Euler scheme for SDEs with non-lipschitz diffusion coefficient: strong convergence

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1 Author manuscript, published in "SAIM: Probability and Statistics 12 (28) 15" INSTITUT NATIONAL D RCHRCH N INFORMATIQU T N AUTOMATIQU uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence Abdel Berkaoui Mireille Bossy Awa Diop No 5637 version 2 version initiale Juillet 25 version révisée Janvier 26 Thème NUM apport de recherche N inria-176, version 2-1 Jan 26

2 inria-176, version 2-1 Jan 26

3 Unité de recherche INRIA Sophia Antipolis uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence Abdel Berkaoui, Mireille Bossy, Awa Diop inria-176, version 2-1 Jan 26 Thème NUM Systèmes numériques Projets Oméga Rapport de recherche no 5637 version 2 version initiale Juillet 25 version révisée Janvier pages Abstract: We consider one-dimensional stochastic differential equations in the particular case of diffusion coefficient functions of the form x α, α [1/2, 1). In that case, we study the rate of convergence of a symmetrized version of the uler scheme. This symmetrized version is easy to simulate on a computer. We prove its strong convergence and obtain the same rate of convergence as when the coefficients are Lipschitz. uler scheme, strong error, CIR process, Hull-White process, SABR pro- Key-words: cesses. Dept of Statistics, University of Warwick, Gibbet Hill road, Coventry CV4 7AL UK INRIA, OMGA Modification du Lemme 3.3 et preuve du Théoreme 2.2

4 Schéma d uler pour des DS à coefficients de diffusion non lipschitziens : convergence forte Résumé : On s intéresse à la discrétisation d DS unidimensionnelles, dans le cas particulier d un coefficient de diffusion de la forme x α, α [1/2, 1). Dans ce cas, on étudie la vitesse de convergence d un schéma d uler symétrisé. Le schéma reste simple à simuler sur un ordinateur. On démontre la convergence forte du schéma : on obtient la même vitesse de convergence que dans le cas de coefficients lipschitziens. Mots-clés : Schéma d uler, erreur forte, processus de CIR, processus d Hull-White, processus SABR. inria-176, version 2-1 Jan 26

5 uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence 3 1 Introduction The aim of this article is to present an original technique for the strong convergence rate analysis of discretization schemes applied to SD s with non Lipschitz diffusion coefficients. To treat the non Lipschitz property of the coefficient, we use a stochastic time change inspired from Berkaoui [2. We restrict ourselves to one dimensional SD s, with diffusion coefficients of the form σ(x) = x α, α [1/2, 1) and we analyze the strong rate of convergence of the symmetrized uler scheme. More precisely, we consider (X t ) t, the R-valued process solution to the following onedimensional Itô stochastic differential equation inria-176, version 2-1 Jan 26 X t = x + b(x s )ds + σ X s α dw s, where x and σ > are given constants and (W t ) t is a one-dimensional Brownian motion defined on a given probability space (Ω, F, P). We denote by (F t ) t the Brownian filtration. To ensure the existence of such process, we state the following hypotheses: (H) α [1/2, 1) and there exists a positive constant K such that b() >, and b(x) b(y) K x y, x R, y R. Under hypotheses (H), strong existence and uniqueness holds for the previous equation (see e.g. [1). Moreover, as b(x) Kx, using comparison results one can show that (X t ) t is valued in [, + ). Then (X t ) t is the unique strong solution to X t = x + b(x s )ds + σ X α s dw s. (1.1) Simulation schemes for (1.1) are motivated by applications in Finance where equations like (1.1) models the dynamics of the short term interest rate: we refer to the Cox, Ingersoll and Ross model [5, the Hull and White model [9 or the SABR models [8. The solution processes are positive. This is one of the reasons to introduce these models in practice. By using the classical uler scheme, one cannot define a positive approximation process. We study a symmetrized uler scheme for (1.1), which preserves the sign of the solution and which can be simulated easily. The symmetrized uler scheme, introduce to treat reflected stochastic differential equations, leads to a weak convergence rate of order one for Lipschitz coefficients (see [4). For quation (1.1), the convergence in the weak sense of the present scheme has already been studied by Bossy and Diop in [3, In this present work, we aim to study the L p -convergence rate, which required original arguments to face the fact that the diffusion coefficient is not globally Lipschitz. In section 2, we describe the time discretization scheme for (X t ) t T. In this particular case, symmetrized uler scheme means that we take the absolute value of the classical uler RR no 5637

6 4 Berkaoui, Bossy & Diop inria-176, version 2-1 Jan 26 scheme. In section 3, we prove the convergence. When the coefficients are globally Lipschitz, results on the convergence in L p (Ω)-norm for the uler scheme are well known. A classical technique to obtain the convergence rate consists in applying the Itô formula to the function x 2p and the error approximation process t = X t X t and using the Lipschitz property of the coefficients (see [7). Here, we replace the Itô formula by the Itô Tanaka formula, in order to deal with the absolute value of the classical uler scheme. When 1/2 < α < 1, we obtain the usual rate of convergence in O( t). We prove the same when α = 1/2 on stronger hypothesis on the drift coefficient. Let us mention two close works. First, Deelstra and Delbaen in [6 give a strong rate of convergence using Yamada s method for a quite similar scheme. Second, Alfonsi in [1 analyses the (strong and weak) rate of convergence of some implicit schemes (with analytical solution) in the special case of the CIR process and compare them numerically with the Deelstra and Delbaen scheme s and the present one. 2 On the symmetrized uler scheme for (1.1) In all what follows, we assume hypotheses (H) even if this is no more explicitly mentioned. 2.1 The scheme and notations Let (X t ) t be given by (1.1). For a fixed time T >, we define a discretization scheme (X tk, k =,..., N) by { X = x >, X tk+1 = X tk + b(x tk ) t + σx α (2.1) t k (W tk+1 W tk ), k =,..., N 1, where N denotes the number of discretization times t k = k t and t > is a constant time step such that N t = T. In the following, we use the time continuous version (X t ) t T : X t = X η(t) + (t η(t))b(x η(t) ) + σx α η(t)(w t W η(t) ), (2.2) where η(s) = sup k {1,...,N} {t k ; t k s}. The process (X t ) t T takes positive values. Using Tanaka s formula, we can easily show, by induction on each subinterval [t k, t k+1 ), for k = to N 1, that (X) is a continuous semimartingale with a continuous local time (L t (X), t T ) at point. Indeed, for any t [, T, if we set then, X t = Z t and Z t = X η(t) + b(x η(t) )(t η(t)) + σx α η(t)(w t W η(t) ), (2.3) X t = x + sgn(z s )b(x η(s) )ds + σ sgn(z s )X α η(s)dw s + L t (X), (2.4) INRIA

7 uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence 5 where sgn(x) := 1 2 ll (x ). The following lemma ensures the existence of all positives moments of (X) and (X) as well as some negative moments of (X): Lemma 2.1 For any x >, for any p 1, there exists a positive constant C, depending on p, but also on the parameters b(), K, σ, α and T, such that ( ) ( ) sup X 2p t t [,T + sup X 2p t t [,T C(1 + x 2p ). (2.5) inria-176, version 2-1 Jan 26 When 1/2 < α < 1, for any p >, ( 1 sup t [,T X p t ) C(1 + x p ). (2.6) When α = 1 2b() 2, for any p such that 1 < p < σ 1 2 ( ) 1 X p C t x p. (2.7) The proof can be found in [ Strong convergence Theorem 2.2 Consider (X t ) t T, the solution of (1.1) and (X t ) t T, the corresponding symmetrized uler scheme given by (2.1). Let t satisfying t 1/(2K). i) For all 1/2 < α < 1, for all p 1, there exists a positive constant C(T, p) depending on α, σ, b, T and x, increasing in T and p, such that [ 1/2p sup X t X t 2p C(T, p) t. (2.8) t T ii) When α = 1/2, the estimate (2.8) holds if we assume also that σ 2 8 with K(p) = K(4p 1) (4σα(2p 1)) 2. ( ) 2 2b() σ 2 1 > K(4p), (2.9) We emphasize the difference between the situation 1/2 < α < 1 and α = 1/2. Let τ = inf{t ; X t = }. When 1/2 < α < 1 and x >, Feller s test on process (X) shows that RR no 5637

8 6 Berkaoui, Bossy & Diop it is enough to suppose b() > as in (H) to ensure that P(τ = ) = 1. When α = 1/2, (X) satisfies the equation X t = x + b(x s )ds + σ Xs dw s, t T. (2.1) inria-176, version 2-1 Jan 26 When b(x) is of the form a bx, with a >, (X) is the classical CIR process. When b(x) = a >, (X) is the square of a Bessel process. When x >, one can show that P(τ = ) = 1 for any drift b(x) satisfying (H) and b() σ 2 /2, using the classical comparison lemma and Feller s test. In addition, if b() σ 2 /2, we are able to control the exponential inverse moment of the CIR like process (X) (see Lemma 3.1 below). The constraint (2.9) on b() and σ is stronger than b() σ 2 /2 due to the particular use of Lemma 3.1. in the proof of Theorem Proof In all the sequel, C denotes a positive constant depending on p or on the parameters b, σ, α and x of the model but not on t. We also use the following notation ( O exp ( t) = exp C ) t β, for some β >. 3.1 Preliminaries In this subsection, we gather a few results on exponential moments of (X) and on the behavior of the approximation processes (X) and (Z) visiting zero. Those results are crucial in the proof of convergence. Lemma 3.1 Let (X t ) t T be the solution of (1.1) with 1/2 < α < 1. For all µ, there exists a positive constant C(T, µ), increasing in µ and T, depending also on b σ, α and x such that ( ) T ds exp µ X 2(1 α) s ( ) When α = 1/2, Inequality (3.1) holds if µ ν2 σ 2 8 and ν = 2b() σ 1 2 C(T, µ). (3.1). The proof can be found in [3. Lemma 3.2 Let (X t ) t T be the approximation process defined by (2.2) and (Z t ) t T given by (2.3). Let t 1/(2K). i) If 1/2 < α < 1, then sup P ( Z t ) O exp ( t). t [,T INRIA

9 uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence 7 ii) If α = 1/2 and t 1/(2K) x, sup P ( Z t ) C t [,T ( t x ) b() σ 2. inria-176, version 2-1 Jan 26 Proof. Under (H), for any x, b(x) b() Kx. Thus, using (2.3), P ( Z t ) ( ) P X η(t) + (b() KX η(t) )(t η(t)) + σx α η(t)(w t W η(t) ) ( = P W t W η(t) X ) η(t)(1 K(t η(t))) b()(t η(t)), X η(t) > 1 2 [exp ( σx α η(t) (X η(t)(1 K(t η(t))) + b()(t η(t))) 2 2σ 2 (t η(t))x 2α η(t) ) ll {Xη(t) >} When 1/2 < α < 1, noting that (1 K t) 1 2, we have P ( Z t ) ( ) 1 2 exp X2(1 α) η(t) 8σ 2 exp b() t 2σ 2 X 2α 1 ll {Xη(t) >}. η(t) ( We distinguish the events X η(t) ) ( t and X η(t) > ) t and easily conclude that P ( Z t ) = O exp ( t). When α = 1/2, we deduce from the previous computation that P ( Z t / ( ) 1 X η(t) 2 exp X ) η(t) 8σ 2. t Lemma 3.5 in [3 ensures that, for any γ 1 ( sup exp X ) t k k N γσ 2 C t ( t x ) 2b() σ 2 (1 1 2γ ),. which ends the proof. 3.2 Proof of Theorem 2.2 We start with estimating the local error of the scheme (2.2): Lemma 3.3 For all p 1 and all 1/2 α < 1, there exists a positive constant C, depending on b(), σ, α, p, K and T but not on t such that { Xt } sup X η(t) 2p C t p. t [,T RR no 5637

10 8 Berkaoui, Bossy & Diop Proof. Indeed, sup X t X η(t) 2p t [,T Using (2.2), we have for t [t k, t k+1, sup k=...n 1 sup X t X tk 2p. t [t k,t k+1 X t X tk 2p b(x tk )(t t k ) + σx α t k (W t W tk ) 2 2p 1 t 2p b(x tk ) 2p + 2 2p 1 σ 2p X 2pα t k W t W tk 2p. 2p inria-176, version 2-1 Jan 26 But b(x tk ) 2p 2 2p 1 ( b() 2p + K 2p X tk 2p) and we easily conclude by using Lemma 2.1 that sup t [tk,t k+1 X t X tk 2p C 2 2p 1 t 2p + C2 2p 1 σ 2p C(p) t p. We define the process (γ(t)) t by γ(t) = ( X 1 α s ds + X 1 α η(s) sup t [t k,t k+1 ( W t W tk 4p ) 1/2 ) 2. (3.2) Note that γ(t) is well defined. Indeed, almost surely the process (X) is positive and under (H) (and the condition b() > σ 2 /2, when α = 1/2), the process (X) is almost surely strictly positive. Let τ λ be the stopping time defined by τ λ = inf{s [, T, γ(s) + s λ}, (3.3) with inf = T. We begin the convergence analysis, considering the strong error at the stopping time τ λ : Lemma 3.4 For all λ and all integer p 1, there exists a positive constant C(p) depending on b(), σ, α, p, K and T but not on t such that with K(p) = K(4p 1) (4σα(2p 1)) 2. X τλ X τλ 2p exp(k(p)λ)c(p) t p (3.4) INRIA

11 uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence 9 inria-176, version 2-1 Jan 26 Proof. The error process ( t ) t T defined by t := X t X t, satisfies t = ( b(xη(s) )sgn(z s ) b(x s ) ) ds + σ ( ) X α η(s)sgn(z s ) Xs α dw s + L t (X). For an arbitrary stopping time τ valued in [,T, we apply the Itô formula to 2p t, between and τ. As τ ( s) 2p 1 dl s(x) = τ ( X s) 2p 1 dl s(x), we obtain τ 2p 2p ( s ) 2p 1 ( b(x η(s) )sgn(z s ) b(x s ) ) ds ( ) + p(2p 1)σ 2 ( s ) 2p 2 X α 2 η(s)sgn(z s ) Xs α ds 2p ( s ) 2p 1 ( b(x η(s) ) b(x s ) ) ds ( ) + 2p(2p 1)σ 2 ( s ) 2p 2 X α 2 η(s) Xs α ds { } + 2 2p s 2p 1 b(x η(s) ) + 2p(2p 1)σ 2 ( s ) 2p 2 X 2α η(s) ll ds {Zs }. Thanks to Lemma 2.1, one can easily check that for any β, ( ) s β b(x η(s) ) + X 2α η(s) ll ds {Zs } C When 1/2 < α < 1, by Lemma 3.2, we conclude that τ 2p 2p ( s ) 2p 1 ( b(x η(s) ) b(x s ) ) ds ( + 2p(2p 1)σ 2 ( s ) 2p 2 X α η(s) Xs α sup t [,T sup t [,T P ( Z t ). (3.5) ) 2 ds + O exp ( t) (3.6) When α = 1/2, in the case of CIR-like processes, from Lemma 3.2 we only have P ( Z t ) C t b() 2σ 2. When we sum up the case α = 1/2 and 1/2 < α < 1, (3.6) becomes τ 2p 2p ( s ) 2p 1 ( b(x η(s) ) b(x s ) ) ds ( ) + 2p(2p 1)σ 2 ( s ) 2p 2 X α 2 η(s) Xs α ds + C t b() 2σ 2 p. (3.7) RR no 5637

12 1 Berkaoui, Bossy & Diop inria-176, version 2-1 Jan 26 In the case α = 1/2, the convergence rate seems to decrease form t p to t b() 2σ 2 p. This also occurs when one analyzes the weak convergence rate in [3. Indeed, for smooth enough test function f(x), one has that f(xt ) f(x T ) b() C t σ 2 1 instead of C t, and such behavior is confirmed by numerical experiments (see [1). Here, due to constraint (2.9), we are in the situation where b() 2σ p and the convergence 2 rate does not change when α = 1/2. The reason to impose this so strong constraint (2.9) will appear later, when we will try to apply Lemma 3.1 in the case α = 1/2, in the proof of Theorem 2.2. In view of (3.2), we have ( s ) 2p 2 (X α η(s) Xs α ) 2 ds As, for all 1/2 α 1, we deduce τ = ( s ) 2p 2 (X α η(s) Xs α ) 2 (Xs 1 α + X 1 α η(s) ) 2 dγ(s). x, y, (x α y α )(x 1 α + y 1 α ) 2α x y, (3.8) ( s ) 2p 2 (X α η(s) X α s ) 2 ds 4α 2 τ ( s ) 2(p 1) (X η(s) X s ) 2 dγ(s), from which (3.7) becomes τ 2p 2pK s 2p 1 Xη(s) X s ds + 2p(2p 1)σ 2 4α 2 ( s ) 2(p 1) (X η(s) X s ) 2 dγ(s) + C t p. We remark that for r = 1, 2, s 2p r Xη(s) X s r s 2p r ( s + X η(s) X s ) r 2 r 1 s 2p r ( s r + X η(s) X s r). By the Young Inequality we also have s 2p r X η(s) X s r 2p r 2p ( s) 2p + r 2p X η(s) X s 2p so that s 2p r X η(s) X s r 2p r r(1 + 2p )( s) 2p + r2 2p X η(s) X s 2p, (3.9) INRIA

13 uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence 11 and τ 2p K(p) ( s ) 2p d(s + γ(s)) + [ K (2p 1)(4ασ) 2 X η(s) X s 2p d(s + γ(s)) + C t p, with K(p) = K(4p 1) (4σα(2p 1)) 2. But, [ T X η(s) X s 2p d(s + γ(s)) X η(s) X s 2p (1 + 1 s X 2 2α )ds. inria-176, version 2-1 Jan 26 Using the local error estimate in Lemma 3.3, for any couple (a, b) such that 1/a + 1/b = 1, we have X η(s) X s 2p d(s + γ(s)) C t p + T C t p (1 + sup t [,T ( Xη(s) X s 2pb) ( ( 1 b ( ( 1 X a(2 2α) s )) 1 a 1 Xs a(2 2α) ) )) 1 a ds We apply Lemma 2.5 to upper-bound the negative moment of (X). By condition (2.9), as 1 > 2, we can choose a = 2, even if α = 1/2. Then 2b() σ 2 τ 2p K(p) ( s ) 2p d(s + γ(s)) + C t p. Now we choose τ = τ λ defined in (3.3). Noting that τ λ + γ(τ λ ) = λ, we apply the change of time u = s + γ(s) in the above integral: [ λ τλ 2p K(p) τu 2p du + C t p. By Gronwall Lemma, we conclude that τλ 2p C t p exp(k(p)λ). RR no 5637

14 12 Berkaoui, Bossy & Diop Proof of Theorem 2.2. We proceed as before to get for all t [, T, inria-176, version 2-1 Jan 26 t 2p 2p ( s ) 2p 1 (b(x η(s) )sgn(z s ) b(x s ))ds ( ) + p(2p 1)σ 2 ( s ) 2p 2 X α 2 η(s)sgn(z s ) Xs α ds + 2pσ ( s ) 2p 1 ( X α η(s)sgn(z s ) X α s ) dw s and by the Burkholder-Davis-Gundy Inequality, [ T sup t 2p C s 2p 1 b(x η(s) )sgn(z s ) b(x s ) ds s T [ T ( ) + C ( s ) 2p 2 X α 2 η(s)sgn(z s ) Xs α ds + C T ( s ) 4p 2 ( X α η(s)sgn(z s ) X α s ) 2 ds. Again we use (3.5) and Lemma 3.2, (together with the condition (2.9) when α = 1/2) to get [ T sup t 2p C s 2p 1 X η(s) X s ds s T [ T ( ) + C ( s ) 2p 2 X α 2 η(s) Xs α ds + C T ( s ) 4p 2 ( X α η(s) X α s ) 2 ds + C t p. Now, we use (3.8) and by definition of γ t in (3.2), it comes: [ T sup t 2p C ( s ) 2p 1 X η(s) X s ds s T [ T + C ( s ) 2p 2 (X η(s) X s ) 2 dγ(s) + C T ( s ) 4p 2 (X η(s) X s ) 2 dγ(s) + C t p. INRIA

15 uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence 13 inria-176, version 2-1 Jan 26 We use the upper bound (3.9) in each of the three terms above: [ T sup t 2p C ( s ) 2p d(γ(s) + s) s T [ T + C X η(s) X s 2p d(γ(s) + s) + C + C T T ( s ) 4p dγ(s) X η(s) X s 4p dγ(s) + C t p, [ T As previously, X η(s) X s 2p d(γ(s) + s) C t p and the same treatment on T X η(s) X s 4p dγ(s) leads to [ T sup s T t 2p C ( s ) 2p d(γ(s) + s) + C Again, we apply the change of time u = s + γ(s): [ γ(t )+T sup s T t 2p C ( τu ) 2p du + C Now, using Lemma 3.4, we have [ γ(t )+T + ( τu ) 2p du = ( ll {γ(t )+T u} ( τu ) 2p) du + We proceed similarly to bound T γ(t )+T [P (γ(t ) + T u) 1/2 [ ( τu ) 4p 1/2 du C t p [T exp (T K(2p)) + γ(t )+T + ( s ) 4p dγ(s) + C t p, ( τu ) 4p du + C t p. [P (γ(t ) u) 1/2 exp (uk(2p)) du. ( τu ) 4p du from above and finally we get sup s T + t 2p C t p [P (γ(t ) u) 1/2 exp (uk(4p)) du. RR no 5637

16 14 Berkaoui, Bossy & Diop To finish the proof, we have to show that u [P (γ(t ) u) 1/2 exp(uk(4p)) L 1 (R + ). By the Markov inequality, we observe that, for µ >, [P (γ(t ) u) 1/2 exp( µu) ( [exp (2µγ(T ))) 1/2. We choose µ > K(4p). Moreover, in view of the definition of γ(t ) and Lemma 3.1, we have ( ) T ds [exp (2µγ(T )) exp 2µ C(T, µ). X 2(1 α) s inria-176, version 2-1 Jan 26 Note that in the case α = 1/2, the constraint (2.9) allows us to choose µ σ2 8 as required in Lemma 3.1. References ( ) 2 2b() σ 1 2 [1 A. ALFONSI. On the discretization schemes for the CIR (and Bessel squared) processes. Preprint, 25. [2 A. BRKAOUI. On the discretization of the solution of one dimensional reflected stochastic differential equation. Preprint, 22. [3 M. BOSSY and A. DIOP. An efficient discretization scheme for one dimensional SDs with a diffusion coefficient function of the form x a, a in [1/2,1). Technical report, INRIA, Décembre 24. preprint RR [4 M. BOSSY,. GOBT, and D. TALAY. A symmetrized uler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab., 41(3): , 24. [5 J. COX, J.. INGRSOLL, and S.A. ROSS. A theory of the term structure of the interest rates. conometrica, 53, [6 G. DLSTRA and F. DLBAN. Convergence of discretized stochastic (interest rate) processes with stochastic drift term. Appl. Stochastic Models Data Anal., 14(1):77 84, [7 O. FAUR. Simulation du Mouvement Brownien et des Diffusions. PhD thesis, cole Nationale des Ponts et Chaussées, [8 P.S. HAGAN, D. KUMAR, A.S. LSNIWSKI, and D.. WOODWARD. Managing smile risk. WILMOTT Magazine, September, 22. [9 J.C. HULL and A. WHIT. Pricing interest-rate derivative securities. Rev. Finan. Stud., 3: , 199. INRIA

17 uler scheme for SDs with non-lipschitz diffusion coefficient: strong convergence 15 [1 I. KARATZAS and S.. SHRV. Brownian Motion and Stochastic Calculus. Springer- Verlag, New York, inria-176, version 2-1 Jan 26 RR no 5637

18 inria-176, version 2-1 Jan 26 Unité de recherche INRIA Sophia Antipolis 24, route des Lucioles - BP Sophia Antipolis Cedex (France) Unité de recherche INRIA Futurs : Parc Club Orsay Université - ZAC des Vignes 4, rue Jacques Monod ORSAY Cedex (France) Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique 615, rue du Jardin Botanique - BP Villers-lès-Nancy Cedex (France) Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu Rennes Cedex (France) Unité de recherche INRIA Rhône-Alpes : 655, avenue de l urope Montbonnot Saint-Ismier (France) Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP Le Chesnay Cedex (France) Éditeur INRIA - Domaine de Voluceau - Rocquencourt, BP Le Chesnay Cedex (France) ISSN