε P -out (τ, y)).
It appears that all the Green's kernels can be written in the following form:

S j k (t, τ, z, y) = T (z)S(t, τ, z, y) (32)
where T is a truncation function. We will choose the three constants M 1 , M 2 , M 3 at the end of the estimates on the error matrix so that they verify

4M 1 ≤ M 2 ≤ 1 4 M 3 .
These inequalities are necessary to have j,k

Π j k ≡ 1 and G app (τ, τ, z, y) = δ y (z) Id .
Under these notations, Sj 0 describes the viscous dynamic at the shock j, Sj 1 the creation of outgoing waves in a vicinity at the right of the shock j, Sj 2 the creation and propagation of outgoing waves away from the shock j, at its right, Sj 3 the creation and propagation of incoming waves at the right of the shock j. Sj 4 describes the propagation of the waves between the shocks j and j + 1 Moreover, the kernels Sj 7 , Sj 6 , Sj 5 are the symmetric of respectively Sj 1 , Sj 2 , Sj 3 for the left of the shock j + 1. We summarize this splitting in Figure 4.

x (j -1) L m j L m viscous Sj 0 Sj+1 0 viscous Sj 1 -→ Sj 2 -→ ←-Sj

Bounds on the error matrix

As said in Subsection 3.1, to prove Theorem 4, it remains to prove that M p converges to 0 when p goes to ∞. Since the coefficients of M are non-negative, it suffices to prove that M is bounded above by an other matrix which has the "good" convergence.

To bound the error terms, we use the same method than F. Rousset in [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF]. We split all the error terms into two parts: the truncation of the error on the kernel and the commutator:

R j k = E j k1 + E j k2
where, with the notation of (32), E j k1 (t, τ, z, y) = T (z)L ε S(t, τ, z, y) and E j 2k (t, τ, z, y) = [L ε , T (z)]S(t, τ, z, y).

Lemma 1. We have the estimates at shock j: Rj

0 1 |y-(j-1)L/m|≤2M 2 ε E j 01 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 1 (T + ε), 1 |y-(j-1)L/m|≤2M 2 ε 1 ±z≥0 P ± out E j 02 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 2 , 1 |y-(j-1)L/m|≤2M 2 ε 1 ±z≥0 P ± in E j 02 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 3 + C 2 T ;
for the outgoing waves: Rj 1 , Rj 2 , Rj 6 , Rj 7 . Let M ≥ M 2 . we have:

1 y-(j-1)L/m≥M ε E j 11 (t, τ, z, y) L ∞ τ,y ,L 1 t,z , 1 y≥M ε E j 21 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 4 (T + ε 2γ-1 ) + C 5 , 1 y-(j-1)L/m≤-M ε E j 61 (t, τ, z, y) L ∞ τ,y ,L 1 t,z , 1 y≤-M ε E j 71 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 4 (T + ε 2γ-1 ) + C 5 , 1 y-(j-1)L/m≥M ε E j 12 (t, τ, z, y) L ∞ τ,y ,L 1 t,z , 1 y≥M ε E j 22 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 5 +C 1 (T +ε), 1 y-(j-1)L/m≤-M ε E j 62 (t, τ, z, y) L ∞ τ,y ,L 1 t,z , 1 y≤-M ε E j 72 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 5 + C 1 (T + ε);
for the incoming waves: Rj 3 , Rj Proof. We do not give here the complete proof of the lemma. Mainly, it deals with terms that are not treated in [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF]: zeroth order terms and terms related to ϕ or η.

5 1 y-(j-1)L/m≥M ε E j 31 (t, τ, z, y) L ∞ τ,y ,L 1 t,z , 1 y≤-M ε E j 51 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 6 (T + ε 2γ-1 ) + C 7 , 1 y-(j-1)L/m≥M ε E j 32 (t, τ, z, y) L ∞ τ,y ,L 1 t,z , 1 y≤-M ε E j 52 (t, τ, z, y) L ∞ τ,y ,L 1 t,z ≤ C 8 + C 1 (T + ε); between two shocks: Rj 4 R j 4 L ∞ τ,y ,L 1 t,z , ≤ C 8 (T + ε), where C 1 is locally bounded in M 2 , M 3 ,C 2 is
First, we consider the error at the shock j. On the support of E j 01 , we have that ϕ z (•, t) ≡ 1 and ϕ t (•, t) ≡ X ′ j (t)εδ j t (t) so we obtain

E j 01 = µ z -(j -1) L m 2η µ z -(j -1) L m M 3 ε h G Sj τ (t -τ, z, y) + df (ũ ε app ) -df V j z -(j -1)/m ε , τ + X ′ j (τ ) -X ′ j (t) + εδ j t G Sj τ z
where h is a bounded function of t, z. Hence, using the fact that df and X j are smooth and the expression of u ε app , we have

|E j 01 | ≤ Cµ z -(j -1) L m 2η µ z -(j -1) L m M 3 ε (|t -τ | + ε)|G Sj τ z | + |G Sj τ |
where C is locally bounded in M 3 . The calculations of [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF] give directly a bound for the first term, in G Sj τ z . So, it only remains to bound the integral:

T τ (j-1)L/m+2M 3 ε (j-1)L/m-2M 3 ε |G Sj τ (t -τ, z, y)| dz dt, T τ (j-1) L m +4η (j-1) L m +M 1 ε |G j i | = T τ (j-1) L m +4η (j-1) L m +M 1 ε |ϕ z (y, τ )| 4πε(t -τ ) exp - (ϕ(z, t) -χ j i (t, τ, ϕ(y, τ ))) 2 4ε(t -τ ) dz dt.
Hence, using the change of variable z ′ = ϕ(z,t)-χ j i (t,τ,ϕ(y,τ ))

√ 4ε(t-τ )
, we obtain

T τ (j-1) L m +4η (j-1) L m +M 1 ε |G j i | ≤ C T τ R e -z ′2 dz ′ dt ≤ CT.
Similarly, we have

T τ (j-1) L m +4η (j-1) L m +M 1 ε |G j iz | = T τ (j-1) L m +4η (j-1) L m +M 1 ε |ϕ z (y, τ )| 4πε(t -τ ) exp - (ϕ -χ j i ) 2 4ε(t -τ ) -ϕ z ϕ -χ j i 2ε(t -τ ) dz dt.
And with the same change of variable, we obtain:

ε T τ (j-1) L m +4η (j-1) L m +M 1 ε |G j iz | ≤ C √ ε T τ 1 √ t -τ R |z ′ |e -z ′2 dz ′ dt ≤ C(T + ε),
which gives the estimate for E j i2 . Since there is not new difficulty in the proof of the estimates on R j 4 , we do not develop it here.

We now use Lemma 1 to bound the matrix M. Since the Green's kernel depends on the shock, we note

σ ij kl (T 1 , T 2 ) = sup T 1 ≤τ ≤T 2 ,y∈supp Π j l T 2 T 1 R |Π i k (t, z)R j l (t, τ, z, y)| dz dt.
Since two shocks do not interact, the error coefficients σ ij kl vanish for i = j and kl = 0 and for |i -j| > 1. So the error matrix M is bounded:

M ≤           M 1 M 2 M 3 M 3 M 1 M 2 . . . . . . . . . . . . . . . . . . . . . . . . M 2 M 2 M 3 M 1          
where M 2 is null except on the first column, and M 3 is null except on the first line. Moreover, using Lemma 1, and it was done in [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF], we can choose α < 1/2 and M 1 , M 2 such that when M 3 → +∞, ε, T → 0, the matrices tend to

M 1 →             • α • C • • • • • α • α • • • • C α • α • • • • • α • α • • • • • α • C • C • α • • • • • α • α • • • • • α • α • • • • • α • α             , M 2 →             M 3 →             • • • • • C • α • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •            
.

So that M → M where the eigenvalues of M are 0 and 2α. We can now conclude that we have M p → 0 when p → ∞. This ends the proof of Theorem 4.

Convergence

The purpose of this section is to prove Theorem 3 and to conclude the proof of Theorem 1. For this sake, it remains to show that solution of

L ε w = -q ε + Q 1 (ũ ε app , w) - 1 ϕ z Q 2 (ũ ε app , w) z , (33)
w(z, 0) = 0 (34) vanishes as ε → 0. In the previous section, we obtain estimate (29) on the Green's function of operator L ε . We recall that Q 1 and Q 2 are at least quadratic in w and qε verifies inequalities ( 6), ( 7), [START_REF] Noble | Persistence of roll waves for the Saint Venant equations[END_REF], and (9).

As in [START_REF] Goodman | Viscous limits for piecewise smooth solutions to systems of conservation laws[END_REF], [START_REF] Grenier | Stability of one-dimensional boundary layers by using Green's functions[END_REF] and [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF], we use standard arguments for parabolic equations. First, we remark that local existence of a smooth solution w for (33)-(34) is classical. Then we define

T ε = sup{T 1 ∈ [0; T * ], ∃ w solution on R × [0; T 1 ), E(T 1 ) ≤ 1}, where E(T 1 ) = T 1 0 L 0 | w| ε 3γ-α + | wz | ε 3γ-α-1/2 + | wt | ε 3γ-2α-1/2 + | wzz | ε 3γ-2α-1 + | wtz | ε 3γ-2α-1 + | wtzz | ε 3γ-3α-3/2 + | wzzz | ε 3γ-2α-2 dz dt
with α > 0 and γ ∈ (2/3, 1), chosen later.

Before estimating the L 1 -norms, we define the notation:

w 1 = w L 1 ((0;T ε )×[0,L)) , w ∞ = w L ∞ ((0;T ε )×[0,L))
First, we use Fourier coefficients c n to estimate w L ∞ . Indeed, w(z, t) = when ε ≤ 1. Hence, w ∞ tends to 0 as ε goes to 0 if γ and α are such that γα > 1/2. We deduce that to prove that the time existence is T * and that we have the convergence, it remains to prove that T ε = T * for ε small enough. In the sequel, we suppose that T ε < T * so E(T ε ) = 1.

Then, we can use the estimate on w ∞ to bound w and its derivatives. First, using (33)-(34), we have Therefore, we deduce

w(t, z) = t 0 R G ε (t, τ, z, y) -q ε + Q 1 (ũ ε app , w) - 1 ϕ z Q 2 (ũ ε app , w) z (τ,
w 1 ≤ Cε 3γ + C w ∞ w 1 + C w ∞ ( w 1 + wz 1 ) so w 1 ε 3γ-α ≤ C(ε α + ε 3γ-3α-3/2 (1 + ε -1/2 )).
This can be made smaller than 1 as ε → 0 if α and γ are such that 3γ -3α -2 > 0.

We now take the z derivative w and obtain an expression of wz :

wz (t, z) = t 0 R G ε z (t, τ, z, y) -q ε + Q 1 (ũ ε app , w) - 1 ϕ z Q 2 (ũ ε app , w) z (τ, y) dy dτ
and the estimate:

wz 1 ε 3γ-α-1/2 ≤ C(ε α + ε 3γ-3α-3/2 (1 + ε -1/2 )).
Differentiating equation (33) with respect to t or z, we obtain equations verified by w t and w z : L ε wt = -q ε + Q 1 (ũ ε app , w) -Q 2 (ũ ε app , w) z t + l 1 ( w, wz , ε wzz ), L ε wz = -q ε + Q 1 (ũ ε app , w) -Q 2 (ũ ε app , w) z z + l 2 ( w, wz , ε wzz ) where l 1 and l 2 are continuous linear forms, uniformly bounded with respect to ε. Thus, using again the Green's function, we get the inequalities: wt 1 ε 3γ-2α-1/2 ≤ C(ε 2α+1/2 + ε 3γ-3α-3/2 (ε α+1/2 + 1 + ε α + ε -1/2 ) + ε α+1/2 + ε α + ε 1/2 ), wtz 1 ε 3γ-2α-1 ≤ C(ε 2α+1/2 + ε 3γ-3α-3/2 (ε α+1/2 + ε -1/2 ) + ε α + ε 1/2 ), wzz 1 ε 3γ-2α-1 ≤ C(ε 2α+1/2-γ + ε 3γ-3α-3/2 (ε α+1/2 + ε -1/2 ) + ε α + ε 1/2 ). In the last inequality, the right-hand side can be made smaller than 1 as ε → 0 if α and γ are such that 2α + 1/2γ > 0.

Differentiating again equation (33) with respect to t, z or z, z, we obtain the equalities:

L ε wtz = -q ε + Q 1 (ũ ε app , w) -Q 2 (ũ ε app , w) z tz + l 3 ( w, wz , wt , wzz , wzt , ε wzzz , ε wzzt ),

L ε wtz = -q ε + Q 1 (ũ ε app , w) -Q 2 (ũ ε app , w) z zz + l 4 ( w, wz , wzz , ε wzzz ) where l 3 and l 4 are continuous linear forms, uniformly bounded with respect to ε. As seen before, we deduce the inequalities: wtzz 1 ε 3γ-3α-3/2 ≤ C(ε 3α+1-γ + ε 3γ-3α-3/2 (ε 2α+1 + ε -1/2 ) + ε 2α+1 + ε α + ε 1/2 ), wzzz 1 ε 3γ-2α-2 ≤ C(ε 2α+3/2-2γ + ε 3γ-3α-3/2 (ε α+3/2 + ε -1/2 ) + ε α+3/2 + ε 1/2 ). The both bounds can be made smaller than 1 as ε → 0 if α and γ are such that 3α + 1γ > 0 and 2α + 3/2 -2γ > 0.

We can now verify that there exist γ and α checking all the previous conditions: they define a non-empty trapeze in the plane α, γ. Hence, we have proved that for ε small enough, we have E(T ε ) ≤ ε β with β > 0. Consequently, we can not have T ε < T * for such an ε. Moreover, using the change of variable ϕ, we return to w = u εu ε app . Thus, we have w ∞ ≤ Cε 3γ-3α-3/2 , and u εu ε app ∞ ≤ Cε 3γ-3α-3/2 . Inequality E(T * ) ≤ 1 also gives

w L ∞ (L 1 ) ≤ w t 1 ≤ C wt 1 ≤ Cε 3γ-α 1 /2 so u ε -u ε app L ∞ (L 1 ) ≤ Cε 3γ-2α-1/2 .
This concludes the proof of Theorem 3. It only remains to prove Theorem 1. Since u ε appu L ∞ (L 1 ) → 0, we have the convergence in L ∞ ((0; T * ), L 1 (0; L)). Moreover, the fast convergences of the viscous shock profiles V j give the last point of the theorem.

Conclusion and perspectives

In this article, we have proved the persistence of solutions of the inviscid equation ( 2) close to roll-waves by adding full viscosity. One of the main assumptions that we have taken is the periodicity of the solution of (2). This one is not necessary in the construction of the approximate solution u ε app , but it gives that u 1 and u stay bounded (because periodic). An idea to weaken this assumption would be that solution u of ( 2) approximates the roll-wave as |x| goes to infinity (in particular, the shock curves would be closer as |x| goes to infinity). Moreover, the periodicity of u allows us to use the method of [START_REF] Grenier | Stability of one-dimensional boundary layers by using Green's functions[END_REF] to construct the Green's function of L ε . Indeed, in this step, the number of Green's functions that we consider is proportional to the number of shocks. In the periodic case, taking into account the repetitions, it returns to a finite number of periodic Green's kernels. Thus, we can write a matrix of errors, and deduce Theorem 4 on the existence of the Green's function relative to L ε and estimates on this Green's function. Another way to hope to obtain a finite number of Green's functions could be to assume that u coincides with the roll-wave outside a bounded domain.

Furthermore, Theorem 1 proved here is valid in the case of an artificial viscosity. Therefore, one should also study the persistence in the case of real viscosity as presented to the system of Saint Venant (3).

Finally, one can also be interested in what happens in the multidimensional case. For this, we could build on work already done in the case of a single multidimensional shock, based on the study of Evans' functions at each shock [START_REF] Guès | Existence and stability of multidimensional shock fronts in the vanishing viscosity limit[END_REF].

Figure 1 :

 1 Figure 1: Allure of solution u over one period when u is scalar and m = 4. The periodic roll-wave is drawn in dotted line. The solution which checks our assumptions is represented by continuous line. One also placed the shocks for the two solutions.

7 Figure 4 :

 74 Figure 4: Summarize of the splitting by Green's kernels.

  locally bounded in M 2 uniformly in M 3 , C 3 depends only on M 2 and M 3 and goes to 0 as M 3 → +∞, C 4 is independent of M 1 , M 2 and M 3 , C 5 depends only on M and goes to 0 as M → +∞, C 6 is locally bounded in M 1 , C 7 goes to 0 as M 1 → +∞, and C 8 is bounded uniformly in M 1 .

  w ∞ ≤ C( wt 1 + wzzt 1 ) ≤ Cε 3γ-3α-3/2

0 G 0 L 0 G

 000 y) dy dτ. Also, we have for periodic function ψ the estimate R L ε (t, τ, z, y)dz ψ(y) dy = O L ε (t, τ, z, y)dz ψ(y) dy .

Acknowledgment. The author wishes to thank Pascal Noble for suggesting the problem and for fruitful discussions.

and, thanks to Proposition 1, to estimate:

Using the classical change of variable

, we obtain the bound

so this concludes the estimate for E j 01 . For E j 02 , the linear term in G Sj τ disappears in the commutator. Hence, there is no change with the proof of F. Rousset.

For the estimates on Rj i , i = 1, 2, 3, 5, 6, and 7, we first remark that ϕ z ≡ 1 on the support of the errors, so E j i1 is bounded as in [START_REF] Rousset | Viscous approximation of strong shocks of systems of conservation laws[END_REF], except for the zeroth order term which is treated as in the case of E j 01 . So, we only consider the estimate on E j i2 . We first remark that the support of this error is not of size ε. Indeed, the truncation µ z-(j-1) L m 2η adds some error terms with support of size 4η. Though, we have to bound:

The two terms in G j are bounded by CT , and the term in (P D +out G j ) z is bounded by C(ε + T ). Indeed, for 1 ≤ i ≤ n,


Stefan Geiss
email: stefan.geiss@uibk.ac.at

Emmanuel Gobet
email: emmanuel.gobet@imag.fr

Fractional smoothness and applications in Finance

This overview article concerns the notion of fractional smoothness of random variables of the form g(XT ), where X = (Xt) t∈[0,T ] is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.

Introduction

From the practitioners one learns that hedging an option which payoff is discontinuous is more difficult compared to the case of smooth payoffs: this feature appears for instance for digital options or barrier options (we refer the reader to [START_REF] Taleb | Dynamic Hedging: Managing Vanilla and Exotic Options[END_REF] among others). On the one hand, for such options the number of assets (i.e. the delta) to incorporate in the hedging portfolio is unbounded, and it may become larger and larger when one gets close to the singularity (i.e. the maturity and the strike for digital options, or the trigger level for barrier options). On the other hand, the numerical estimation of this delta becomes less and less accurate, leading to global stability issues. These heuristic observations are the starting point for deeper mathematical investigations about the concept of irregular payoffs, in order to formalize it and to quantify the payoff irregularity (with the notion of fractional smoothness). In the current contribution, we aim to give an overview of this concept and some applications in stochastic finance. Actually, the applications go beyond the financial framework and more generally, they concern the theory of stochastic differential equations and their approximations.

The discrete time hedging error as an important application.

Since most of the results presented here are applied to the aforementioned example of hedging possibly irregular options, we start with a brief presentation of this problem, in order to emphasize the issues to handle and to raise some natural questions. Take for instance an European-style option exercised at maturity T > 0, with a payoff of the form h(S T ) where S t := [S 1 t , • • • , S d t ] denotes the price of a d-dimensional underlying asset at time 0 ≤ t ≤ T . Sometimes, we will use the notation X i t = log(S i t ) for the log-asset, and g(x) = h(e x1 , • • • , e x d ) for the payoff in the logarithmic variables. In what follows, we assume a Markovian dynamics without jumps for the asset (solution to a SDE defined below), we suppose that the interest rate is equal to 0 (to simplify the presentation) and that the market is complete (for details about this standard framework, see [START_REF] Karatzas | Methods of mathematical finance[END_REF]). Thus, under some regularity assumptions, the payoff h(S T ) can be replicated perfectly by a continuous time strategy, where δ S t = ∇ x H(t, S t ) defines the vector of number of assets to hold at time t. Here, H is the fair price of the option, that is

H(t, x) = ❊ ◗ (h(S T )|S t = x)
where ◗ is the (unique) risk-neutral measure. In practice, only discrete-time hedging is possible at some times τ = (t i ) n i=0 with 0 = t 0 < t 1 < • • • < t n-1 < t n = T . Thus, at time t ∈ [0, T ] the option seller is left with the tracking error

C t (h(S T ), τ ) = H(t, S t ) -H(0, S 0 ) - i ∇ x H(t i , S ti ) • (S ti+1∧t -S ti∧t ) = t 0 (∇ x H(s, S s ) -∇ x H(φ(s), S φ(s) )) • dS s (1.1)
with φ(s) = t i when t i < s ≤ t i+1 . We expect the tracking error (1.1) to converge to 0 as the number n of re-balancing dates goes to infinity. With the above formulation (1.1), the tracking error is naturally associated to the problem of approximation of a stochastic integral using piece-wise constant integrated processes. But the delta process (∇ x H(s, S s )) s∈[0,T ) may exhibit very different behaviors from payoff to payoff: if the payoff is smooth enough, then the delta might be bounded as time goes to maturity, while an irregular payoff usually yields an exploding delta as s → T . This gives rise to the first question.

(Q1) Is there an intrinsic way to relate the growth rate (as s → T ) of the derivatives of H to the irregularity of the payoff h?

The answer will be yes via the notion of fractional smoothness introduced below, see Theorems 1 and 2.

The estimation of stochastic integrals is usually performed with L 2 -norms, but in our financial setting, both measures P and ◗ can be considered. For practitioners, errors under the historical probability P are presumably more relevant, while the mathematical treatment under the risk-neutral measure ◗ is simpler in our context (because the tracking error process (1.1) is a ◗-local martingale).

(Q2) Is the definition of fractional smoothness affected by the choice of a specific measure? Do the L 2 -convergence rates depend on the choice of the probability measures P or ◗?

In the context we consider the answer concerning the fractional smoothness is usually no in the sense of the comments after Theorem 2. Concerning the approximation rates the same is checked for examples so far (see the remarks after Theorem 9).

Beyond the approach to measure tracking errors in L 2 , we could alternatively identify the weak limit of the re-normalized tracking error.

(Q3) Do the weak convergence rates coincide with those in the L 2 sense?

The answer is not necessarily, as there are counter-examples in which the convergence in L 2 and in distribution hold at different rates, see Section 1.5.

Finally, through an efficient choice of re-balancing dates τ , one can expect to reduce tracking errors and improve the risk management of options.

(Q4) Which time-nets τ = (t i ) n i=0 lead to optimal convergence rates? And how to relate them to the fractional smoothness of the payoff?

As answer we get that according to the index of fractional smoothness of the payoff, one can define explicitly re-balancing times achieving the optimal convergence rates, see Section 1.5. These preliminary questions serve as references for the reader when reading the next sections.

Organization of the paper.

First, we define the probabilistic framework and the assumptions used throughout this work. Then in Section 1.2, we define the fractional smoothness and provide basic properties: we choose a presentation that is quite illuminating regarding the previous preliminary questions. In Section 1.3, we take another view on fractional smoothness using the interpolation theory. In Section 1.4, we consider examples of terminal conditions and identify their fractional smoothness. Then, in Section 1.5, we go back to the analysis of discrete time hedging errors and state the main results. We close by further developments and applications of the fractional smoothness in Section 1.6.

Assumptions.

Let us define the probabilistic setting used in the following. We fix a ddimensional Brownian motion W = (W t ) t∈[0,T ] defined on a complete probability space (Ω, F T , P) and we let (F t ) t∈[0,T ] be the augmentation of the natural filtration of W . The log-asset X is the solution of the d-dimensional forward diffusion

X t = x 0 + t 0 b(s, X s )ds + t 0 σ(s, X s )dW s .
To state the results, we mainly consider two types of assumptions:

(SDE) d ≥ 1 and b, σ ∈ C ∞ b ([0, T ] × | d ) and σσ * ≥ δI | d for some δ > 0. (GBM) d = 1 and X t = ln(S t ) = W t -(t/2).
The smoothness conditions in (SDE) are too strong, they are chosen to simplify the presentation. Whenever useful to simplify even more, we may consider the very simple case of the geometric Brownian motion (GBM) (here, the asset is a martingale, meaning that P = ◗). The reader is referred to the corresponding original papers for the possible weaker conditions.

In the following | • | stands for the Euclidean norm and A ∼ c B for A/c ≤ B ≤ cA if c ≥ 1 and A, B ≥ 0. Expectations and conditional expectations under P are simply denoted by ❊(.) and ❊(.|F t ), while under ◗, we indicate explicitly the dependency w.r.t. the probability measure by writing ❊ ◗ (.) and ❊ ◗ (.|F t ).

Definition of fractional smoothness and basic properties

Fractional smoothness on the Wiener space can be defined in various ways, see [START_REF] Watanabe | Fractional order Sobolev spaces on Wiener space[END_REF][START_REF] Hirsch | Lipschitz functions and fractional Sobolev spaces[END_REF]. Our approach is motivated by the questions discussed in Section 1.1. Since we consider only random variables of the form Z = g(X T ) = h(S T ) (a function of the process at maturity T ), the time T plays a specific role in our definition. It would be necessary to modify our definition for more general dependencies like Z = g(X t1 , • • • , X tn ), see [START_REF] Geiss | Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition[END_REF].

Definition 1. Assume that Z ∈ L 2 (P). (i) For 0 < θ ≤ 1 we let Z ∈ ❇ θ 2,∞ provided that, for all 0 ≤ t < T , Z -❊(Z|F t ) L2(P) ≤ c(T -t) θ 2 . (ii) For 0 < θ < 1 we let Z ∈ ❇ θ 2,2 provided that T 0 (T -t) -1-θ Z -❊(Z|F t ) 2 L2(P) dt < ∞.
The spaces ❇ θ 2,q above will always be obtained by the conditional expectation and the L 2 -norm under the measure P. Therefore we omit the dependency on P in the notation.

The following properties follow straight from the definition: Proposition 1. For 0 < θ < η < 1 and p, q ∈ {2, ∞} we have that

❇ 1 2,∞ ⊆ ❇ η 2,p ⊆ ❇ θ 2,q and ❇ θ 2,2 ⊆ ❇ θ 2,∞ .
Given a bounded3 measurable g :

| d → | and (t, x) ∈ [0, T ) × | d , we let u(t, x) := ❊(g(X T )|X t = x), D 2 u(t, x) := ∂ 2 u ∂x i ∂x j (t, x) d i,j=1
.

The following equivalences are useful to exploit properties of ❇ θ 2,2 and ❇ θ 2,∞ .

Theorem 1 ([GM10a, Proposition 4]). Under the condition (SDE), for 0 < θ < 1 and a bounded g, the following assertions are equivalent:

(i) g(X T ) ∈ ❇ θ 2,2 . (ii) T 0 (T -t) -θ ❊ |∇ x u(t, X t )| 2 dt < ∞. (iii) T 0 (T -t) 1-θ ❊ D 2 u(t, X t ) 2 dt < ∞.
Theorem 2 ([GM10a, Lemma 6]). Under the condition (SDE), for 0 < θ ≤ 1 and a bounded g, the following assertions are equivalent:

(i) g(X T ) ∈ ❇ θ 2,∞ . (ii) sup t∈[0,T ) (T -t) 1-θ ❊ |∇ x u(t, X t )| 2 < ∞. (iii) For 0 < θ < 1 we have that sup t∈[0,T ) (T -t) 2-θ ❊ D 2 u(t, X t ) 2 < ∞.
Theorems 1 and 2 generalize results obtained in [START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF] and [START_REF] Geiss | Interpolation and approximation in L2(γ)[END_REF]. We see that the fractional smoothness index θ measures exactly the growth rate of the derivatives of the associated PDE solved by u (see question (Q1) in the introduction).

The two above theorems are also valid if u is computed using the risk-

neutral measure ◗ (i.e. u ◗ (t, x) = ❊ ◗ (g(X T )|X t = x))
, while the other L 2 -norms are computed under P. For instance, for 0 < θ < 1 the equivalence of (i) and (ii) of Theorem 1 becomes g(X T ) ∈ ❇ θ,P 2,2 if and only if

T 0 (T - t) -θ ❊ P |∇ x u ◗ (t, X t )| 2 dt < ∞
where we have indicated explicitly if the L 2norms or conditional expectations are computed under P or ◗. This property can be established following [START_REF] Gobet | L2-time regularity of BSDEs with irregular terminal functions[END_REF] and the proof of [START_REF] Gobet | The tracking error rate of the Delta-Gamma hedging strategy[END_REF]Lemma 7]. This accommodates well the fact that the price functions are usually computed under the risk-neutral measure, while hedging is made under the historical probability (see question (Q2) in the introduction).

Proof (Simplified proof of Theorem 2).

We sketch the proof in the simple case where X = W is a linear Brownian motion, d = 1 and θ ∈ (0, 1). First, (u(t,

W t ) = ❊(g(W T )|F t )) t≤T is
a martingale in L 2 (P). In addition, for any fixed 0 < δ < T the processes (∇ x u(t, W t )) t≤T -δ and (D 2 u(t, W t )) t≤T -δ are L 2 (P)-martingales. This property is obtained by checking that ∇ x u and D 2 u both solve the parabolic heat equation and that certain integrability assumptions are satisfied. Then by Itô's formula, one obtains for 0 ≤ s ≤ t < T that

g(W T ) -u(t, W t ) = T t ∇ x u(s, W s )dW s , (1.2) ∇ x u(t, W t ) -∇ x u(s, W s ) = t s D 2 u(r, W r )dW r . (1.3) From the Itô isometry, one deduces from (1.2) that ❊|g(W T ) -u(t, W t )| 2 = T t ❊|∇ x u(s, W s )| 2 ds and it follows that (ii) ⇒ (i). Similarly from (1.3) one obtains ❊|∇ x u(t, W t )| 2 ≤ 2❊|∇ x u(0, W 0 )| 2 + 2 t 0 ❊|D 2 u(r, W r )| 2 dr which proves (iii) ⇒ (ii). Finally, we show (i) ⇒ (iii). Standard computations give that (D 2 u)(t, W t ) = D 2 z | g(x) e -(x-z) 2 2(T -t) 2π(T -t) dx z=Wt = | g(x) (x -z) 2 -(T -t) (T -t) 2 e -(x-z) 2 2(T -t) 2π(T -t) dx z=Wt = ❊ g(W T ) (W T -W t ) 2 -(T -t) (T -t) 2 |F t = ❊ [g(W T ) -❊(g(W T )|F t )] (W T -W t ) 2 -(T -t) (T -t) 2 |F t which implies that D 2 u(t, W t ) L2(P) ≤ W 2 1 -1 L2(P) T -t g(W T ) -❊(g(W T )|F t ) L2(P)
so that we are done.

Connection to real interpolation

Let us connect Definition 1 to the classical notion of fractional smoothness which also explains the notation we have used. In particular, this connection will make clear the difference between ❇ θ 2,∞ and ❇ θ 2,2 . Definition 2 ([BL76, BS88]). Assume a couple of Banach spaces (E 0 , E 1 ) so that E 1 is continuously embedded into E 0 . Given x ∈ E 0 and 0 < λ < ∞, the K-functional is given by

K(x, λ; E 0 , E 1 ) := inf{ x 0 E0 + λ x 1 E1 : x = x 0 + x 1 }.
Moreover, given 0 < θ < 1 and 1 ≤ q ≤ ∞ we define the real interpolation norm

x θ,q := λ -θ K(x, λ; E 0 , E 1 ) Lq((0,∞), dλ λ ) and the space (E 0 , E 1 ) θ,q := {x ∈ E 0 : x θ,q < ∞}.

With our setting (E 1 is continuously embedded into E 0 ) we obtain the following lexicographical ordering of the interpolation spaces:

E 1 ⊆ (E 0 , E 1 ) θ,p ⊆ (E 0 , E 1 ) θ,q ⊆ (E 0 , E 1 ) η,r ⊆ E 0 for all 0 < η < θ < 1, 1 ≤ p ≤ q ≤ ∞ and all 1 ≤ r ≤ ∞.
We apply this concept to the analysis on the Wiener space, which needs to introduce some standard notation (see [Nu06, Sections 1.1 and 1.2]). Let H be a separable real Hilbert space with the scalar product denoted by ., . H and (M, Σ, µ) be a complete probability space. We assume an isonormal family g = {g h : h ∈ H} of centered Gaussian random variables, i.e.

❊ µ (g h g k ) = h, k H for all h, k ∈ H, and that Σ is the completed σ-field generated by the random variables {g h : h ∈ H}.

For each n ≥ 1, we denote by H n the closed linear subspace of L 2 (µ) generated by the random variables {H n (g h ) : h ∈ H, h H = 1} where

H n (x) = (-1) n √ n! e x 2 2 d n dx n (e -x 2 2 ), (1.4)
i.e. the n-th Hermite polynomial. H 0 is the set of constants. H n is the so-called Wiener chaos of order n and we define by J n : L 2 (µ) → L 2 (µ) the orthogonal projection onto H n . The following orthogonal decomposition is known as the Wiener chaos decomposition:

L 2 (µ) = ∞ n=0 H n .
Now, we are in a position to define the Malliavin Sobolev space and Malliavin Besov space.

Definition 3. The Malliavin Sobolev space ❉ 1,2 (µ) ⊆ L 2 (µ) is given by

❉ 1,2 (µ) :=    Z ∈ L 2 (µ) : Z ❉1,2(µ) := ∞ n=0 (n + 1) J n Z 2 L2(µ) 1 2 < ∞    .
Moreover, given 0 < θ < 1 and 1 ≤ q ≤ ∞, we define the Malliavin Besov space

❇ θ 2,q (µ) := (L 2 (µ), ❉ 1,2 (µ)) θ,q
of fractional smoothness θ with fine parameter q.

We use this construction in the case that H = ℓ d

g h (x) := x, h for x ∈ M = | d and h ∈ H = ℓ d 2 .
To make the connection between the definitions of ❇ θ 2,q and ❇ θ 2,q (γ d ) for q ∈ {2, ∞} we let, as before, (W t ) t∈[0,1] be the standard d-dimensional Brownian motion on (Ω, F, P, (F t ) t∈[0,1] ). Then we have

Theorem 3 ([GH07, Corollary 2.3]). For 0 < θ < 1, 1 ≤ q ≤ ∞, and g ∈ L 2 (γ d ) one has g ❇ θ 2,q (γ d ) ∼ c g L2(γ d ) + (1 -t) -θ 2 M 1 -M t L2(P) Lq([0,1), dt 1-t )
where M t := ❊ (g(W 1 )|F t ) and c ≥ 1 depends on (θ, q) only.

Applying this theorem to q = ∞ gives that

g ❇ θ 2,∞ (γ d ) ∼ c g L2(γ d ) + sup 0≤t≤1 (1 -t) -θ 2 M 1 -M t L2(P) , whereas q = 2 gives that g ❇ θ 2,2 (γ d ) ∼ c g L2(γ d ) + 1 0 (1 -t) -1-θ M 1 -M t 2 L2(P) dt 1 2
which brings us back to Definition 1.

Multi-dimensional Black-Scholes-Samuelson model.

This is a log-normal model which dynamics on the price and the log-price can be written as

dS i t = S i t   d j=1 σ ij dW j t + µ i dt   , 1 ≤ i ≤ d, X i t = log(s i 0 ) + d j=1 σ ij W j t + (µ i - 1 2 σ 2 i )t,
where

σ i := j σ 2 ij . Assume that (σ ij ) d i,j=1 is invertible. To the payoff func- tion S → h(S), we associate g(x 1 , ..., x d ) := h   s i 0 e d j=1 σij xj +µi- σ 2 i 2 d i=1   .
From this we see that

g ∈ ❇ θ 2,q (γ d ) if and only if h(S 1 ) ∈ ❇ θ 2,q
for q ∈ {2, ∞} and g ∈ L 2 (γ d ).

Remark 1. In the case θ = 1 we get that

g ∈ ❉ 1,2 (γ d ) if and only if h(S 1 ) ∈ ❇ 1 2,∞
for all g ∈ L 2 (γ d ). This can be checked by using arguments from the proof of [GH07, Corollary 2.3].

Examples

In this section, we provide examples of random variables Z = g(X T ) for which we determine the fractional smoothness.

Example 1 (Lipschitz function). The case, where the fractional smoothness is obvious, is the Lipschitz case. Assume a Lipschitz function g : | d → | with constant L ≥ 0, i.e. |g(x)g(y)| ≤ L|x -y| and assume (SDE). Then one has that

❊ |g(X T ) -❊(g(X T )|F t )| 2 ≤ ❊ |g(X T ) -g(X t )| 2 ≤ L 2 ❊ |X T -X t | 2 ≤ L 2 c 2 (T -t),
using standard estimates on the increments of X. Hence, g(X T ) ∈ ❇ 1 2,∞ . This example includes call and put payoffs, i.e. g(x) = (x-K) + or g(x) = (K -x) + . Exactly the same argument as above yields for θ-Hölder functions g with θ ∈ (0, 1) that g(X T ) ∈ ❇ θ 2,∞ . But the situation is here not as clear as one expects as shown by Example 2. Assume the setting (GBM) and that h θ (x) := (x -K) θ + for some K > 0 and 0 < θ < 1/2. Then it is shown in [GT01, Lemma 2] (under more general assumptions) that ❊|D 2 u(t, X t )| 2 ≤ c(Tt) -3/2+θ so that Theorem 2 gives that

h θ (S T ) ∈ ❇ θ+ 1 2 2,∞ . For 1/2 < θ < 1 one gets h θ (S T ) ∈ ❇ 1 2,∞ .
Example 3 (Binary option). Generally, indicator functions yield to a fractional smoothness of order 1 2 . In the case X = W , d = 1 and g

(x) = 1 [L,∞) (x) with L ∈ | one has u(t, x) = P(x + W T -W t ≥ L) = N x -L √ T -t , ∇ x u(t, x) = 1 2π(T -t) exp - (x -L) 2 2(T -t) , so that ❊|∇ x u(t, W t )| 2 ∼ c 1 √ T -t and g(W T ) ∈ ❇ 1 2
2,∞ because of Theorem 2. This can be extended to the (SDE) case as follows: Our assumption guarantees that X has a transition density Γ such that

Γ (s, x; t, y) ≤ κ 2π(t -s) e -1 2 (x-y) 2 κ(t-s) = κγ κ(t-s) (x -y)
for some κ > 0 and all 0 ≤ s < t ≤ T , where γ t is the Gaussian density with zero expectation and variance t (see [START_REF] Friedman | Partial Differential Equations of Parabolic Type[END_REF]). Then we can compute that

❊ 1 [L,∞) (X T ) -❊ 1 [L,∞) (X T )|F t 2 ≤ ❊ 1 [L,∞) (X T ) -1 [L,∞) (X t ) = P(X T < L ≤ X t ) + P(X t < L ≤ X T ) ≤ κ 2 [P(W κT < L -x 0 ≤ W κt ) + P(W κt < L -x 0 ≤ W κT )] ≤ c √ T -t where X 0 = x 0 so that 1 [L,∞) (X T ) ∈ ❇ 1 2
2,∞ . The application in financial mathematics is done via S t = e Xt which gives, for a positive strike K > 0,

1 {S T ≥K} = 1 {X T ≥log K} ∈ ❇ 1 2 2,∞ .
In our context the fractional smoothness of jump functions (under different assumptions) was considered in [START_REF] Gobet | Discrete time hedging errors for options with irregular pay-offs[END_REF][START_REF] Geiss | Quantitative approximation of certain stochastic integrals[END_REF][START_REF] Geiss | On approximation of a class of stochastic integrals and interpolation[END_REF]. In certain multidimensional settings one can deduce for g(x) = 1 {x1≥K1,•••,x d ≥K d } (or variants of it) the same fractional smoothness from the 1-dimensional case. Finally, the indicator function g

(x) = 1 D (x) of a C 2 -domain D also leads to g(X T ) ∈ ❇ 1 2 2,∞ (see [GM05, Proposition 1.2]).
Example 4 (an extreme case). By the choice of the previous examples, we emphasize that random variables g(X T ) = h(S T ), usually used in financial applications, belong to a space ❇ θ 2,∞ for θ ∈ (0, 1]. However, it is not true that ∪ θ∈(0,1] ❇ θ 2,∞ = L 2 (P). The following result gives a way to construct g(W 1 ) belonging to L 2 (P) (here W is the linear Brownian motion) but g(W 1 ) / ∈ ❇ θ 2,∞ for all θ ∈ (0, 1]:

Proposition 2 ([GH07]). Let 0 < θ < 1, g = ∞ k=0 α k H k ∈ L 2 (γ 1 )
, where (H k ) k≥0 is the orthogonal basis of Hermite polynomials defined in (1.4). Then

g(W 1 ) ∈ ❇ θ 2,∞ if and only if sup 0≤t<1 (1 -t) 1-θ ∞ k=1 kt k-1 α 2 k < ∞.
Approximation properties as described in Section 1.5.2 for g with g(W 1 ) ∈ L 2 (P) \ 0<θ≤1 ❇ θ 2,∞ were studied in [START_REF] Hujo | Is the approximation rate for European pay-offs in the Black-Scholes model always 1/ √ n?[END_REF] and [START_REF] Seppälä | Optimal approximation rate of certain stochastic integrals[END_REF].

Applications

In this section we discuss some applications in stochastic finance which lead us to the fractional smoothness as introduced above. As mentioned at the beginning, a central role is played by the tracking error that arises when discrete time hedging is used, instead of a continuous time strategy. For the sake of convenience, we briefly recall the notation:

• the option payoff at maturity T is Z = h(S T );

• the fair price function is

H(t, x) = ❊ ◗ (h(S T )|S t = x);
• the n re-balancing dates are defined by a deterministic time-net τ = (t i )

n i=0 with 0 = t 0 < t 1 < • • • < t n-1 < t n = T ; • the resulting tracking error process C(Z, τ ) = (C t (Z, τ )) t∈[0,T ] is given by C t (Z, τ ) := ❊ ◗ (Z|F t ) -❊ ◗ Z - n-1 i=0 ∇ x H(t i , S ti ) • (S ti+1∧t -S ti∧t ).

Weak limits of error processes

Weak limits of stochastic processes have been intensively studied in the literature; see, for instance, [START_REF] Kurtz | Weak limit theorems for stochastic integrals and stochastic differential equations[END_REF][START_REF] Jacod | On continuous conditional Gaussian martingales and stable convergence in law[END_REF][START_REF] Jacod | Limit theorems for stochastic processes[END_REF]. For the particular problem of the weak convergence of the tracking error the reader is referred to [Roo80, GT01, HM05, GT09]. To formulate our results, we let W = ( W t ) t≥0 be a standard Brownian motion starting at zero defined on some auxiliary probability space, where we may and do assume that all paths are continuous. In the following =⇒ C[0,s] stands for the weak convergence in C[0, s] for some s > 0.

In this paragraph we assume that T = 14 and that S is the standard geometric Brownian motion, i.e. the setting of (GBM) and P = ◗. The following result is the starting point of this section:

Theorem 4 ([GT01]
). Let τ n = (i/n) n i=0 be the equidistant time-nets and let Z := 1 [K,∞) (S 1 ) be the payoff of a digital option with strike price K > 0. Then one has that

√ nC 1 (Z, τ n ) =⇒ W 1 2 1 0 S 2 t ∂ 2 H ∂x 2 (t,St) 2 dt
where =⇒ denotes the weak convergence as n goes to infinity.

The remarkable fact is that the weak limit is not square-integrable. In the following we describe a way to increase the integrability of the weak limit. This is of particular interest for risk management purposes, as a higher integrability gives better tail-estimates. The idea is to use adapted time-nets that are more concentrated close to maturity. They are defined as follows: Given a parameter θ ∈ (0, 1], we define the nets τ n,θ by

t n,θ k := 1 -1 - k n 1 θ
.

For θ = 1 we have the equidistant time-nets, i.e. t n,1 k = k n . Now we have

Theorem 5 ([GT09]). Let 0 < θ ≤ 1, Z = h(S 1 ) ∈ L 2 (P) and 0 ≤ s < 1. Then ( √ nC t (Z, τ n,θ )) t∈[0,s] =⇒ C[0,s] W t 0 (1-r) 1-θ 2θ S 2 r ∂ 2 H ∂x 2 (r,Sr) 2 dr t∈[0,s]
.

Moreover, the following assertions are equivalent:

(i) One has h(S 1 ) ∈ ❇ θ 2,2 for 0 < θ < 1 or h(S 1 ) ∈ ❇ 1
2,∞ for θ = 1. (ii) On some stochastic basis there exists a continuous square-integrable martingale

M = (M t ) t∈[0,1] such that √ nC(Z, τ n,θ ) =⇒ C[0,1] M . (iii) For A := 1 0 (1 -t) 1-θ 2θ S 2 t ∂ 2 H ∂x 2 (t, S t ) 2 dt one has that ❊A < ∞ and √ nC(Z, τ n,θ ) =⇒ C[0,1] W 1 {A<∞} t 0 (1-r) 1-θ 2θ S 2 r ∂ 2 H ∂x 2 (r,Sr) 2 dr t∈[0,1]
.

The theorem above gives us one way to consider the L p -setting for 2 ≤

p < ∞. Given a differentiable function ψ : (0, ∞) → | we let (Aψ)(x) := xψ ′ (x) -ψ(x).
In the following AH(t, x) means that A acts on the x-variable of the function H(t, x).

Definition 4. For h(S 1 ) ∈ L 2 (P), 0 < θ < 1, and 0 ≤ t < 1 we let

D S,θ t h(S 1 ) := 1 -θ 2 1 0 (1 -u) -1+θ 2 [AH(u ∧ t, S u∧t ) -AH(0, S 0 )]du.
For θ = 1 and t ∈ [0, 1) we let D S,1 t h(S 1 ) := AH(t, S t ) -AH(0, S 0 ).

The process D S,θ h(S 1 ) = (D θ t h(S 1 )) t∈[0,1) is a quadratic integrable martingale on the half open time interval [0, 1). Using the Riemann-Liouville operator of partial integration the process D S,θ h(S 1 ) can be interpreted as a fractional differentiation of order θ in x (see [START_REF] Geiss | Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces[END_REF]). The point of the construction of D S,θ h(S 1 ) is that we may have L p -singularities of S t ∂H ∂x (t, S t ) as t ↑ 1 whereas D S,θ h(S 1 ) remains L p -bounded.

Theorem 6 ([GT09]

). For 2 ≤ p < ∞, 0 < θ ≤ 1, and Z = h(S 1 ) ∈ L 2 (P) the following assertions are equivalent:

(i) On some stochastic basis there exists a continuous L p (P)-integrable martingale M such that

√ nC(Z, τ n,θ ) =⇒ C[0,1] M .
(ii) The martingale D S,θ h(S 1 ) is bounded in L p (P).

L 2 -estimates of the tracking error

In this section we work in the 1-dimensional martingale case assuming (SDE) with σ(t, x) = σ(x) and b(t, x) = -1 2 σ 2 (x) (meaning P = ◗). The payoff function h is polynomially bounded and the option maturity is T > 0. We remind the reader about the time-nets τ n,θ given by

t n,θ k := T 1 -1 - k n 1 θ
and that for θ = 1 we obtain the equidistant nets. Let us first check what quadratic hedging error one can expect at all if the portfolio is re-balanced n-times. The answer is the rate 1/ √ n as shown by

Theorem 7 ([GG04, Theorem 2.5]). Assume that there are no constants

c 0 , c 1 ∈ | such that h(S T ) = c 0 + c 1 S T a.s. Then inf n=1,2,... 0=t 0 <•••<tn=T n 1 2 C(h(S T ), (t k ) n k=0 ) L2(P) > 0
where the infimum is taken over deterministic time-nets.

This was extended to the case of random time-nets in [START_REF] Geiss | On an approximation problem for stochastic integrals where random time nets do not help[END_REF] in the case of the geometric Brownian motion. Now we continue with the case of equidistant time-nets which are often used in discretizations.

Equidistant time-nets.

Here a starting point is the following result of Zhang: This is the result one would expect: Given a Lipschitz payoff, the L 2 -rate of the error is 1/2 for equidistant nets. But this is not the case in general as shown in

Theorem 9 ([GT01, Theorem 1]). For h(x) = 1 [K,∞) (x) for some K > 0 we have that lim n n 1 4 C(h(S T ), τ n,1 ) L2(P) ∈ (0, ∞).
This means that the L 2 -approximation rate for the binary option is n 1/4 if one uses equidistant nets. The two above results also hold true for appropriate ◗ = P (i.e. S is not martingale) where the outer L 2 -norm is computed w.r.t. the historical probability P (cf. the remarks after Theorem 2).

Theorems 8 and 9 lead naturally to two questions: What is the reason for the rate 1/4 and, secondly, can one improve the rate 1/4? Both questions can be answered by the usage of the concept of fractional smoothness.

Theorem 10 ([GG04, Theorems 2.3 and 2.8]). For 0 < θ ≤ 1 and a polynomially bounded h : (0, ∞) → | the following assertions are equivalent:

(i) h(S T ) ∈ ❇ θ 2,∞ . (ii) sup n n θ 2 C(h(S T ), τ n,1 ) L2(P) < ∞.
In particular, it turns out that h(S T ) ∈ ❉ 1,2 if and only if

sup n n 1 2 C(h(S T ), τ n,1 ) L2(P) < ∞, see [GG04, Theorem 2.6],
where ❉ 1,2 is the Malliavin Sobolev space obtained from the construction in Section 1.3 with H = L 2 [0, T ] and g h := T 0 h(t)dW t .

For the binary option one has in Theorem 10 that θ = 1/2 (cf. Example 3 in Section 1.4). This recovers the rate 1/4 obtained in Theorem 9.

Non equidistant time-nets.

Next we show how to obtain the optimal rate n 1/2 by a suitable choice of the trading dates (see question (Q4) in Section 1.1). We can combine [GG04, Lemmas 3.2 and 5.3] and [GH07, Lemma 3.8] to get Theorem 11. For 0 < θ ≤ 1 and a polynomially bounded h : (0, ∞) → | the following assertions are equivalent:

(i) T 0 (T -t) 1-θ ❊ S 2 t ∂ 2 H ∂x 2 (t, S t ) 2 dt < ∞. (ii) sup n n 1 2 C(h(S T ), τ n,θ ) L2(P) < ∞.
For 0 < θ < 1 (and at least a bounded h) the condition Theorem 11(i) is equivalent to

(i') h(S T ) ∈ ❇ θ 2,2
which can be checked by using Theorem 1. For the binary option this gives that sup

n n 1 2 C(1 [K,∞) (S T ), τ n,η ) L2(P) < ∞
for any strike K > 0 and 0 < η < 1/2.

For the next two theorems we assume that T = 1, that S t = e Wt-t 2 and that h might be general, i.e. not polynomially bounded. The formulation of Theorem 11 in the language of the interpolation spaces introduced in Section 1.3 gives Theorem 12 ([GH07, Theorem 3.2]). For 0 < θ ≤ 1 and h(S 1 ) ∈ L 2 (P) the following assertions are equivalent:

(i) h(e •-(1/2) ) ∈ ❇ θ 2,2 (γ 1 ) if 0 < θ < 1 and h(e •-(1/2) ) ∈ ❉ 1,2 (γ 1 ) if θ = 1. (ii) sup n n 1 2 C(h(S 1 ), τ n,θ ) L2(P) < ∞.
And Theorem 10 can be extended in this context to the full scale of real interpolation spaces as Theorem 13 ([GH07, Theorem 3.5]). For 1 ≤ q ≤ ∞, 0 < θ < 1 and h(S 1 ) ∈ L 2 (P) the following assertions are equivalent:

(i) h(e •-(1/2) ) ∈ ❇ θ 2,q (γ 1 ). (ii) n θ 2 -1 q a n ∞ n=1 ℓq < ∞ for a n := C(h(S 1 ), τ n,1 ) L2(P) .
Concluding remarks (i) The higher dimensional case for X was considered in the literature as well. Roughly speaking, one can analogously obtain upper bounds, however precise lower bounds as in the one-dimensional case are still missing. This is due to the fact that a characterization of the L 2 -error proved in [Gei02, Theorem 4.4] and [GG04, Lemma 3.2] is missing for higher dimensions. However, after Zhang [START_REF] Zhang | Couverture approchée des options Européennes[END_REF] started with the regular case, Temam [START_REF] Temam | Analysis of error with Malliavin calculus: application to hedging[END_REF] extended results from [START_REF] Gobet | Discrete time hedging errors for options with irregular pay-offs[END_REF] to higher dimensions and Hujo [START_REF] Hujo | On discrete time hedging in d-dimensional option pricing models[END_REF] used non-uniform time-nets to improve the approximation rates for certain irregular payoffs to the optimal rate 1/ √ n in this setting. (ii) Seppälä [START_REF] Seppälä | Optimal approximation rate of certain stochastic integrals[END_REF] found a criterion to characterize under certain conditions that there is a constant c > 0 such that inf

τ =(t i ) n i=0 0=t 0 <•••<tn=1 C(h(S 1 ), τ ) L2(P) ≤ c √ n
where deterministic time-nets are taken. It should be noted that one has a non-linear approximation problem as the time-nets may change for fixed n from payoff to payoff h. (iii) In the above discussion, the time-nets τ are deterministic. Alternatively, one can allow the time-nets to be stochastic and adapted. This issue has been handled by [START_REF] Martini | Variance optimal hedging in the Black-Scholes model for a given number of transactions[END_REF] using optimal stopping tools. The estimation of convergence rates is an open question. However, it was shown in [START_REF] Geiss | On an approximation problem for stochastic integrals where random time nets do not help[END_REF] that the random time-nets do not improve the best possible approximation rate 1/ √ n in the case (GBM) when in the n-th approximation a sequence of n stopping times is used. (iv) Similar studies can be performed when studying the Delta-Gamma hedging strategies. Instead of hedging the payoff using only the asset, we use other traded options written on the same asset. For a one-dimensional asset, if the price of the additional option is (P (t, S t )) 0≤t≤T , the numbers of options P and assets to hold at time t i are respectively equal to

δ P ti := ∂ 2 S H(t i , S ti ) ∂ 2 S P (t i , S ti ) and δ S ti := ∂ S Ht i , S ti ) - ∂ 2 S H(t i , S ti ) ∂ 2 S P (t i , S ti ) ∂ S P (t i , S ti ).
In [GM10b, Theorem 6], considering a multi-dimensional Black-Scholes model, it is established that for an exponentially bounded payoff such that g(X T ) ∈ ❇ θ 2,∞ for some 0 < θ < 1, the use of equidistant timenets leads to the same convergence rate 1/n θ/2 as for the delta hedging strategy. On the contrary, the use of non equidistant time-nets τ n,η with 0 < η < θ/2 enables us to obtain the improved convergence rate 1/n.

Further developments

Backward stochastic differential equations

Makhlouf and the second author applied in [START_REF] Gobet | L2-time regularity of BSDEs with irregular terminal functions[END_REF] the concept of fractional smoothness to backward stochastic differential equations of the type Y t = g(X T ) + T t f (s, X s , Y s , Z s )ds -T t Z s dW s where X = (X t ) t∈[0,T ] is our forward diffusion and the generator f is continuous in its four arguments, continuously differentiable in (x, y, z) with uniformly bounded derivatives. These equations are particularly useful in stochastic finance, since they allow to take into account market frictions and constraints (we refer to [START_REF] Karoui | Backward stochastic differential equations in finance[END_REF] for a more complete account on this subject).

Solving numerically this type of equation is a challenging issue since it concerns a non-linear problem (due to the generator f ), generally defined in a multi-dimensional setting. One possible approach consists in approximating the BSDE using a discrete-time dynamic programming equation (see [START_REF] Zhang | A numerical scheme for BSDEs[END_REF][START_REF] Bouchard | Discrete time approximation and Monte Carlo simulation of backward stochastic differential equations[END_REF] LGW06] among others). One of the main error contribution is related to the L 2 -regularity on Z, defined by

E(Z, τ ) = n i=1 ti ti-1 Z t -Z ti-1 2 L2(P) dt.
If f were equal to 0, then the Z-component is given by z t = ∇ x u(t, X t )σ(t, X t ) where u(t, x) = ❊(g(X T )|X t = x). Studying the L 2 -regularity of z is thus very similar to the analysis of the tracking error presented in Section 1.5. Additionally, using BSDE techniques, one can prove explicit upper bounds for the difference Zz Theorem 14 ([GM10a, Corollary 14]). Assume (SDE) and g(X T ) ∈ ❇ θ 2,∞ for 0 < θ ≤ 1. Then, for some c > 0, one has that

|Z t -z t | ≤ c T t ❊ |g(X T ) -❊(g(X T )|F s )| 2 |F t T -s ds + c(T -t), ❊ |Z t -z t | 2 ≤ c(T -t) θ .
Taking advantage of this approximation result close to the time singularity, we can prove that the estimate of E(z, τ ) (linear case) transfers to E(Z, τ ) (non-linear case) and get

Theorem 15 ([GM10a, Theorem 21]). Assume (SDE), g(X T ) ∈ ❇ θ 2,∞ and that 0 < η < θ < 1 or η = θ = 1. Then one has that E(Z, τ n,η ) ≤ c n .

In [START_REF] Geiss | Generalized fractional smoothness and Lp-variation of BSDEs with non-Lipschitz terminal condition[END_REF], extensions of the above in different directions are discussed.

Lévy processes

An extension of the results of [START_REF] Geiss | Interpolation and approximation in L2(γ)[END_REF] to Lévy Processes is done by C. Geiss and Laukkarinen in [START_REF] Geiss | A note on Malliavin fractional smoothness and discrete time hedging in the Lévy model[END_REF]. Moreover, Tankov and Brodén proved in [START_REF] Tankov | Errors from discrete hedging in exponential Lévy models: the L2 approach[END_REF] results along the line of [START_REF] Gobet | Discrete time hedging errors for options with irregular pay-offs[END_REF].

Multigrid Monte-Carlo Methods

In the context of Multigrid Monte-Carlo Methods it turned out that the concept of fractional smoothness is useful as well. The reader is referred to the papers of Avikainen [START_REF] Avikainen | On irregular functionals of SDEs and the Euler scheme[END_REF][START_REF] Avikainen | On generalized bounded variation and approximation of SDEs[END_REF].

2

  and M = | d , Σ is the completion of the Borel σ-algebra on | d and µ = γ d is the d-dimensional standard Gaussian measure. The family of Gaussian random variables is given by

1 2

 1 Theorem 8 ([Zha99, Theorem 2.4.1]). Assume that h : | → | is a Lipschitz function. Then we have that lim n n C(h(S T ), τ n,1 ) L2(P) ∈ [0, ∞).

Here again, the boundedness assumptions on g can be weakened and we refer to the original papers.

With T = 1 we are in accordance with the quoted literature that used Hermite polynomials. Of course, we could do a re-scaling to T > 0 afterwards.