L A Abbas-Turki

B Lapeyre

AMERICAN OPTIONS BASED ON MALLIAVIN CALCULUS AND NONPARAMETRIC VARIANCE REDUCTION METHODS

Keywords: American Options, Malliavin Calculus, Monte Carlo, GPU AMS subject classifications. 60G40, 60H07

This paper is devoted to pricing American options using Monte Carlo and the Malliavin calculus. Unlike the majority of articles related to this topic, in this work we will not use localization fonctions to reduce the variance. Our method is based on expressing the conditional expectation E[f (St)/Ss] using the Malliavin calculus without localization. Then the variance of the estimator of E[f (St)/Ss] is reduced using closed formulas, techniques based on a conditioning and a judicious choice of the number of simulated paths. Finally, we perform the stopping times version of the dynamic programming algorithm to decrease the bias. On the one hand, we will develop the Malliavin calculus tools for exponential multi-dimensional diffusions that have deterministic and no constant coefficients. On the other hand, we will detail various nonparametric technics to reduce the variance. Moreover, we will test the numerical efficiency of our method on a heterogeneous CPU/GPU multi-core machine.

Introduction and objectives. To manage CPU (Central Processing Unit) power dissipation, the processor makers have oriented their architectures to multicores. This switch in technology led us to study the pricing algorithms based on Monte Carlo (MC) for multi-core architectures using CPUs and GPUs (Graphics Processing Units) in [START_REF] Abbas-Turki | High dimensional pricing of exotic european contracts on a GPU cluster, and comparison to a CPU cluster[END_REF] and [START_REF] Abbas-Turki | American options pricing on multicore graphic cards[END_REF]. In the latter articles we basically studied the impact of using GPUs instead of CPUs for pricing European options using MC and American options using the Longstaff and Schwartz (LS) algorithm [START_REF] Longstaff | Valuing American options by simulation: A simple leastsquares approach[END_REF]. The results of this study proves that we can greatly decrease the execution time and the energy consumed during the simulation.

In this paper, we explore another method to price American Options (AO) and which is based on MC using the Malliavin calculus (MCM). Unlike the LS method that uses a regression phase which is difficult to parallelize according to [START_REF] Abbas-Turki | American options pricing on multicore graphic cards[END_REF], the MCM is a square 1 Monte Carlo method which is more adapted to multi-cores than the LS method. Moreover, using MCM without localization does not depend on parametric regression, we can increase the dimensionality of the problem without any constraints except for adding more trajectories if we aim at more accurate results.

American contracts can be exercised at any trading date until maturity and their prices are given, at each time t, by [START_REF] Glasserman | Monte Carlo Methods in Financial Engineering[END_REF] P t (x) = sup θ∈Tt,T E t,x e -r(θ-t) Φ(S θ ) , (0.1) where T t,T is the set of stopping times in the time interval [t, T ], E t,x is the expectation associated to the risk neutral probability knowing that S t = x and r and φ(S t ) are respectively the risk neutral interest rate and the payoff of the contract.

With Markovian models (which is the case in this article), to evaluate numerically the price (0.1), we first need to approach stopping times in T t,T with stopping times taking values in the finite set t = t 0 < t 1 < ... < t n = T . When we do this approximation, pricing American options can be reduced to the implementation of the dynamic programming algorithm [START_REF] Glasserman | Monte Carlo Methods in Financial Engineering[END_REF]. Longstaff and Schwartz consider the stopping times formulation of the dynamic programming algorithm which allows them to reduce the bias by using the actual realized cash flow. We refer the reader to [START_REF] Clément | An analysis of a least squares regression algorithm for American option pricing[END_REF] for a formal presentation of the LS algorithm and details on the convergence. In (0.2), we rewrite the dynamic programming principle in terms of the optimal stopping times τ k , for each path, as follows

τ n = T, ∀k ∈ {n -1, ..., 0}, τ k = t k 1 A k + τ k+1 1 A c k , (0.2)
where the set A k = {Φ(S t k ) > C(S t k )} and C(S t k ) is the continuation value whose expression is given by C(S t k ) = E e -r(t k+1 -t k ) P t k+1 (S t k+1 ) S t k . (0.3) Thus, to evaluate the price (0.1), we need to estimate C(S t k ). Algorithms devoted to American pricing and based on Monte Carlo, differ essentially in the way they estimate and use the continuation value (0.3). For example the authors of [START_REF] Tsitsiklis | Regression methods for pricing complex American-style options[END_REF] perform a regression to estimate the continuation value, but unlike [START_REF] Longstaff | Valuing American options by simulation: A simple leastsquares approach[END_REF], they use C(S t k ) instead of the actual realized cash flow P t k+1 (S t k+1 ) to update the price in (0.2). Other methods use the Malliavin Calculus [START_REF] Bally | Pricing American options by Monte Carlo methods using a Malliavin calculus approach[END_REF] or the quantization method [START_REF] Bally | A quantization algorithm for solving multidimensional discrete-time optimal stopping problems[END_REF] for C(S t k ) estimation. In [START_REF] Abbas-Turki | American options pricing on multicore graphic cards[END_REF], we implement the LS method because it is gaining widespread use in the financial industry. As far as this work is concerned, we are going to implement MCM but unlike [START_REF] Bally | Pricing American options by Monte Carlo methods using a Malliavin calculus approach[END_REF] we use the induction (0.2) for the implementation and we reduce the variance differently, without using localization.

Formally speaking, if r = 0, we can rewrite the continuation using the Dirac measure ε x (•) at the point x C(x) = E P t k+1 (S t k+1 )ε x (S t k )

E (ε x (S t k )) = E P t k+1 (S t k+1 )1 St k ≥x (S t k )π t k ,t k+1 E 1 St k ≥x (S t k )π t k ,t k+1 . (0.4)
The second equality is obtained using the Malliavin calculus and we will specify, in section 2 expression (2.3), the value of π t k ,t k+1 by an integration by part argument for the Multi-dimensional Exponential Diffusions with deterministic Coefficients (MEDC) model

dS t = S t σ(t)dW t , S 0 = y,
in the case of deterministic non-constant triangular matrix σ(t) and when σ ij (t) = σ ij δ(ij) with a fixed constant σ ij (The latter case will be used as a benchmark). Instead of simulating directly the last term in (0.4), in section 3 we project 1 St k ≥x (S t k )π t k ,t k+1 using a conditioning as follows

C(x) = E P t k+1 (S t k+1 )E 1 St k ≥x (S t k )π t k ,t k+1 { t k+1 0 σ ij (u)dW j u } 1≤j≤i≤d E E 1 St k ≥x (S t k )π t k ,t k+1 { t k+1 0 σ ij (u)dW j u } 1≤j≤i≤d . (0.5)
Then, in section 4, we estimate (0.5) by Monte Carlo simulation and we use the approximation

C(x) ≈ 1 N ′ N ′ l=1 P l t k+1 (S t k+1 )h(x, { t k+1 0 σ ij (u)dW j u } l 1≤j≤i≤d ) 1 N N l=1 h(x, { t k+1 0 σ ij (u)dW j u } l 1≤j≤i≤d ) , (0.6) h(x, {y ij } j≤i ) = E(1 St k ≥x (S t k )π t k ,t k+1 { t k+1 0 σ ij (u)dW j u } 1≤j≤i≤d = {y ij } 1≤j≤i≤d
) and N = N ′ . Thus, in section 4, we provide another method to accelerate the convergence based on a choice of the appropriate relation between N and N ′ that reduces the variance of the quotient (0.6). Note that, even if one can reduce the variance by an "appropriate" control variable, we choose here not to implement this kind of method because it is not standard for American options.

In the last section, on the one hand, we provide the numerical result comparison of LS and MCM. On the other hand, we study the results of using the two variance reduction methods (0.5) and (0.6). Finally, we test the parallel capabilities of MCM on a desktop computer that has the following specifications: Intel Core i7 Processor 920 with 9GB of tri-channel memory at frequency 1333MHz. It also contains one NVIDIA GeForce GTX 480.

Let us begin with section 1 in which we establish the notations, the Malliavin calculus tools and the model used.

1. Notations, hypothesis and key tools. Let T be the maturity of the American contract, (Ω, F , P ) a probability space on which we define an d-dimensional standard Brownian motion W = (W 1 , ..., W d ) and F = {F s } s≤T the P -completion of the filtration generated by W until maturity. Moreover, we denote by {F i,...,d s } s≤t the P -completion of the filtration generated by (W i , ..., W d ) until the fixed time t ∈ [0, T ]. The process S t models the price of a vector of assets S 1 t , ..., S d t which constitute the solution of the following stochastic differential equation ( ' is the transpose operator)

dS i t S i t = (σ i (t)) ′ dW t , S i 0 = z i , i = 1, .., d, (1.1)
where σ(t) = {σ ij (t)} 1≤i,j≤d is a deterministic triangular matrix ({σ ij (t)} i<j = 0). We suppose that the matrix σ(t) is invertible, bounded and uniformly elliptic which insures the existence of the inverse matrix ρ(t) = σ -1 (t) and its boundedness.

We choose the dynamic (1.1) because it is largely used for equity models, HJM interest rate models and variance swap models. Moreover, the case of σ ij (t) = σ ij δ(ij) (σ ij is a constant) will be easily tested in the section 5. One should notice that in the case where the dynamic of S is given by

dS i t S i t = (σ i ) ′ (t, S t )dW t , S i 0 = z i , i = 1, .., d,
we can use, for instance, the following Euler scheme to reduce this model to the model (1.1)

d log(S i t ) = n-1 k=0 1 t∈[t k ,t k+1 [ (σ i ) ′ (t k , S t k )dW t - 1 2 [(σ i ) ′ σ i ](t k , S t k )dt , S i 0 = z i , i = 1, .., d, t k = kT n .
Note that this scheme does not discretize the process S but the process log(S). Throughout this article, we will use two operators: The Malliavin derivatives D and the Skorohod integral δ and we define them in the same way as in [START_REF] Bally | An elementary introduction to Malliavin calculus[END_REF]. For a fixed m ∈ N, we define the subdivision {t k m } k≤2 m of the finite interval [0, T ] by: t k m = kT /2 m . Then we introduce S(R 2 m ) the Schwartz space of infinitely differentiable and rapidly decreasing functions on R 2 m . Let f ∈ S(R 2 m ), we define the set S m of simple functionals by the following representation

F ∈ S m ⇔ F = f W t 1 m -W t 0 m , W t 2 m -W t 1 m , ..., W t 2 m m -W t 2 m -1 m .
One can prove that S = m∈N S m is a linear and dense subspace in L 2 (Ω) and that the Malliavin derivatives D i F of F ∈ S defined by

D i t F = 2 m -1 k=0 ∂f ∂x i,k W t 1 m -W t 0 m , ..., W t 2 m m -W t 2 m -1 m 1 [t k m ,t k+1 m [ (t)
represents a process of L 2 (Ω × [0, T ]) with values in L 2 ([0, T ]). We associate to S the norm || • || 1,2 defined by

||F || 2 1,2 = E|F | 2 + d i=1 E T 0 (D i t F ) 2 dt.
Finally, the space D 1,2 is the closure of S with respect to this norm and we say that

F ∈ D 1,2 if there exists a sequence F m ∈ S that converges to F in L 2 (Ω) and that D u F m is a Cauchy sequence in L 2 (Ω × [0, T ]
). Now we use the duality property between δ and D to define the Skorohod integral δ. We say that the process

U ∈ Dom(δ) if ∀F ∈ D 1,2 E T 0 U t • D t F dt ≤ C(U )||F || 1,2 ,
where C(U ) is a positive constant that depends on the process U . If U ∈ Dom(δ), we define the Skorohod integral δ(U ) = U t δW t by

∀F ∈ D 1,2 , E F T 0 U t • δW t = E (F δ(U )) = E T 0 U t • D t F dt , (1.2) (•) is the inner scalar product on R d .
Below, we give some standard properties of the operators D and δ:

1. If the process U t is adapted, δ(U ) = U t δW t coincides with the It integral U t dW t . 2. The Chain Rule:

Let F = (F 1 , F 2 , ..., F k ) ∈ (D 1,2
) k and φ : R k → R a continuously differentiable function with bounded partial derivatives. Then φ(F 1 , F 2 , ..., F k ) ∈ D 1,2 and:

D t φ(F 1 , F 2 , ..., F k ) = k i=1 ∂φ ∂x i (F 1 , F 2 , ..., F k )D t F i .

The Integration by Parts:

The IP formula will be extensively used in the next section on the time intervals I = (0, s) and I = (s, t) with s < t ∈]0, T ]: Let F ∈ D 1,2 , an adapted process U ∈ Dom(δ) and that F U ∈ Dom(δ). For each 1 ≤ i ≤ d we have the following equality

I F U u δ i W u = F I U u dW i u - I U u D i u F du. (1.3)
To simplify the notations, we denote H i (S i s ) = H(S i sx i ) for the heaviside function of the difference between the i th stock and the i th coordinate of the positive vector x.

Throughout this article, we will suppose that g ∈ E b is a measurable function with polynomial growth

E b (R d ) = f ∈ M(R d ) : ∃C > 0 and m ∈ N; f (y) ≤ C(1 + |y| d ) m ) ,
where M(R d ) is the set of measurable functions on R d . The elements of the set E b (R d ) satisfy the finiteness of the expectations computed in this article.

2. The expression of the continuation value . The first theorem of this section provides the expression of the continuation (0.3) when using Malliavin calculus for MEDC models. This theorem can be considered as an extension of the log-normal multi-dimensional model detailed in [START_REF] Bally | Pricing American options by Monte Carlo methods using a Malliavin calculus approach[END_REF]. In Theorem 2.4, we provide the expression of Γ k s,t , introduced in Theorem 2.1, without using Malliavin derivatives and this expression can be computed using the relation (2.15). The last theorem is a special case of the first one because we take σ ij (t) = σ ij δ(ij) (σ ij is a constant) that will be used to test numerically our nonparametric variance reduction methods detailed in section 3 and section 4.

Theorem 2.1.

For any s ∈]0, t[, g ∈ E b and x > 0 E g(S t ) S s = x = T s,t [g](x) T s,t [1](x) , (2.1) with T s,t [f ](x) = E f (S t )Γ s,t d k=1 H k (S k s ) S k s , (2.2)
where Γ s,t = Γ 1 s,t and Γ 1 s,t can be computed by the following induction scheme

Γ d s,t = π d,d s,t , for k ∈ {1, ..., d -1}: Γ k s,t = Γ k+1 s,t π k,d s,t - d j=k+1 t 0 D j u Γ k+1 s,t D j u π k,d s,t du with π k,d s,t = 1 + d j=k t 0 ϕ jk (u)dW j u , ϕ jk (u) = 1 s ρ jk (u)1 u∈]0,s[ - 1 t -s ρ jk (u)1 u∈]s,t[ , where ρ is the inverse matrix ρ(u) = σ -1 (u).
From this theorem the value of π t k ,t k+1 in (0.4) is given by

π t k ,t k+1 = Γ t k ,t k+1 d i=1 1 S i t k . (2.3)
To prove Theorem 2.1, we need the following two lemmas which are proved in the appendix. Lemma 2.2 expresses the independence of the sum d i=k ρ ik (u)D i u g(S t ) from the variable u.

Lemma 2.2. For any

u ∈]0, t[ and f ∈ C 1 (R d ) then d i=k ρ ik (u)D i u f (S t ) = S k t ∂ x k f (S t ), ρ(u) = σ -1 (u). (2.4)
The following lemma constitutes with equality (2.4) the two keys of the proof of Theorem 2.1.

Lemma 2.3. For any

I ⊂]0, t[, h ∈ C ∞ b (R), x ∈ R d + and F ∈ D 1,2 , we have E I F D k u h(S k s ) σ kk (u) du = E h(S k s )F d i=k I ρ ik (u)dW i u -E h(S k s ) d i=k I ρ ik (u)D i u F du , (2.5)
where ρ is the inverse matrix ρ(u) = σ -1 (u).

Proof of Theorem 2.1. To prove Theorem 2.1, it is sufficient to prove the following recursive relation on the parameter k for each

h i ∈ C ∞ b (R) and f ∈ C 1 (R d ) ∩ E b (R d ) E f (S t ) d i=1 h ′ i (S i s ) = E f (S t )Γ k+1 s,t k i=1 h ′ i (S i s ) d i=k+1 h i (S i s ) S i s . (2.6)
Indeed, if it is the case then by density of S in L 2 (Ω), one can approximate f (S t ) ∈ L 2 (Ω) by F m ∈ S and pass to the limit on the left and on the right term of (2.6) using Cauchy-Schwarz inequality and the dominated convergence theorem. Let us now consider the singularity due to the heaviside, let φ ∈ C ∞ c (R) be a mollifier function with support equal to [-1, 1] and such that R φ(y)dy = 1, then for any y ∈ R we define

h mk (y) = (H k * φ m )(y) ∈ C ∞ b (R), φ m (y) = m -1 φ(m -1 y).
If the equality (2.6) is correct for any k, then

E f (S t ) d k=1 h ′ mk (S k s ) = E f (S t )Γ s,t d k=1 h mk (S k s ) S k s . (2.7)
On the one hand, h mk (y) converges to H k (y) except at y = 0 and the absolute continuity of the law of S k s ensures that h mk (S k s ) converges almost surely to H k (S k s ). Using the dominated convergence theorem, we prove the convergence of h mk (S k s ) to H k (S k s ) in L p (Ω) for p ≥ 1. By Cauchy-Schwarz inequality, we prove the convergence

E f (S t )Γ s,t d k=1 h mk (S k s ) S k s -→ E f (S t )Γ s,t d k=1 H k (S k s ) S k s . On the other hand, h ′ mk (y k ) = R H k (z k )φ ′ m (y k -z k )dz k = φ m (y k -x k ).
Moreover, we observe that, according to our assumption, the distribution of the vector (S 1 s , ..., S d s , S 1 t , ..., S d t ) admits a density with respect to the Lebesgue mesure on R d ×R d we denote it by p(y, z) with y = (y 1 , ..., y d ) and z = (z 1 , ..., z d ), thus

E f (S t ) d k=1 h ′ mk (S k s ) = R d f (z) R d d k=1 φ m (y k -x k )p(y, z)dy 1 ...dy d dz 1 ...dz d Because R d d k=1 φ m (y k -x k )p(y, z)dy 1 ...dy d converges to p(x, z), we have E f (S t ) d k=1 h ′ mk (S k s ) -→ E f (S t ) d k=1 ε x k (S k s ) ,
which concludes the first part of this proof.

To prove the induction (2.6), we introduce the following notations:

h d k (x) = d i=k h i (x i ) x i , h ′ k (x) = k i=1 h ′ i (x i ), x = (x 1 , ..., x d ).
The case k = d is given by

E f (S t ) h ′ d (S s ) = E 1 s s 0 f (S t ) h ′ d-1 (S s ) D d u h d (S d s ) D d u S d s du = E 1 s s 0 f (S t ) h ′ d-1 (S s ) D d u h d (S d s ) σ dd (u)S d
F = f (S t ) S d s d-1 i=1 h ′ i (S i s ) = f (S t ) S d s h ′ d-1 (S s )
and the fact that h ′ d-1 (S s ) does not depend on the d th coordinate of the Brownian motion yields

E 1 s s 0 f (S t ) h ′ d-1 (S s ) D d u h d (S d s )du σ dd (u)S d s = E F h d (S d s ) 1 s s 0 dW d u σ dd (u) -E h d (S d s ) 1 s s 0 D d u h ′ d-1 (Ss)f (St) S d s du σ dd (u) = E F h d (S d s ) 1 s s 0 dW d u σ dd (u) -E h ′ d-1 (S s )h d (S d s ) 1 s s 0 D d u f (St) S d s du σ dd (u) .
(2.8)

Besides using Lemma 2.2 for the Malliavin derivative of f (S t ), we get for v ∈]s, t[

1 σ dd (u) D d u f (S t ) S d s = 1 S d s σ dd (v) D d v f (S t ) - f (S t ) S d s .
Thus, the value of the last term of (2.8) is given by

E h ′ d-1 (S s )h d (S d s ) 1 s s 0 D d u f (St) S d s du σ dd (u) = -E h ′ d-1 (S s )h d (S d s ) f (St) S d s +E h ′ d-1 (S s ) h d (S d s ) S d s 1 t-s t s D d v f (S t ) dv σ dd (v) .
And by duality (1.2) we remove the Malliavin derivative of f (S t ) in the last term of the previous equality

E h ′ d-1 (Ss)h d (S d s ) S d s 1 t-s t s D d v f (St)dv σ dd (v) = E h ′ d-1 (Ss)h d (S d s ) S d s E 1 t-s t s D d v f (St)dv σ dd (v) F s = E h ′ d-1 (Ss)h d (S d s ) S d s E f (S t ) 1 t-s t s dW d v σ dd (v) F s .

Regrouping all terms together

E f (S t ) h ′ d (S s ) = E f (S t )Γ d s,t h ′ d-1 (S s ) h d d (S s ) , Γ d s,t = π d,d s,t .
Let us suppose that (2.6) is satisfied for k and prove it for k -1, thus

E f (S t ) h ′ d-1 (S s ) = E f (S t )Γ k+1 s,t h d k+1 (S s ) h ′ k (S s ) = E 1 s s 0 f (S t )Γ k+1 s,t h d k+1 (S s ) h ′ k-1 (S s ) D k u h k (S k s ) σ kk (u)S k s du = E 1 s s 0 f (St)Γ k+1 s,t h d k+1 (Ss) h ′ k-1 (Ss) S k s D k u h k (S k s ) σ kk (u) du ,
where we replaced h ′ k (S k s ) by

D k u h k (S k s ) D k u S k s
in the second equality. Using Lemma 2.3 with

F = f (S t )Γ k+1 s,t h d k+1 (S s ) h ′ k-1 (S s ) S k s
and the fact that h ′ k-1 (S s ) does not depend on the j th coordinate (j ≥ k) of the Brownian motion yields

E 1 s s 0 F D k u h k (S k s ) σ kk (u) du = d j=k E F h k (S k s ) 1 s s 0 ρ jk (u)dW j u - d j=k E h k (S k s ) h ′ k-1 (S s ) 1 s s 0 D j u f (St) h d k+1 (Ss)Γ k+1 s,t S k s ρ jk (u)du .
(2.9)

Besides, if for x = (x 1 , ..., x d ) we denote Π(x) = h d k+1 (x) x k
, the Malliavin derivative of the last term of (2.9) provides

D j u Γ k+1 s,t Π(S s )f (S t ) = D j u Γ k+1 s,t Π(S s )f (S t ) + Γ k+1 s,t D j u Π(S s )f (S t ) + Γ k+1 s,t Π(S s )D j u f (S t
). Using Lemma 2.2 for the Malliavin derivative in the two last terms, we get

d j=k ρ jk (u)D j u Π(S s ) = S k s ∂ x k Π(S s ) = -Π(S s ), (2.10) d j=k ρ jk (u)D j u f (S t ) = S k t ∂ x k f (S t ). (2.11)
From (2.10), we deduce that

h ′ k-1 (S s )h k (S k s )f (S t )Γ k+1 s,t 1 s s 0 d j=k ρ jk (u)D j u Π(S s )du = - h ′ k-1 (S s ) h d k (S s )f (S t )Γ k+1 s,t S k s .
Thus, introducing the random variable π k,d s,t = 1 + d j=k s 0 ρ jk (u)dW j u and using (2.9)

E 1 s s 0 F D k u h k (S k s ) σ kk (u) du = E F h k (S k s ) π k,d s,t - E h d k (Ss) h ′ k-1 (Ss)f (St) S k s 1 s s 0 d j=k ρ jk (u)D j u Γ k+1 s,t du -E h d k (Ss) h ′ k-1 (Ss)Γ k+1 s,t S k s 1 t-s t s d j=k ρ jk (u)D j u f (S t )du , (2.12)
where we used the fact (2.11) that d j=k ρ jk (u)D j u f (S t ) does not depend on u. Let us develop the last term of (2.12)

E h d k (Ss) h ′ k-1 (Ss)Γ k+1 s,t S k s 1 t-s t s d j=k ρ jk (u)D j u f (S t )du = E h d k (Ss) h ′ k-1 (Ss) S k s d j=k E 1 t-s t s Γ k+1 s,t ρ jk (u)D j u f (S t )du F s = E h d k (Ss) h ′ k-1 (Ss) S k s d j=k E f (S t ) 1 t-s t s Γ k+1 s,t ρ jk (u)δW j u F s = d j=k E F h k (S k s ) 1 t-s t s ρ jk (u)dW j u - d j=k E f (St) h d k (Ss) h ′ k-1 (Ss) S k s 1 t-s t s ρ jk (u)D j u Γ k+1 s,t du .
We applied (1.2) in the third equality to remove the Malliavin derivative of f (S t ). We also used (1.3) in the last equality. To complete the proof, we should remark that

1 s s 0 D j u Γ k+1 s,t ρ jk (u)du - 1 t -s t s D j v Γ k+1 s,t ρ jk (v)dv = - t 0 D j y Γ k+1 s,t D j y π k,d s,t dy
and because Γ k+1 s,t is an F k+1,...,d t -measurable random variable

Γ k s,t = Γ k+1 s,t π k,d s,t - d j=k t 0 D j u Γ k+1 s,t D j u π k,d s,t du = Γ k+1 s,t π k,d s,t - d j=k+1 t 0 D j u Γ k+1 s,t D j u π k,d s,t du.
Theorem 2.4 provides the expression of Γ k s,t in (2.16) without using the Malliavin derivatives {D j u } j>k and which can be efficiently computed using (2.15). We will use in Theorem 2.4 the set of the second order permutations S k,d defined as the following

S k,d = {p ∈ S k,d , p • p = Id}, (2.13)
where S k,d is the set of permutations on {k, ..., d} and Id is the identity application. By induction, one can easily prove that

S k,d = {τ k k • p, p ∈ S k+1,d } ∪ {τ l k • p, p ∈ S k+1,d
, p(l) = l, l ∈ {k + 1, ..., d}}, (2.14) with τ j i : i → j as the transition application on {k, ..., d}. We also denote by ∆ the determinant that involves only the permutations of S k,d , that is to say, the ∆ associated to the matrix C = {C i,j } k≤i,j≤d is given by

∆ = p∈S k,d ǫ(p) d i=1 C i,p(i)
Using (2.14), we can easily prove that

∆ = C k,k ∆ k,k + d i=k+1 ǫ(τ i k )C i,k C k,i ∆ k,i (2.15)
where ∆ k,i is the ∆ associated to the C i,k obtained from C by suppressing the line and the column i as well as the line and the column k. Based on the development according to the first line, relation (2.15) provides a recursive formula even more efficient than the determinant formula. Of course, we can generalize the relation (2.15) to the one that involves the development according to a j th line or a j th column with k ≤ j ≤ d.

Theorem 2.4. Based on the assumptions and the results of Theorem 2.1, for k ∈ {1, ..., d} the value of Γ k s,t is given by

Γ k s,t = p∈S k,d ǫ(p)A k,p(k) A k+1,p(k+1) ...A d,p(d) = p∈S k,d ǫ(p) d i=k A i,p(i) , (2.16)
with ǫ(p) as the signature of the permutation p ∈ S k,d , S k,d defined in (2.13) and

A =         π 1,d s,t C 1,2 C 1,3 • • • C 1,d 1 π 2,d s,t C 2,3 • • • C 2,d . . . . . . . . . . . . . . . 1 • • • 1 π d-1,d s,t C d-1,d 1 1 • • • 1 π d,d s,t        
, where C k,l is the covariance of π k,d s,t and π l,d s,t . Proof. We prove (2.16) by a decreasing induction. For k = d, the expression (2.16) is clearly satisfied. We suppose that (2.16) is satisfied for k + 1 and we prove it for k. According to Theorem 2.1, Γ k s,t = Γ k+1 s,t π k,d s,t -

d j=k+1 t 0 D j u Γ k+1 s,t D j u π k,d s,t du, but D j u Γ k+1 s,t = d l=k+1 p∈S k+1,d ǫ(p) d i=k+1,i =l A i,p(i) D j u A l,p(l) = d l=k+1 p∈S k+1,d ,p(l)=l ǫ(p) d i=k+1,i =l A i,p(i) D j u A l,l ,
the second equality is due to the fact that A l,p(l) is a constant except for p(l) = l. Subsequently

-d j=k+1 t 0 D j u Γ k+1 s,t D j u π k,d s,t du = -d l=k+1 p∈S k+1,d ,p(l)=l ǫ(p) d i=k+1,i =l A i,p(i) d j=k+1 t 0 D j u A l,l D j u π k,d s,t = - d l=k+1 p∈S k+1,d ,p(l)=l ǫ(p) d i=k+1,i =l A i,p(i) C k,l . Finally Γ k s,t = Γ k+1 s,t π k,d s,t - d j=k+1 t 0 D j u Γ k+1 s,t D j u π k,d s,t du = π k,d s,t p∈S k+1,d ǫ(p) d i=k+1 A i,p(i) - d l=k+1 C k,l p∈S k+1,d ,p(l)=l ǫ(p) d i=k+1,i =l A i,p(i) = p∈S k,d ǫ(p) d i=k A i,p(i) .
The last equality is due to the development of p∈S k,d ǫ(p) d i=k A i,p(i) according to the k th line of A which can be justified by (2.14).

As a corollary of Theorem 2.1 and Theorem 2.4, we obtain the following result. Corollary 2.5.

For any s ∈]0, t[, g ∈ E b and x > 0, if σ ij (t) = σ ij δ(i -j) then E g(S t ) S s = x = T s,t [g](x) T s,t [1](x) , with T s,t [f ](x) = E f (S t ) d k=1 H k (S k s )W k s,t σ k s(t -s)S k s , (2.17) and W k s,t = (t -s)(W k s + σ k s) -s(W k t -W k s ), k = 1, ..., d.
3. Variance reduction method based on conditioning . In this section, we show that one can reduce the variance by a projection on

L 2 t 0 σ ij (u)dW j u i,j
and by using a closed formula of T s,t [1](x). Like in section 2, we give in Theorem 3.1 the results of the special case σ ij (t) = σ ij δ(ij) (σ ij is a constant) that will be used to test our variance reduction method.

We begin with T s,t [START_REF] Abbas-Turki | High dimensional pricing of exotic european contracts on a GPU cluster, and comparison to a CPU cluster[END_REF](x), we can compute the explicit value of this function of x. The T s,t [START_REF] Abbas-Turki | High dimensional pricing of exotic european contracts on a GPU cluster, and comparison to a CPU cluster[END_REF](x) closed formula can be got, for instance, from a change of probability. Indeed, we define the probability

P = N coef f ( d k=1 S k 0 /S k s )P which yields T s,t [1](x) = 1 N coef f E d k=1 H k (S k s ) Γ s,t , N coef f is a deterministic normalization coefficient such that M s = N coef f ( d k=1 S k 0 /S k s
) is an exponential martingale with E(M s ) = 1. Under P, Γ s,t has the same law as a polynomial of Gaussian variables which is sufficient to conduct the computations.

Let us now denote

h(x, {y ij } j≤i ) = E Γ s,t d k=1 H k (S k s ) S k s t 0 σ ij (u)dW j u 1≤j≤i≤d = {y ij } 1≤j≤i≤d
In what follows, we are going to prove that the function h(x, {y ij } 1≤j≤i≤d ) can be explicitly known if, for each j, the (d

-k) × (d -k) matrix Σ jt = Σ ik jt j≤i,k≤d = t 0 σ ij (u)σ kj (u)du j≤i,k≤d
is invertible. First, please remark that according to our notations ij + 1 and kj + 1 are the indices of the element Σ ik jt in the matrix Σ jt (we will use similar convention also for A j , B j , Ψ jt and Φ jt ). Also we notice that the condition of invertibility of Σ jt is not an important constraint, because one can choose a time discretization {t m } such that the matrices {Σ jtm } k≤d fulfill this condition2 .

The computation of h(x, {y ij } 1≤j≤i≤d ) is based on a regression of Gaussian variables according to the Gaussian variables Y ij = t 0 σ ij (u)dW j u . First, we perform a linear regression of

t 0 ϕ jk (u)dW j u according to Y ij t 0 ϕ jk (u)dW j u = d i=j a j i,k Y ij + X jk , (3.1)
with {X jk } 1≤k≤j≤d as a Gaussian vector N (0, C X ) which is orthogonal to Y . Using It isometry twice and the orthogonality of Y and X, we obtain

E t 0 ϕ jk (u)dW j u Y lj = t 0 ϕ jk (u)σ lj (u)du = n j=k Σ li jt a j i,k . If we denote A j = {a j i,k } j≤i,k≤d and Ψ jt = t 0 ϕ jk (u)σ lj (u)du k,l
, we get

A j = Σ -1 jt Ψ jt .
In the same way, we perform a linear regression of

s 0 σ kj (u)dW j u according to Y ij s 0 σ kj (u)dW j u = d i=j b j i,k Y ij + Z kj , (3.2)
with {Z kj } 1≤j≤k≤d as a Gaussian vector N (0, C Z ) which is orthogonal to Y . Using It isometry twice and the orthogonality of Y and Z, we obtain

E s 0 σ kj (u)dW j u Y lj = s 0 σ kj (u)σ lj (u)du = d i=j Σ li jt b j i,k .
If we denote B j = {b j i,k } j≤i,k≤d , we get B j = Σ -1 jt Σ js . Now using (3.1), (3.2) and the value of A and B, the covariance matrices C X , C Z and

C XZ = E(XZ) are given by (Φ i,k jt = t 0 ϕ ji (u)ϕ jk (u)du) [C X ] j i,k = E(X ji X jk ) = Φ i,k jt -(A j k ) ′ Ψ i jt -(A j i ) ′ Ψ k jt + (A j k ) ′ Σ jt A j i , [C Z ] j i,k = E(Z ij Z kj ) = Σ i,k js -(B j k ) ′ Σ i js -(B j i ) ′ Σ k js + (B j k ) ′ Σ jt B j i , [C XZ ] j i,k = E(X ji Z kj ) = Ψ i,k js -(A j k ) ′ Σ i js -(B j i ) ′ Ψ k jt + (A j k ) ′ Σ jt B j i .
Using (3.1) and (3.2), we express Γ s,t and S k s according to Y ij , Z ij and X ji then we conduct standard Gaussian computations to obtain the expression of h(x, y ij ) 3 . In Theorem 3.1, we give an explicit expression of T s,t [START_REF] Abbas-Turki | High dimensional pricing of exotic european contracts on a GPU cluster, and comparison to a CPU cluster[END_REF](x) and h(x, y ij ) in the case of multi-dimensional B&S models with independent coordinates. We can see that now that we know the explicit value of T s,t [START_REF] Abbas-Turki | High dimensional pricing of exotic european contracts on a GPU cluster, and comparison to a CPU cluster[END_REF](x) and h(x, {y ij } 1≤j≤i≤d ), subsequently, we should choose between the simulation of: P1) N paths of g(S t )h x, { t 0 σ ij (u)dW j u } i,j then set the continuation to the value

C(x) := 1 N N l=1 g l (S t )h x, { t 0 σ ij (u)dW j u } l 1≤j≤i≤d T s,t [1](x) .
P2) N ′ paths of g(S t )h x, { t 0 σ ij (u)dW j u } i,j and N paths of h x, { t 0 σ ij (u)dW j u } i,j then set the continuation to the value

C(x) := 1 N ′ N ′ l=1 g l (S t )h x, { t 0 σ ij (u)dW j u } l 1≤j≤i≤d 1 N N l=1 h x, { t 0 σ ij (u)dW j u } l 1≤j≤i≤d .
Based on a variance reduction argument, Theorem 4.3 will indicate the preferable method to use. Theorem 3.1. For any s ∈]0, t[, g ∈ E b and x > 0, if σ ij (t) = σ ij δ(ij) then the conditional expectation given in Theorem 2.5 can be reduced to

E g(S t ) S s = x = E g(S t ) d k=1 √ t exp -sσ k t sσ k t + W k t - (d 2k (W k t )+m k ) 2 2 d k=1 (t -s)e - d 2 1k 2
, with

m k = σ k s(t -s) t , d 2k (W k t ) = t s(t -s) β k - sW k t t , d 1k = β k + σ k s √ s ,
where

β k = 1 σ k ln x k S k 0 + σ 2 k 2 .
Proof. We simplify the constant σ k s(ts) in (2.17) from the denominator and the numerator of the conditional expectation E g(S t ) S s = x , then we use the independence of the coordinates to obtain

E g(S t ) S s = x = E g(S t ) d k=1 H i (S k s )W k s,t /S k s d k=1 E H k (S k s )W k s,t /S k s .
Afterwards, we use the independence of the increments to obtain

E H k (S k s ) S k s W k s,t = E H k (S k s ) S k s [(t -s)(W k s + σ k s) -s(W k t -W k s )] = (t -s)E H k (S k s ) S k s (W k s + σ k s) -sE H k (S k s ) S k s E(W k t -W k s ) = (t -s)E H k (S k s ) S k s ( √ sG + σ k s) ,
where the random variable G has a standard normal distribution. Moreover we have the following equality in distribution

S k s . = S k 0 exp - σ 2 k 2 s + σ k √ sG .
Computing the expectation we obtain

E H k (S k s ) S k s ( √ sG + σ k s) = α k (t -s) s 2π e -d 2 1k 2 , (3.3) with α k = e σ 2
k s . Regarding the numerator, we condition according to W k t = w k and we use the independence of coordinates

E g(S t ) d k=1 H k (S k s )W k s,t /S k s = E g(S t ) d k=1 h k (W k t ) , with h k (w k ) = E H k (S k s )W k s,t /S k s W k t = w k . (3.4) Knowing W k 0 = 0 and W k t = w k , when we fix s the random variable W k s . = sw k t + s(t-s) t
G and G has a standard normal distribution. Also, we have the follow-

ing equality in distribution for W k s,t : W k s,t . = (t -s)σ k s + ts(t -s)G and S k s . = S k 0 exp - σ 2 k 2 s + σ k sw k t + σ k s(t-s) t
G . Then we compute (3.4) which yields:

h k (w k ) . = α k ts(t -s) 2π exp -sσ k t sσ k t + w k - (d 2k (w k ) + m k ) 2 2 , (3.5)
with α k = e σ 2 k s . Using (3.3) and (3.5) we obtain the requested result.

4. Advanced variance reduction method. We present, in this section, a less intuitive idea of variance reduction that is based on an appropriate relation between N and N ′ in (0.6). This method can be applied independently from conditioning detailed in previous section.

Lemma 4.1. Let (X k ) k∈N * be a sequence of independent R n -valued random variables that have the same law. We suppose that X k is square integrable and we denote

µ = E(X k ), C i,j = Cov(X i , X j ). Let r > 0, V µ = {x ∈ R n , ||x -µ|| R n < r} and g : R n → R such that g ∈ C 1 (V µ ), then we have the following limits when N → ∞ g(X N ) -→ g(µ) a.s., √ N (g(X N ) -g(µ)) -→ N (0, Σ) in law,
such that

X N = 1 N N i=1 X i , Σ = ∂g ∂x 1 , ..., ∂g ∂x n x=µ C ∂g ∂x 1 , ..., ∂g ∂x n t x=µ . (4.1)
Proof. The almost sure convergence of g(X N ) results from the law of the large numbers and from the continuity of g in µ. For the same reasons, the gradient vector

∂g ∂x (X N ) converges a.s. to ∂g ∂x (µ). Besides √ N (g(X N ) -g(µ)) = ∂g ∂x (X N ) • √ N (X N - µ) + √ N (X N -µ) • ǫ(X N -µ) and using the Slutsky Theorem, with G ∼ N (0, C) • ( ∂g ∂x (X N ), √ N (X N -µ)) converges in law to ( ∂g ∂x (µ), G). • (ǫ(X N -µ), √ N (X N -µ))
converges in law to (0, G). Finally, because (x, y) → xy and (x, y) → x + y are continuous, then √ N (g(X N )g(µ)) converges in law to ∂g ∂x (µ)G. Let us denote Q as the quotient given by

Q = 1 N ′ N ′ i=1 X i 1 N N i=1 Y i (4.2) If |E(Y i )| ≥ ε > 0, according to Lemma 4.1 Q converges to E(X i )/E(Y i ).
In the following two theorems we will prove that we can accelerate the speed of convergence when acting on the relation between N and N ′ . We analyze the two cases: case 1:

N ′ = λ 1 N with λ 1 ∈ [1/N, 1] and we normalize (4.2) Q = 1 N ′ N ′ i=1 X i 1 N N ′ N ′ N ′ i=1 Y i + N -N ′ N -N ′ N -N ′ i=1 Y i = A N ′ λ 1 B N ′ + (1 -λ 1 )B N,N ′ ,
where

A N ′ = 1 N ′ N ′ i=1 X i , B N ′ = 1 N ′ N ′ i=1 Y i , B N,N ′ = 1 -N ′ N -N ′ i=1 Y i .
We set g 1 (x, y, z) = x/(λ 1 y + (1λ 1 )z) and (4.1) provides

Σ 1 (λ 1 ) = 1 B 2 (2λ 2 1 -2λ 1 + 1) A 2 B 2 σ 2 2 + σ 2 1 - 2λ 1 A B σ 1 σ 2 ρ , (4.3) with
A = E(X), B = E(Y ), σ 2 1 = V ar(X), σ 2 2 = V ar(Y ) and ρ = Cov(X, Y )/(σ 1 σ 2 ). case 2: N = λ 2 N ′ with λ 2 ∈ [1/N ′ , 1] and we normalize (4.2) Q = 1 N ′ N N N i=1 X i + N ′ -N N ′ -N N ′ -N i=1 X i 1 N N i=1 Y i = λ 2 A N + (1 -λ 2 )A N ′ ,N B N ,
where

A N = 1 N N i=1 X i , A N ′ ,N = 1 N ′ -N N ′ -N i=1 X i , B N = 1 N N i=1 Y i . We set g 2 (x, y, z) = (λ 2 x + (1 -λ 2 )y)/z and (4.1) provides Σ 2 (λ 2 ) = 1 B 2 (2λ 2 2 -2λ 2 + 1)σ 2 1 + A 2 B 2 σ 2 2 - 2λ 2 A B σ 1 σ 2 ρ , (4.4) with A = E(X), B = E(Y ), σ 2 1 = V ar(X), σ 2 2 = V ar(Y ) and ρ = Cov(X, Y )/(σ 1 σ 2 ). Theorem 4.2.
Based on what we defined above:

1. If A 2 σ 2 2 ≥ B 2 σ 2 1 the minimum variance Σ min = Σ 1 (λ min 1
), with

λ min 1 = 1 2 + Bσ 1 ρ 2Aσ 2
, Σ 1 given in (4.3).

Simulation and numerical results.

In this section we test our simulations on a geometric average payoff that has the following payoff

Φ d geo (S T ) = K - d i=1 (S iT ) 1/d + . (5.1)
In addition, we will test the American put on minimum and the American call on maximum that have the following payoffs

Φ min (S T ) = K -min(S 1 T , S 2 T ) + , Φ max (S T ) = max(S 1 T , S 2 T ) -K + . (5.2)
The parameters of the simulations are the following: The strike K = 100, the maturity T = 1, the risk neutral interest rate r = ln(1.1), the time discretization is defined using the time steps that is given as a parameter in each simulation, S i 0 = 100 and σ ij (t) = σ ij (t)δ(ji) with σ ii = 0.2. The model considered is a multidimensional log-normal model

dS i t S i t = rdt + (σ i ) ′ dW t , S i0 = y i , i = 1, .., d.
All the prices and the standard deviations are computed using a sample of 16 simulations. Besides, the true values, to which we compare our simulation results, are set using:

• the one-dimensional equivalence and a tree method [START_REF] Broadie | American option valuation: new bounds, approximations, and a comparison of existing methods securities using simulation[END_REF], available in Premia [11], for options with Φ n geo (S T ) as payoff, • the Premia implementation of a finite difference algorithm [START_REF] Villeneuve | Parabolic ADI methods for pricing American options on two stocks[END_REF] in two dimensions for Φ min (S T ) and Φ max (S T ).

In Figure 5.1, we compare the P2 (N = N ′ ) version of MCM with a standard LS algorithm. The LS is implemented using linear regression for multidimensional contracts and using up to three degree monomials for the one-dimensional contract. The reason behind the choice of linear regression in the multidimensional case is the fact that the regression phase of LS can really increase the execution time without a significant amelioration of the prices tested.

In Figure 5.1, even if all the prices are sufficiently good, we see that MCM provides better prices than those of LS. Also when we increase the time steps, MCM is more stable than LS. However, for n = 10 and time steps > 10, we remark that one should simulate 2 14 trajectories to stablize MCM. This fact is expected due to the important variance of the ten dimension contract and that one should simulate more trajectories, on the one hand, to have an asymptotically good approximation of the relation between N and N ′ and, on the other, to have a sufficient number of trajectories for the approximation of the continuation. The executions of MCM and LS with 2 10 trajectories are carried out in less than one second. Moreover, using 2 14 trajectories the LS and MCM are executed within seconds (< 5s). As a conclusion from this figure, MCM provides better results than LS in approximately the same execution time. When we increase the simulated trajectories to 2 14 , the MCM prices are stabilized for high dimensions and are always better than LS prices. In Table 5.1, we remain with the same payoff Φ d geo (S T ) but this time we compare the different nonparametric methods of implementing MCM. In P2(=) and P2(Opt), we use the same P2 method but with N = N ′ for the first one and N = N ′ for the second (The relation between N and N ′ is detailed in pages 16 and 17). First, we remark that P2(=) is not stable in the multidimensional case and can give wrong results if the time steps > 10. However the P2 method is stabilized when we im-plement the version N = N ′ of the advanced variance reduction method detailed in section 4. Also when we use 2 10 trajectories, P1 and P2(Opt) are almost similar. Nevertheless, with 2 14 trajectories, P2(Opt) outperforms P1 which indicates that we fill the conditions of Theorem 4.3 and we have an asymptotically good approximation of the relation between N and N ′ . As far as the execution time is concerned, the time consumed by P2(Opt) is not much different from P1 when we use 2 10 trajectories. In addition, using 2 14 trajectories, the computations of the relation between N and N ′ can be performed on the CPU when the rest of the simulation is done on the GPU. The latter fact allows a similar overall execution time for P2(Opt) and P1 (within seconds). Because of the bad results obtained previously with P2(=), we eliminate this method and we only consider P2(Opt) and P1. In Table 5.2, we analyze the American put on minimum and the American call on maximum in two dimensions. As far as Φ min is concerned, P2(Opt) outperforms P1 even when we use only 2 10 .Regarding Φ max , P1 performs better than P2(Opt) for 2 10 trajectories which indicates that, because of the big variance produced by Φ max relatively to Φ min , the relation between N and N ′ is not well estimated. Simulating 2 14 trajectories, we obtain similar results for P1 and P2(Opt) for Φ max .

Let us now study the parallel adaptability of MCM for parallel architectures. In Figure 5.2, we present the speedup of parallelizing4 MCM on the four cores of the CPU instead of implementing it on only one core. We notice that the speedup increases quickly according to the number of the simulated trajectories and it reaches a saturation state for > 9000 trajectories. For a large dimensional problem, the maximum speedup obtained is approximately equal to the number of logical cores on the CPU which indicates that MCM is very appropriate for parallel architectures. We point out, however, that our parallelization of MCM is done on the trajectories5 , so the speedup is invariable according to dimensions and time steps.

Regarding GPU implementation, we also use a path parallelization of simulations. In Figure 5.3, we present the speedup of parallelizing6 MCM on the GPU instead of implementing it on the four cores of the CPU. The speedup increases quickly not only according to the number of simulated trajectories, but also according to the dimension of the contract. The latter fact can be easily explained by the memory hierarchy of the GPU. The speedups provided in Figure 5.3 prove, once again, the high adaptability of MCM on parallel architectures.

Conclusion.

In this article we provided, on the one hand, theoretical results that deal with the continuation computations using the Malliavin calculus and how one can reduce the Monte Carlo variance when simulating this continuation. On the other hand, we presented numerical results related to the accuracy of the prices obtained and the parallel adaptability of the MCM method on multi-core architectures.

As far as the theoretical results are concerned, based on the Malliavin calculus, we provided a generalization of the value of the continuation for the multi-dimensional models with deterministic and non a constant triangular matrix σ(t). Moreover, we pointed out that one can effectively reduce the variance by a simple conditioning method. Finally, we presented a less intuitive but very effective variance reduction method based on an appropriate choice of the number of trajectories used to approximate the quotient of two expectations.

Regarding the numerical part, we proved that the one who looks for instantaneous simulations can obtain better and sufficiently good prices with MCM than with LS using only 2 10 trajectories. Also, unlike LS, our nonparametric variance reduction implementation of MCM does not require parametric regression. Thus we improve the results of the simulation by only increasing the number of trajectories. Finally, increasing the number of trajectories is time consuming but MCM can be effectively parallelized on multi-core CPUs and GPUs. Indeed, the MCM simulation of 2 14 trajectories using the GTX 480 GPU can be performed within seconds (< 5s).

Appendix. Proof of Lemma 2.2. The equality (2.4) can be easily proved. Indeed, using the chain rule

D k u f (S t ) = n p=k σ pk (u)S p t ∂ xp f (S t )
Besides, we assumed that ρ(u) = σ -1 (u) which completes the proof.

Proof of Lemma 2. Moreover, the fact that σ(u) and ρ(u) are two triangular matrices such that ρ kk (u) = 1/σ kk (u) simplifies the last term which can be also rewritten using the Malliavin derivative

E F I h ′ (S k s )S k s du = E F I D k u h(S k s ) σ kk (u) du
This provides the required result.

Fig. 5 . 1 .

 51 Fig. 5.1. MCM Vs. LS for Φ d geo (S T ): PR is the real price. PM and PL are the prices obtained respectively by MCM and LS represented with their standard deviations.

Fig. 5 . 2 .

 52 Fig. 5.2. The speedup of using all the CPU cores according to the number of trajectories.

Fig. 5 . 3 .

 53 Fig. 5.3. The speedup of using the GPU instead of the CPU cores according to the number of trajectories.

3 .

 3 Using duality (1.2) we haveE h(S k s )F n i=k I ρ ik (u)dW i u = E n i=k I D i u h(S k s )F ρ ik (u)du = E h(S k s ) n i=k I D i u F ρ ik (u)du + E F n i=k I h ′ (S k s )σ ki (u)ρ ik (u)S k s du

Table 5 .

 5 1 P1 Vs. P2 for Φ d geo (S T ): The real values are equal to 4.918, 1.583 and 0.890 for dimensions one, five and ten respectively

	Simulated Dim Time		Price			Std Deviation
	Paths	n	Steps	P1	P2(=) P2(Opt)	P1	P2(=) P2(Opt)
	2 10	1	10	4.750	4.826	4.789 0.213	0.167	0.160
	2 10	1	20	4.729	4.880	4.800 0.270	0.226	0.216
	2 10	1	30	4.679	4.909	4.853 0.270	0.179	0.190
	2 10	5	10	1.548	1.681	1.526 0.071	0.073	0.067
	2 10	5	20	1.632	> 2.0	1.588 0.070		0.048
	2 10	5	30	1.650	> 2.3	1.619 0.074		0.069
	2 10	10	10	0.900	1.112	0.869 0.039	0.045	0.044
	2 10	10	20	0.921	> 1.3	0.936 0.043		0.047
	2 10	10	30	0.908	> 1.5	0.949 0.035		0.046
	2 14	1	10	4.738	4.812	4.807 0.057	0.046	0.047
	2 14	1	20	4.675	4.869	4.825 0.047	0.044	0.043
	2 14	1	30	4.638	4.876	4.856 0.072	0.059	0.058
	2 14	5	10	1.487	1.526	1.506 0.057	0.012	0.012
	2 14	5	20	1.504	1.639	1.534 0.047	0.021	0.016
	2 14	5	30	1.508	> 1.8	1.543 0.072		0.015
	2 14	10	10	0.845	0.938	0.842 0.013	0.015	0.012
	2 14	10	20	0.901	> 1.2	0.893 0.012		0.014
	2 14	10	30	0.923	> 1.3	0.916 0.015		0.016

Table 5 .

 5 2 P1 Vs. P2 for Φ min and Φmax: The real values are equal to 8.262 and 21.15 respectively

	Simulated	The	Time		Price	Std Deviation
	Paths	Payoff Steps	P1	P2(Opt)	P1	P2(Opt)
	2 10 2 10 2 10	Φ min Φ min Φ min	10 20 30	7.734 7.618 7.564	7.986 0.190 7.895 0.257 7.920 0.224	0.248 0.270 0.263
	2 10 2 10 2 10	Φ max Φ max Φ max	10 20 30	21.03 20.46 19.73	20.33 19.38 18.13	0.66 0.61 0.73	0.86 0.73 0.93
	2 14 2 14 2 14	Φ min Φ min Φ min	10 20 30	7.755 7.584 7.467	8.088 0.058 8.098 0.098 8.087 0.082	0.067 0.052 0.043
	2 14 2 14 2 14	Φ max Φ max Φ max	10 20 30	20.96 20.58 20.36	20.91 20.56 20.05	0.09 0.16 0.15	0.24 0.16 0.22
	3.5
	3
	2.5
	0 2	2000 4000 6000 8000 10000 12000 14000 16000 18000

Nevertheless, this is a difficult task when the dimension is sufficiently big.

One can use Mathematica to compute it formally.

We use OpenMP directives.

which is the most natural procedure of parallelizing Monte Carlo.

We use CUDA language.

Acknowledgment: We started this work in the ANR GCPMF project, and it is supported now by CreditNext project. The authors want to thank Professor Damien Lamberton for his review of our work and Professor Vlad Bally for his valuable advice.

Proof. We almost proved this theorem, indeed, one can easily verify that λ min 1 is the minimum of Σ 1 (λ 1 ) and λ min 2 is the minimum of Σ 2 (λ 2 ). To conclude we verify that Σ

What is really appealing, in this theorem, is the fact that even if ρ = 0, one should use N = (1/2)N ′ or N ′ = (1/2)N depending on whether A 2 σ 2 2 ≥ B 2 σ 2 1 or not. Nevertheless, in order to apply the results of either this theorem or Theorem 4.3, we should have a "sufficiently good" approximation of σ 1 , σ 2 , A, B and ρ. With our model B = T s,t [START_REF] Abbas-Turki | High dimensional pricing of exotic european contracts on a GPU cluster, and comparison to a CPU cluster[END_REF](x) is explicitly known and we can have σ 2 in the same fashion as T s,t [START_REF] Abbas-Turki | High dimensional pricing of exotic european contracts on a GPU cluster, and comparison to a CPU cluster[END_REF](x). In section 5, procedure P2 is implemented by using the closed expression of B and σ 2 and simulating σ 1 , A, ρ to get an approximation of λ min 1 or λ min 2 that we use to re-simulate Q. In the case where B and σ 2 are not known, we can implement one of the two methods that are also efficient:

M1) Using all the simulated paths N max , we approximate the values of σ 1 , σ 2 , A, B and ρ then we compute λ min 1 or λ min 2 that we use to re-simulate Q. M2) A fixed point alike method: Using all the simulated paths N max , we approximate the values of σ 1 , σ 2 , A, B and ρ then fix a threshold ǫ and test the condition

Use the previous approximations except A that will be simulated using λ 1 N max paths, such that λ min 1 is reached when

Use the previous approximations except B that will be simulated using λ 2 N max paths, such that λ min 2 is reached when

In the following theorem, we will answer on whether we should implement the simulation procedure P1 or P2.

) = Σ 1 (ρ) then we look for the condition on ρ that allows that the trinomial B 2 Σ 1 (ρ)σ 2 1 is negative. 2. We go through the same argument as in 1.

Theorem 4.3 tells us that, even though we can compute explicitly the expression of T s,t [START_REF] Abbas-Turki | High dimensional pricing of exotic european contracts on a GPU cluster, and comparison to a CPU cluster[END_REF](x), according to the correlation, one can accelerate the convergence when using the quotient of two Monte Carlo estimators.


Bernard Dussardier
email: bernard.dussardier@unice.fr

Vipul Rastogi

Ajeet Kumar

Gérard Monnom

Large mode area leaky optical fiber fabricated by MCVD

A large mode area single-mode optical fiber based on leaky mode filtering has been prepared by modified chemical vapor deposition (MCVD) technique. The fiber has a leaky cladding, which discriminates the fundamental mode from the higher order ones. A preliminary version has 25-µm core diameter and 0.11 numerical aperture. A Gaussian-like mode with 22-µm mode field diameter has been observed after 3-m propagation, in agreement with modeling.

Introduction

Single-mode Large Mode Area (LMA) fibers are of great interest for high power transportation due to their high threshold of nonlinearities such as stimulated Brillouin scattering, and for the high energy storage capacity in applications such as fiber lasers systems and power-amplifiers for telecommunications. In this aim we address here only solid core optical fibers that can be doped with rare-earth (RE) elements. LMA fibers exist in both the index guidance and photonic bandgap (PBG) guidance principles. Fabrication techniques include MCVD both in index guidance [START_REF] Jeong | Ytterbium-doped large-core fiber laser with 1.36 kW, continuous-wave output power[END_REF] and PBG [START_REF] Février | Very large effective area singlemode photonic bandgap fiber[END_REF], external direct nanoparticle deposition [START_REF] Tammela | Direct Nanoparticle Deposition process for manufacturing very short high gain Er-doped silica glass fibers[END_REF], and air-silica microstructurization based on a so-called "stack-and-draw" process [START_REF] Limpert | Extended single-mode photonic crystal fiber lasers[END_REF]. Generally all these fibers guide several modes causing detrimental instabilities of the output spot shape and beam divergence and pointing [START_REF] Nicholson | Spatially and spectrally resolved imaging of modal content in large-mode-area fibers[END_REF] in applications where singlemode operation is required. To address this issue, specific set-ups and fibre packaging are used to minimize the power carried by high order modes (HOM) at the expense of reducing the effective fundamental mode area (A eff ) or the flexibility of the fiber system. To obtain rigorous single mode operation, irrespective of the fabrication technique, the most difficult specification to meet is the stringent control of the refractive index difference between core and cladding over a large diameter and/or of the cladding microstructure. These stringent specifications impose complex fabrication steps in order to achieve very low core-cladding refractive index difference, and hence a numerical aperture (NA) as low as 0.04 to obtain a really single mode 20 µm-diameter core (at 1064 nm), for example. This is even more difficult when high level of rare earth (RE) ions are added in the core, causing further index elevation. Still, some demonstrations of LMA fibers with record mode area are based on leaky HOM, using only air-silica microstructured fabrication techniques [6]. Form medium size LMA (<1000 µm 2 ) and medium average laser power, it is interesting to propose a simpler fiber concept and manufacture approach, such as those implementing a CVD process.

Recently, new single mode LMA structures were proposed, which were composed of an otherwise multimode core and a structured cladding acting as a modal filter and guiding through total internal reflection of leaky modes. The cladding structuring may be azimuthal [START_REF] Rastogi | Analysis of segmented-cladding fiber by the radialeffective-index method[END_REF][START_REF] Millo | Silver-halide segmented cladding fibers for the middle infrared[END_REF] or radial [START_REF] Rastogi | Leaky optical fiber for large mode area single mode operation[END_REF]. The most relevant parameter allowing effective single mode operation is the strong discrimination between the propagation loss of the fundamental mode and any higher order mode (HOM). The discrimination must be effective within a length of fiber compatible with the expected application. For example, in applications of kilometer-long beam transportations, a 1-dB/km loss on the leaky fundamental mode is acceptable. Instead, for amplifiers and laser applications, a loss less than 0.1 dB/m is acceptable. There are advantages of this type of fiber, because the cylindrical mode geometry (when low birefringence is needed) inherent of this fiber structure is accessible to straightforward CVD fabrication techniques and the imposed fabrication constraints on the size and shape of the core index of refraction are sensibly relaxed. Numerical studies have shown that an effective mode area as large as 1000 µm 2 are achievable in such fibers using a standard fabrication technique such as MCVD [START_REF] Kumar | Design and analysis of a multilayer cladding large-mode-area optical fiber[END_REF]. Further, careful design of the cladding structure allows even stronger mode discrimination by implementing resonant coupling of some HOM from the core to the lossy cladding structure [START_REF] Kumar | Co-axial dual-core resonant leaky fibre for optical amplifiers[END_REF].

In this paper, we report on the preparation by MCVD and the characterization of an optical fiber designed using the principle detailed in [START_REF] Kumar | Design and analysis of a multilayer cladding large-mode-area optical fiber[END_REF]. To experimentally validate the proposed principle, we have designed a simplified refractive index profile (RIP) characterized by two annular, low refractive index trenches in the cladding surrounding a multimode core.

We have prepared the fiber, characterized the mode filtering action of the cladding and measured the loss of the fundamental mode. We have observed numerically and experimentally that the effective single mode guidance is achieved beyond 3 meter long propagation.

Design and Analysis

To experimentally validate the principle of operation of the multilayer cladding leaky LMA fiber we proposed a simplified design having only two low index trenches in the cladding region. The proposed design is shown in Fig. 1a: it is designed to operate around the C-band of the telecommunucations. However it could be easily adapted to other spectral windows.

The core and other parts with same refractive index (n 1 ) are of pure silica whereas the low refractive index trenches are doped with fluorine (F). For applications requiring a high rare-earth doping in the core, the refractive index of the core may increase above n 1 . However, this increase would not disturb the filtering as long as the HOM effective indices stay lower than n 1 , which is easily achievable. n 2 is the refractive index of the zone with highest F-content, in direct contact with the core.

Numerical study on this fiber has been carried out by transfer matrix method (TMM) [START_REF] Thyagarajan | Simple numerical technique for the analysis of cylindrically symmetric refractive-index profile optical fibers[END_REF]. TMM is a powerful and convenient method for modal analysis of an optical fiber with arbitrary refractive index profile fiber. In TMM an arbitrary refractive index profile is approximated by a staircase profile by dividing it into a large number of homogeneous layers of sufficiently small thickness. The electric field solutions of scalar wave equation are written in each layer. The field coefficients of the two consecutive layers can be related through a 2 × 2 matrix by applying suitable boundary conditions at the interface of the two layers. The field coefficients of the first and the last layer of the profile can then be connected by simply multiplying the transfer matrices of all the intermediate layers. Application of suitable boundary conditions in the first and the last layer results in a complex eigenvalue equation.

The eigenvalue equation can be solved by a suitable root searching algorithm to obtain complex propagation constant. The real part of the propagation constant gives information about the effective indices of the modes while leakage losses can be calculated by the imaginary part. For the proposed fiber structure the leakage losses of the modes increase with their mode number. It is therefore sufficient to calculate the leakage losses of the first two modes LP 01 and LP 11 for the determination of the single-mode condition. The leakage losses of the LP 01 and LP 11 modes of the fiber proposed in Fig. 1a have been calculated to be 0.35 dB/m and 8.3 dB/m. The simulated results suggest that the fiber should filter out higher order modes after propagation of 2.4 m and should show single mode operation. The mode field diameter has been calculated to be 26 µm.

Fabrication and characterization

A preform with diameter 11.4 ± 0.2 mm was fabricated by MCVD and drawn into a 110.0 ± 0.2 µm-fiber. The refractive index profile of the drawn fiber, measured with a S14 apparatus (NetTest), is shown in Fig. 1b. The relative refractive index difference Δ 1 ~ (n 1 -n 2 )/ n 1 = 0.28 % at 1.55 µm (n 1 = 1.444) is relatively low. The fluorine concentration could be increased to obtain Δ ~ 0.5 % (by MCVD). Plasma Chemical Vapour Deposition could also be used to increase Δ up to 2 %. The diameter and the numerical aperture of the core, bounded by the first trench with index n 2 , are 25.4 ± 0.3 µm and 0.11 ± 6%, respectively. It may be noted that the numerical aperture is relatively high for a single mode LMA fiber. With such characteristics, the normalized frequency V ~ 5.6 at 1.55 µm, six LP-modes would be guided by a conventional fiber. The second low refractive index trench (Δ 2 ~ 0.1 %) is 25 µm away from the fiber axis, and has a ~3 µm width.

The RIP of the fabricated fiber deviates slightly from the fiber proposed in Fig. 1a.

The numerical simulations have therefore been performed for the refractive index profile of the exact fiber by using the transfer matrix method [START_REF] Thyagarajan | Simple numerical technique for the analysis of cylindrically symmetric refractive-index profile optical fibers[END_REF]. As the fibre structure was slightly elliptical, simulations were performed on RIP taken at right angle to each other across the fiber section. The calculated intensity diameter (at 1/e 2 of maximum intensity) of the LP 01 mode has been found to be 26 µm (corresponding to an effective area equal to 530 µm 2 ). The calculated propagation loss for the LP 01 modes evaluated on each axis are 3.25 and 1.63 dB/m respectively, whereas the calculated loss for the LP 11 mode are 56 and 37 dB/m, respectively.

All other HOM are expected to suffer higher loss compared to the LP 11 mode. Therefore, after propagation through a short length of fiber one expects that LP 11 mode is stripped-off while the LP 01 mode suffers from nominal loss. Note that the slight ellipticity of the fiber can be corrected by further process optimization, and is not a limitation of the fabrication method to obtain readily cylindrical geometry.

The characterization of the modal properties of the fiber at 1.55 µm was performed by exciting all possible modes with a strongly convergent beam obtained from a fiber pigtailed laser diode and a high numerical aperture lens (60x-microscope objective). The output near-field was observed using another 60x-microscope objective and imaged on a camera. In all intensity profiles the transverse dimensions have been calibrated using the same apparatus and a standard single-mode fiber (SMF, Draka, type G.652.B, mode diameter 10.1 ± 0.5 µm at 1.55 µm). Images and intensity profiles were captured at various lengths of the fiber, using the cut-back method. Some examples are shown in Fig. 2. For 5 cm, the multimode excitation is visible. The intensity profile was extremely sensitive to launching conditions and fiber bending. For longer lengths of the fiber, the profiles became smoother and their sensitivity to launching conditions diminished. For the fiber longer than 2 m, the profile became Gaussian-like, and the launching conditions influenced only the output power and not the profile. The measured 1/e 2 -mode diameter after 4 m propagation was measured to be 22 ± 2 µm (effective area ~380 µm 2 ), relatively in good agreement with numerical simulations (Fig. 2). Also, good agreement between the measured and simulated mode intensity profiles has been observed. Note that none of the leaky modes are properly guided; therefore no cut-off wavelength is defined.

The propagation loss of the fundamental LP 01 mode was measured using the multiple cut-back method between 1500 and 1570 nm wavelengths. A fiber pigtailed tunable laser diode was connected to the input of the test fiber. A 4 cm-loop was made on the input end of the fiber to reject more strongly any HOM and therefore preferentially excite the fundamental LP 01 mode. The output power measurements were done using a power-meter.

The multiple cut-back technique consists of measuring the output power for different fiber lengths (without changing the coupling conditions). The fiber was cut several times (typically 50 to 80 cm) and the output power was measured each time. The starting length was 8.78 m, and the shortest length was 2.0 m, therefore more than ten measurements could be done. The loss coefficient was obtained by fitting the curve of output power versus length with an exponential function. The results are shown in Fig. 3. The loss of the LP 01 mode increases with the wavelength, from ~1 dB/m at 1500 nm up to 3 dB/m at 1560-1570 nm. The calculated losses of the LP 01 mode for the RIP at 0˚ and 90˚ are also plotted in Fig. 3. It can be seen that the measured values always lie between the calculated values corresponding to the two RIPs. To have significance of departures of measured LP 01 mode loss from the calculated values, we have also plotted the numerically calculated spectral variation of LP 11 mode. It can be seen that the departures of LP 01 mode loss from the calculated values are very small compared to the differential leakage loss between the LP 01 and LP 11 modes and does not significantly affect the single-mode operation of the fiber. The measured loss for the LP 01 mode is higher than acceptable even for a laser or an amplifier. However a more complex structure in the cladding involving a larger number of low-index trenches, and accessible to CVD processes, would lower the confinement loss of the LP 01 mode, as already shown by numerical simulations. The effective single-mode operation of this fiber, having a very simple RIP structure, is in good agreement with numerical simulations. It is experimentally shown that a large and high NA core can support only the LP 01 mode after a couple of meters of propagation. A mode effective area of 380 µm 2 was obtained. A more complex version of the RIP, comprising up to four trenches with low-refractive index may allow a mode effective area up to 700 µm 2 together with 40 dB-mode discrimination between LP 01 and LP 11 modes [START_REF] Kumar | Design and analysis of a multilayer cladding large-mode-area optical fiber[END_REF]. Although the value of accessible mode area is less than state-of-art using complex microstructured leaky fiber, it has potentials for medium power fiber laser, and implements a mature and straightforward manufacturing technique.

Conclusions

We have successfully demonstrated the principle of operation of an MCVD LMA fiber based on higher-order mode discrimination. There are several advantages of this kind of fiber in laser and amplifier applications : low bending loss is achievable thanks to the high numerical aperture of the central core; preforms and fiber are easy to prepare because standard fabrication techniques can be implemented, such as MCVD or OVD; it is possible to dope the core with rare-earth ions for amplification; and the waveguide geometry could be strictly cylindrical for applications where low birefringence is required. Cladding pumping for high power amplification should not be altered by the structured cladding, because the latter would be transparent to short wavelength pump modes. These advantages altogether render this class of single-mode, solid-core, large mode area optical fibers interesting for future applications.

Figure captions list

Foreign

  Affairs and the Centre National de la Recherche Scientifique (France) and the Department of Science and Technology (India). Laboratoire de Physique de la Matière Condensée is part of the CNRS Groupement d'Intérêt Scientifique 'GRIFON' (www.unice.fr/GIS).

Fig. 1 :

 1 Fig. 1: Refractive index profile of the LMA fiber proposed (a), fabricated (b). The ripples

Fig. 2 :

 2 Fig. 2 : Top: Near-field mode intensity images captured at various lengths of the fiber.

Fig. 3 :

 3 Fig. 3 : Leakage losses of the first two modes of the fiber. The top curve: calculated loss of

Acknowledgements

M. Ude and S. Trzésien are thanked for preparing the fiber, and S. Trzésien for measuring the mode intensity profiles. This work has been partially supported by an Indo-French Networking Project ("Projet de Recherche en Réseau", P2R) « R&D on specialty optical fibers and fiber-based components for optical communications » funded by the Ministry of


archive ouverte pluridisciplinaire HAL, est destinée au dépôt

  The mafic-silicic layered intrusions of Saint-Jean-du-Doigt (France) and North-Guernsey (Channel Islands), Armorican Massif: Gabbro-diorite layering and mafic cumulate-pegmatoid association Martial Caroff, Nolwenn Coint, Erwan Hallot, Cédric Hamelin, Jean-Jacque Peucat, Gilles Charreteur To cite this version: Martial Caroff, Nolwenn Coint, Erwan Hallot, Cédric Hamelin, Jean-Jacque Peucat, et al.. The mafic-silicic layered intrusions of Saint-Jean-du-Doigt (France) and North-Guernsey (Channel Islands), Armorican Massif: Gabbro-diorite layering and mafic cumulate-pegmatoid association. Lithos, 2011, 125, pp.675-692. 10.1016/j.lithos.2011.03.019. insu-00589245

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Bernard Dussardier
email: bernard.dussardier@unice.fr

Jerome Maria

Pavel Peterka

Jérôme Maria

Passively Q-Switched Ytterbium-and Chromium-doped All-Fiber Laser

A chromium-doped saturable absorber fiber stabilizes an otherwise spontaneously chaotically self-pulsing ytterbium-doped fiber laser . This original passively Q-switched all-fiber laser produces sustained and relatively stable trains of smooth pulses at high repetition rate.

Introduction

The development of Q-switched fiber lasers is an alternative to bulk pulsed solid-state lasers operating in the ns-to µs-range for many applications such as material processing, medicine, aerial or space communications, remote sensing in free space or in fiber systems. Fiber laser systems benefit from the high versatility and reliability of integrated, factory-aligned and fusion spliced fiber devices, relying on fiberized components produced at low cost and having high damage threshold. However most Q-switched fiber lasers use externally driven intracavity bulk modulators, or bulk passive components such as a bulk polarizer [START_REF] Hideur | Mode-lock, Qswitch and CW operation of an Yb-doped double-clad fiber ring laser[END_REF], or a semiconductor saturable absorber mirror [START_REF] Paschotta | Passively Q-switched 0.1-mJ fiber laser system at 1.53 µm[END_REF]. These must be located in a free space section of the cavity, causing alignment and reliability problems.

The development of "all-fiber" passively Q-switched (PQS) fiber lasers is interesting because, as all components are pre-aligned, the device stability, damage threshold and reliability are high. Also, no external electronic driver is needed, though some may need additional cheap and slow feedback control systems. Transition metal-doped bulk crystals have been used as saturable absorber (SA) within PQS fiber lasers [START_REF] Laroche | Compact diode-pumped passively Q-switched tunable Er-Yb doubleclad fiber laser[END_REF][START_REF] Filippov | All-fiber passively Qswitched low-threshold erbium laser[END_REF], however alignment and optical damage were critical. We have for the first time proposed a core-pumped, low power, all-fiber PQS laser [5] using a newly developed chromium-doped fiber [START_REF] Felice | Chromiumdoped silica optical fibres : influence of the core composition on the Cr oxidation states and crystal field[END_REF] as integrated SA (CrSA). Since, few PQS fiber lasers have implemented alternative SA ions like rare-earth ions (holmium [START_REF] Kurkov | All fiber Yb-Ho pulsed laser[END_REF], thulium [START_REF] Kurkov | All fiber Er-Tm Q-switched laser[END_REF], samarium [START_REF] Fotiadi | Dynamics of All-Fiber Self-Qswitched Ytterbium/ Samarium Laser[END_REF]) or metals (bismuth [START_REF] Dvoyrin | Yb-Bi pulsed fiber lasers[END_REF]). However, some of these SA have drawbacks: low saturation intensity, long response time, absorption band narrower than the gain curve or wavelength-shifted relative to the gain medium. These characteristics necessitate additions to the laser cavity, resulting in more complex devices [START_REF] Kurkov | All fiber Yb-Ho pulsed laser[END_REF][START_REF] Dvoyrin | Yb-Bi pulsed fiber lasers[END_REF]. As for bulk SA, some transition metals from the 3d series, like Cr 4+ ions, are in principle the most attractive SA in silica in the near infrared, because of their high transition strength over broad wavelength ranges [START_REF] Felice | Cr4+-doped silica optical fibres : absorption and fluorescence properties[END_REF] and short response time [START_REF] Dussardier | Cr4+-doped silicabased optical fibers fluorescence from 0.8 µm to 1.7 µm[END_REF]. A waveguidesaturable absorber was fabricated by femtosecond pulses in YAG:Cr 4+ to implement PQS in a hybrid all guided device [START_REF] Okhrimchuk | Waveguide-saturable absorber fabricated by femtosecond pulses in YAG:Cr 4+ crystal for Q-switched operation of Yb-fiber laser[END_REF].

Other fiber laser systems produce self-pulsing in the sub-µs domain. Regular selfpulsing in erbium highly-doped fibers was studied, based on energy transfers within the gain medium [START_REF] Sanchez | Effect of ion pairs on the dynamics of erbium-doped fiber lasers[END_REF]. However this principle allows only a limited potential for engineering and optimization. More recently, double-clad ytterbium-doped fiber (DCYF) lasers have raised a great interest. However DCYF lasers have the particularity to randomly self-pulse in some cases. In particular, linear cavities with high loss and/or non-uniform population inversion induce chaotic pulse generation [START_REF] Upadhyaya | Self-pulsing characteristics of a high-power single transverse mode Ybdoped CW fiber laser[END_REF]. Two main operating regimes are observed: the sustained self-pulsing (SSP), having a period longer than the cavity round-trip time, and the self modelocking (SML), of which period is equal to the cavity round-trip time) [START_REF] Brunet | A Simple Model Describing Both Self-Mode Locking and Sustained Self-Pulsing in Ytterbium-Doped Ring Fiber Lasers[END_REF].

In this paper, we take advantage of the self-pulsing phenomenon in DCYF lasers and study the effect of a CrSA fiber inserted in the cavity on the laser dynamics. We show that the "DCYF alone" laser has a chaotic pulsed behaviour (both SSP and SML) at any pump level above threshold, whereas in presence of the Cr-doped fiber SA, relatively stable and sustained PQS pulses trains are produced.

Experimentals

Characteristics of the Ytterbium-and Chromium-Doped Fibers

The DCYF and the single-mode CrSA fiber were prepared at LPMC by MCVD and solution doping technique. The DCYF has a D-shaped cladding (125 x 100 µm) for high pump absorption (3.8 dB/m at 975 nm). The diameter and numerical aperture (NA) of its singlemode core are 7 µm and 0.11, respectively. The cladding NA is 0.38. The CrSA was doped with Cr 4+ ions only [START_REF] Felice | Cr4+-doped silica optical fibres : absorption and fluorescence properties[END_REF], having a broadband absorption band across the visible and infrared wavelength range [START_REF] Felice | Chromiumdoped silica optical fibres : influence of the core composition on the Cr oxidation states and crystal field[END_REF] and a composite SA relaxation behaviour [START_REF] Dussardier | Cr4+-doped silicabased optical fibers fluorescence from 0.8 µm to 1.7 µm[END_REF]. Its core absorption is 3.5 dB/m at 1064 nm and its mode-size is adapted to that of the DCYF.

Fiber Laser Setups

Two laser setups were studied. The first used only the DCYF (length 1.2 m, pump absorption < 5 dB). The cavity was formed by a fiber Bragg grating (FBG) with high reflectivity (> 99,5 %) at 1064 nm (bandwidth 0.3 nm) and the perpendicular cleaved end of the DCYF (Fig. 1). The cavity was ~ 6.1 m long. The collimated pump beam from a 20-W, 105-µm diameter and 0.2 NA, multimode fiberized 975-nm laser diode was launched into the DCYF cladding using a x1 magnifying aspherical lenses system through the cleaved output end of the DCYF. The polarization controller (PC) was used to partially control the laser stability.

The FBG was limiting the number of longitudinal modes. Note that the pump coupling was intentionally not optimized: the NA and the area of the pump beam did not match that of the DCYF D-shaped cladding. Therefore pumping of the Yb 3+ populations was not homogeneous, providing favorable conditions for sustained self-pulsing (SSP). Also, this cavity was designed to self generate short pulses, therefore a short cavity length, a relatively high unsaturable loss and high Yb 3+ inversion factor were chosen. As a consequence, the laser slope efficiency was not optimized. That also increased the threshold of appearance of stimulated Brillouin scattering (SBS) far above the achieved output power in this study.

Therefore SBS had not effect on the reported results. In the second setup, 0.3 m of CrSA was fusion-spliced between the FBG and the PC. (Fig. 1). The output beam was collected backward relative to the pump beam and was directed using a dichroic mirror toward a powermeter or a 1 GHz InGaAs detector connected to a fast oscilloscope or an optical spectrum analyser.

Results

The characteristics of both laser configurations are shown on Fig. 2. The 'DCYF only' laser threshold and slope efficiency vs. the incident pump power were 0.45 W and 22 %, respectively. The DCYF laser had a chaotic behaviour randomly switching between CW and Q-switched operating modes over the whole pumping range. A characteristic time trace of the output is shown on Fig. 3(a), typical of such cavity [START_REF] Laroche | Compact diode-pumped passively Q-switched tunable Er-Yb doubleclad fiber laser[END_REF][START_REF] Filippov | All-fiber passively Qswitched low-threshold erbium laser[END_REF][5][START_REF] Kurkov | All fiber Yb-Ho pulsed laser[END_REF][START_REF] Kurkov | All fiber Er-Tm Q-switched laser[END_REF][START_REF] Fotiadi | Dynamics of All-Fiber Self-Qswitched Ytterbium/ Samarium Laser[END_REF][START_REF] Dvoyrin | Yb-Bi pulsed fiber lasers[END_REF][START_REF] Okhrimchuk | Waveguide-saturable absorber fabricated by femtosecond pulses in YAG:Cr 4+ crystal for Q-switched operation of Yb-fiber laser[END_REF]. The few µs-long pulse envelopes were strongly modulated by short pulses with the period of one cavity return-trip time (Fig. 3(b)). Here we observe a combination of SSP and SML [START_REF] Brunet | A Simple Model Describing Both Self-Mode Locking and Sustained Self-Pulsing in Ytterbium-Doped Ring Fiber Lasers[END_REF]. Using the PC to stabilize the output was quite inefficient, and at best produced a very noisy laser output. The chaotic behavior was always dominant. Another short section of a commercially available Yb 3+ -heavily doped fiber, implemented in the same conditions, produced the same results.

By inserting a piece of CrSA fiber in the cavity, the characteristics of the laser changed drastically : first, the threshold of the laser increased and its efficiency diminished, as usually observed when inserting a SA in a cavity (Fig. 2). The CrSA fiber caused the threshold to increase up to 1.0 W, and the slope efficiency to lower down to 6 %. Although the obtained slope efficiency is less than in systems implementing non-linear polarization rotation in fibers (~30 %) [START_REF] Hideur | Mode-lock, Qswitch and CW operation of an Yb-doped double-clad fiber ring laser[END_REF] or bulk Co 2+ -ZnSE crystal (~15 %) [START_REF] Laroche | Compact diode-pumped passively Q-switched tunable Er-Yb doubleclad fiber laser[END_REF], is compares well with all-fiber systems implementing rare-earth or Bi ions (up to ~10%) [START_REF] Kurkov | All fiber Yb-Ho pulsed laser[END_REF][START_REF] Kurkov | All fiber Er-Tm Q-switched laser[END_REF][START_REF] Fotiadi | Dynamics of All-Fiber Self-Qswitched Ytterbium/ Samarium Laser[END_REF][START_REF] Dvoyrin | Yb-Bi pulsed fiber lasers[END_REF]. Our result constitutes in improvement in all-fiber PQS systems using transition metal SA, compared to former reports in fiber [5] or external guided-wave Cr-doped saturable absorbers [START_REF] Okhrimchuk | Waveguide-saturable absorber fabricated by femtosecond pulses in YAG:Cr 4+ crystal for Q-switched operation of Yb-fiber laser[END_REF] (around 1%).. More importantly, the µs-long pulse trains were efficiently stabilized over the whole available pump range. On Fig. 4(a) is shown a typical output at 7.6 times the threshold. The peak RMS fluctuations were less than 10 % at 9.4 times the threshold. The higher the pump, the better the stability. Up to the maximum available pump (16 W, measured at the fiber input end), the stable passive Q-switching was still present, and no transition to CW operation was observed. The Q-switched envelopes were smoothed compared to the 'DCYF only' laser. No sub-modulation at the return trip period was visible anymore (Fig. 4(b)). The pulse duration and repetition rate versus incident pump power of the DCYF+CrSA laser are shown on Fig. 5, with a 480 ns minimum pulse width, a repetition rate as high as 350 kHz, a 2 µJ pulse energy and peak power equal to 6.4 W. Cr-doped saturable absorber fiber, FBG: highly reflecting fiber Bragg grating at 1064 nm, FWHM= 0.3 nm. The pump is a 20-W multimode fiber coupled CW laser diode. The HR dichroic mirror at 1.06 µm, transmits 90% at pump wavelength. The 11-mm focal length lenses system has a unit magnification.

Figure captions list

Fig. 2 : Fiber laser integrated output vs incident pump power. : 'DCYF only', : 'DCYF + CrSA'. Thresholds = 0.45 and 1.7 W, slope efficiencies = 22% and 6%, maximum output power = 3 W and 0.9 W, respectively.

Fig. 1 :

 1 Fig. 1: Laser setup. DCYF: double-clad Yb 3+ -doped fiber, PC: polarization controler, CrSA:Cr-doped saturable absorber fiber, FBG: highly reflecting fiber Bragg grating at 1064 nm, FWHM= 0.3 nm. The pump is a 20-W multimode fiber coupled CW laser diode. The HR dichroic mirror at 1.06 µm, transmits 90% at pump wavelength. The 11-mm focal length lenses system has a unit magnification.

Fig. 3 :

 3 Fig. 3 : (a) 'DCYF only' laser output, for 1W of pump (2.2 times threshold). (b) zoom on one typical pulse envelope.

Fig. 4 :

 4 Fig. 4 : (a) 'DCYF+CrSA' laser output, for 7.6 W of pump (7.6 times threshold). (b) zoom on a typical pulses series.

Fig. 5 :

 5 Fig. 5 : Pulse characteristics of the 'DCYF + CrSA' laser.

Fig. 1

 1 Fig.1

Fig. 2

 2 Fig. 2

Fig. 3 (

 3 Fig. 3

Acknowledgements

The author acknowledge partial support from: "Fédération Doeblin" (CNRS, France), grant MSMT-Kontakt ME10119 (Czech Rep.) and the Czech-French 'Barrande' program 17360VA (EGIDE-France and Czech Academy of Sciences). LPMC is with GIS 'GRIFON' (CNRS, http://grifon.xlim.fr/).

Discussion

The behaviour of the DCYF+CrSA laser is typical of a PQS mode of operation. It must be noted that this results were obtained with a set of cavity parameters causing self-pulsing or chaotic behaviour in the 'DCYF only' laser. Indeed, when the CW or a relatively stable output beam was set in the DCYF laser (by choosing parameters or setting the PC), then adding the CrSA provided only a quasi-CW regime. It is evident near the threshold of the DCYF+CrSA laser, when the pulse peak power is still too weak to induce the CrSA fiber to saturate : then the behavior is fully chaotic. When the pump increases, the laser steadily shifts toward a stable Q-switched regime. The capacity of our CrSA to stabilize SSP and cancel out SML phenomena in an Yb 3+ -doped fiber laser has never been reported, to the best of our knowledge. The reasons of the stabilization are not fully understood yet. It is tentatively proposed that SA relaxation time constants spread over decades within the population of SA ions in the glass (as observed in Cr 3+ -doped glasses [START_REF] Grinberg | Continuous function decay analysis of a multisite activated solid[END_REF]) act in different ways: "slow" saturable absorbers would maintain the SSP process, whereas "fast" ones would dampen the rapid optical fluctuations from the SML phenomenon. Modelling the laser behaviour according to this proposal is under progress, using a fully space-and time-resolved model.

In this particular cavity design and parameters, the shortest possible pulses were achieved, downto ~480 ns. To further shorten them, a more compact cavity comprising only the active fibers (1.5 m) would produce sub-100-ns-pulses. Then the pulse peak power would be increased, causing a stronger CrSA saturation, and hence the pulse energy and the slope efficiency would be larger. In order to stay below the threshold of non-linear effects, mode size enlargement will be required by introducing large-mode area fiber technology. The obtained pulsed laser source reported here could serve as a seed in a MOPA configuration using a power DCYF amplifier, so that much higher pulse energy and/or peak power, as well as shorter duration would be accessible for applications, such as long haul ranging or precision machining.

Conclusions

In conclusion, we have shown the potential of a CrSA fiber to produce stable passively Qswitched pulses trains out of a DCYF laser. This is to our knowledge, the first demonstration of such a laser with the active elements Yb 3+ and Cr 4+ . This laser architecture could be integrated and power upgraded, with great potentials for sub-microsecond, high energy pulsed applications.


Bernard Dussardier
email: bernard.dussardier@unice.fr

Jérôme Maria

Pavel Peterka

Ytterbium-and chromium-doped fibre laser: from chaotic self-pulsing to passive Q-switching

A spontaneously chaotic, self-pulsing ytterbium-doped fibre laser is partially stabilized into the passively Qswitched mode of operation using a chromium-doped saturable absorber fibre. This original all-fibre laser produces sustained and stable trains of smooth pulses at high repetition rate.

Introduction: Q-switched fibre lasers are alternatives to bulk lasers operating in the ns-to µs-range for many applications such as material processing, medicine, remote sensing. However they use externaly driven bulk modulators or passive components in free space sections,causing alignment and reliablity problems. All-fibre self-pulsed laser are prefered, because they are versatile, reliable, factory-aligned and integrated. Some double-clad ytterbium-doped fibre lasers (DCYFL) produce random self-pulsing along two main regimes : the sustained selfpulsing (SSP) and self mode locking (SML) [i]. Developing "all-fibre" passively Q-switched fibre lasers (PQSFL) is interesting with transition metal (TM) ions-doped fibres used as saturable absorbers (SA). TM such as Cr 4+ offer strong, broad and fast relaxing absorption bands. We had proposed a core-pumped PQSFL [ii] using a Cr-doped fibre [iii] as SA (CrSAF). We take advantage of the self-pulsing in DCYFL and a CrSAF as stabilizer, resulting in an "all fibre" PQSFL based on Yb 3+ and Cr 4+ ions.

Experiments and results

: The laser setup is described on Fig. 1. Without the CrSAF, the laser is chaotic over the pumping range (up to 16 W on the cleaved fiber) (fig. 2,left). The pulse envelops are strongly modulated by short pulses at a cavity round-trip frequency. Drastic changes appear when inserting the CrSAF, including stable passive Qswitching (peak RMS fluctuations <10%) upto 16 W of pump (fig. 2 right). The Q-switched envelopes are smoothed compared to the DCYFL without CrSAF. Typical characteristics are: 400 ns width, 350 kHz repetition, 2.6 µJ/pulse and 6.4 W peak power.

Discussion :

The laser output is typical of PQSFL, although the set of cavity parameters were causing chaotic behaviour without the CrSAF. The capacity of the CrSAF to stabilise SSP and cancel out SML in an DCYF laser has never been reported, to the best of our knowledge, and needs further studies including comprehensive, spatially and temporally resolved modelling. The device can be optimized to further saturate the CrSAF and reduce pulse duration. It could be further integrated and power upgraded (MOPA configuration) to permit application. Acknowledgements : "Fédération Doeblin" (CNRS); grant MSMT-Kontakt ME10119 (Czech Rep.) (Czech Rep.); EGIDE "Barrande" 17360VA. LPMC is with GIS "GRIFON" (CNRS, http://grifon.xlim.fr/).

[i] F. Brunet et al., "A Simple Model Describing Both Self-Mode Locking and Sustained Self-Pulsing in Ytterbium-Doped Ring Fiber Lasers", J. Light. Technol. 23 2131(2005) [ii] L. Tordella et al., "High repetition rate passively Q-switched Nd 3+ :Cr 4+ all-fibre laser", Electron. Lett., 39, 1307Lett., 39, (2003) ) [iii]

V. Felice et al., "Chromium-doped silica optical fibres : influence of the core composition on the Cr oxidation states and crystal field", Opt. Mat., 16 269 ( 2001)

Fig. 1 :

 1 Fig.1: Laser setup. DCYF: double-clad Yb 3+ -doped fiber, PC: polarization controler, CrSA: Cr-doped saturable absorber fiber, FBG: highly reflecting fiber Bragg grating at 1064 nm, FWHM= 0.3 nm. Pump: 20-W multimode fiberized laser diode.

Fig. 2 :

 2 Fig.2: Left: chaotic DCYDFL output. Right: stablilized PQSFL output


B Dussardier
email: bernard.dussardier@unice.fr

W Blanc

Novel Dopants for Silica-Based Fiber Amplifiers

Keywords:

We review on some studies of the potential of new dopants in silica fibers for ultrabroadband amplification coverage in the 800-1700 nm range, including transition metal and rare-earth ions, and sensitized nano-particles.

Introduction

Optical telecommunications systems already benefit from well developped and efficient optical fiber amplifiers operating on broad bands. Reliable, low energy consumption, broadband, flat gain erbium-doped amplifiers (EDFA) are commercially available and installed in the C-band (1535-1565 nm) and in the L-band (1565-1625 nm), where silica losses are the lowest (<0.2 dB/km). Even in the S-band (1460-1530 nm), thulium-doped fluoride glass fiber amplifiers (TDFA) are available. Although the total available amplified bandwidth is not fully used, long term basic research is of great importance in the aim of identifying innovative, cheap, reliable fiber amplifiers on the alreadycovered bands, and offer amplification over a broader spectral width, again using cost effective and very reliable fiber materials.

Thanks to recent improvements of silica fibers fabrication processes, their transmission window is now very wide, allowing telecommunications almost anywhere within the 800-1700 nm window, depending on the transmission distance and application. Some ew telecommunication bands not yet covered by EDFAs (ie unexploited or using passive systems) may in the future require amplification due to the ever growing number of users and applications. Developping active (i.e. amplified) links is therefore interesting in several bands, such as the 1 st historical telecommunication window (850 nm, fiber attenuation ~-2 dB/km), the O-band or 2 nd telecommunication window (1260-1360 nm, ~-0.5 dB/km). Even the U-band (1625-1675 nm, ~-0.5 to -1.0 dB/km) would be exploitable (Figure 1). Alternative technologies under development are Raman fiber amplifiers (RFAs) and semiconductor amplifiers (SOAs), with some limitations including complex pumping schemes and inter-band crosstalk for RFAs, limited bandwidth per individual SOA, etc… Besides silica-based EDFAs, other glasses doped with appropriate rare-earth dopants have been proposed to cover the S and C+L bands, sometimes with good characteristics (high quantum efficiency (QE), gain flatness, gain red-shifting,…). Some are even commercially available, such as fluoride-or tellurite-or even mixed coumpounds oxides-glass based EDFAs and TDFAs. However, many issues are still actual in fabricating cheap and reliable fibers using these materials, and fusion splicing to silica components is most of the cases impossible. Therefore silica glass is preferred as glass for cost effectiveness and reliability. The aim of this paper is to review on-going research projects on innovative amplifying dopants implemented in silica-based fibers. Issues that are usually addressed in reports are (1) identifying potential dopants and (2) examine improved manners to incorporate them in silica fibers to obtain spectrally broadened amplification. Investigations concern either conventional bands in the aim of proposing simpler architectures than existing devices, using less components, or amplification over yet uncovered bands. These research activities benefit of contributions from several science fields and technologies: process engineering, glass science, spectroscopy and high resolution material characterization.

Status of the silica transparent window coverage by rare earth doped amplifiers.

We review here on amplification either experimentally demonstrated and/or commercially available in the 800-1700 nm range, starting from the long wavelength side. The C-and L-bands are fully covered by silica and nonsilica EDFAs. Some low-phonon energy glasses (oxide and non-oxides) EDFAs offer smoother and red-shifted gain curves, allowing a slight extension of the overall gain curve [START_REF] Minelly | Applications of antimony-silicate glasses for fiber optic amplifiers[END_REF][START_REF] Lin | Near-infrared emissions with widely different widths in two kinds of Er 3+ -doped oxide glasses with high refractive indices and low phonon energies[END_REF]. S-band is covered only by TDFAs [START_REF] Komukai | Upconversion pumped Thulium-doped fluoride fiber amplifier and laser operating at 1.47 m[END_REF], using fluoride or other low-phonon energy glasses [START_REF] Minelly | Applications of antimony-silicate glasses for fiber optic amplifiers[END_REF], except for silica S-band EDFAs covering 1480-1520 nm, using long wavelength amplified spontaneous emission (ASE) filtering scheme [START_REF] Arbore | 30 dB gain at 1500 nm in S-band Erbium-doped silica fiber with distributed ASE suppression[END_REF] or gain clamping complex architecture. The O-band around 1310 nm is partially covered by praseodymium, only in low-phonon glasses [START_REF] Yamada | 15,1-dB-gain Pr 3+ -doped fluoride fiber amplifier pumped by high power laser diode modules[END_REF], however the QE is low (4-6 % in fluorides) and Pr 3+ is difficult to pump (weak absorption at ~1030 nm). At shorter wavelengths, both ytterbium and neodymium ions have gain in the 1050-1150-nm range, whereas using ASE filtering [START_REF] Arbore | Application of fundamental-mode cutoff for novel amplifiers and lasers[END_REF], signal amplification should be possible in the 980-1050-nm and 900-950-nm ranges, respectively. At last, thulium shows a narrow 20-nm gain band around 840 nm in silica [START_REF] Watekar | A Nd-YAG laser-pumped Tm-doped silica glass optical fiber amplifier at 840 nm[END_REF], although a QE close to unity would be found only in lowphonon energy glasses. Therefore, the main gaps which are not covered yet, even in laboratory experiments, are approximately: 1150-1300 nm, 1350-1460 nm, 1625-1700 nm. Regarding silica-based only fiber amplifiers, these gaps are larger: 800-900 nm, 1150-1490 and 1625-1700 nm, respectively.

Alternative dopants with potential broad band amplification

Alternative dopants to erbium, ytterbium and neodymium in silica suffer from multi-phonon non-radiative (MP-NR) relaxations (giving low QE) and/or gain quenching due to bottlenecking on the lowest energy level of the emission transition. Two main causes induce NR decays: (1) a high material phonon energy associated with a relatively small energy gap between the dopant emitting energy level and the level just underneath (as for Tm 3+ in both 1st telecom. window and S-band) and/or (2) a strong electron-phonon coupling to the material, caused by site symmetry distorsion imposed to the dopant, as for transition metals (TM). Silica has a high phonon energy (1100 cm -1 , higher than in fluoride glasses ~550 cm -1 ) and offers distorded dopant sites. Therefore it is necessary to develop some kind of local environment engineering for the dopants within the silica glass network. Some examples based exclusively on silica-based doped fibers are discussed below. Thulium: Tm 3+ ion has three emission bands at 820, 1470 and 1900 nm respectively. The first two bands both originate from the same 3 H 4 energy level, suffering strong MP-NR decay. In the 1 st telecom. window, 11dB of gain was measured with 300 mW of 1064-nm pumping [START_REF] Watekar | A Nd-YAG laser-pumped Tm-doped silica glass optical fiber amplifier at 840 nm[END_REF] whereas Cole et al. have measured only 7 dB in the S-band [START_REF] Cole | S-band amplification in a thulium doped silicate fiber[END_REF]. Both studies concerned MCVD-produced fibres. By simply codoping with aluminium, a 4-fold QE improvement was demonstrated by us and the overall lifetime decay was increased up to 60 s. Looking closer to the decay data analysis, we have shown that Tm 3+ ions were located in two very different types of sites. One of them is characterized by a almost 100 s-lifetime (QE=15%), ever increasing when the Al-content was increased [START_REF] Blanc | Impact of aluminium codoping on the 1.47 m emission efficiency in a thulium-doped silica fibre[END_REF]. A better control of the composition and process would increase the overall QE. In the U-band, Tm 3+ has also some potential using the strong 3 F 4 => 3 H 6 emission band usually exploited for 1.9 m laser emission. However, ASE filtering and gain flattening can be applied, as it was theoretically proposed for fluorides U-band TDFAs [START_REF] Kakkar | Segmented-Clad Fiber Design for Inherently Gain-Flattened L+-Band TDFA[END_REF][START_REF] Kakkar | Inherently gain flattened L+ band TDFA based on W-fiber design[END_REF].

Chromium: In glasses, TM ions have potentially ten-fold spectrally larger and stronger emission cross-sections than rare-earth ions. Most importantly, their emission bands characteristics are extremely sensitive to local environment [START_REF] Dussardier | Cr 4+ -doped silica-based optical fibers fluorescence from 0.8 m to 1.7 m[END_REF]. Cr 4+ in alumino-silicate fibres has a very broad band from 1000 to 1600 nm [START_REF] Felice | Cr 4+ -doped silica optical fibres : absorption and fluorescence properties[END_REF], although its QE is very low, due to highly distorded site symmetry [START_REF] Felice | Chromium-doped silica optical fibres : influence of the core composition on the Cr oxidation states and crystal field[END_REF]. By tuning the fibre composition (particularly in gallo-silicate fibers) and post-drawing heat treating, all chromium ions were stabilized into Cr 3+ in a less distorded local environment. QE as high as 10% and lifetimes equal to 20 s were found [START_REF] Dvoyrin | Effective room-temperature luminescence in annealed chromium-doped silicate optical fibers[END_REF]. The expected gain curve extends from 800 to 1000 nm, but necessitates visible pump wavelength, around 600-700 nm. It was recently found that the Cr 3+ ions were located in nanocrystallites grown during the heat treatment. [START_REF] Dvoyrin | Core nanocrystallization and broadband amplification in chromium-doped silica-based fibers[END_REF].

Bismuth: Recently, Dvoyrin et al. obtained preliminary results of amplification (0.2 dB/m) at 1300 nm in a Bilow-doped fiber pumped at 1064 nm at low power. The potential gain band exceeds 1150-1350 nm, as assumed from laser experiments, showing very promosing 20-24 % slope efficiencies [START_REF] Dianov | Bi-Doped Silica Fiber Laser[END_REF][START_REF] Bigot | Broadband and efficient bismuth fiber laser around 1.2 m[END_REF]. The exact nature of the active center responsible for this gain curve is still under study [START_REF] Mashinsky | Bismuth-doped optical fibres -spectroscopic study and interpretation of spectra[END_REF].

OMN1.pdf

Activated nanoparticles in silica-based fibers: In order to improve some spectroscopic properties of selected amplifying ions, several research groups are trying to insert them into silica fiber via nanoparticles (NPs) selected for the local composition and site structure they would provide to the dopant. Studies performed in bulk glasses or planar waveguides are encouraging. In silica-based fibers, dielectric particles doped with Co 2+ ions were grown by post-drawing heat-treatment in ZnO-Al 2 O 3 -SiO 2 core glass. Improvement of the absorption and luminescence properties were observed in the 900-nm range, but on a much narrower bandwidth than that of Cr 3+ [START_REF] Yoo | Optical properties of the optical fiber containing Co 2+ -doped ZnO-Al2O3-SiO2 glass-ceramics[END_REF]. As mentioned above, Cr 3+ ions were stabilized in nanoparticles in gallium-codoped silica fibers, providing luminescence in the 700-1000 nm range, more efficient and broader than in as-drawn Cr 3+ & Cr 4+ codoped fibers [START_REF] Dvoyrin | Core nanocrystallization and broadband amplification in chromium-doped silica-based fibers[END_REF].

Conclusion

We have reviewed recent reports on alternative dopants for possible amplification range broadening. Although some of the studied alternative dopants suffer MP-NR decays, systematic local-environment improvement must be performed. Also new formulation and local structuration around more standard dopants are of great interest, particularly isolating ions in nanoparticles within the silica glass. Considering all possible dopants, even those of low QE may contribute to fill some spectral gaps in specific applications.

Figure 1 :

 1 Figure 1: ITU telecommunication bands and possible spectral ranges covered by standard dopants (rare-earth ions operating on high-QE transitions; top line) and alternative dopants (TM and RE ions operating on lower-QE transitions; bottom line) in silicabased fibers.