2 results about the trunk load process which is the solution of the Bene equation. In the second section, we recall briey the SCV for the classical Wi

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1 The Bene Equation and Stochastic Calculus of Variations L. Decreusefond and A.S. st nel Ecole Nationale Sup rieure des T l communications, Paris, France Keywords : Bene Equation, Malliavin calculus, Reection Theory. 1. Introduction With the emergence of new technologies, new problems arise in the performance evaluation of queueing networks. Actually, it is believed that the Quality of Service of future networks will be determined mainly by the ability of switches to predict and hence control their input ow. The main model used to derive analytical properties of these systems is the =D=1 queue with possibly nite buers. In this setting, one often faces the problem of characterizing the input ow. This seems rather complicated because of the large amount of highly correlated and versatile sources. Several studies show that one can reasonably approximate the input streams with diusion processes (possibly with jumps). This approximation works well in case of a large time scale, cf. Roberts (1991), or in case of superposition of several sources (see for instance Ren-Kobayashi (1993)). In these uid approximations, the buer of the switch is modelized as a trunk which is continuously fed by a diusion process and emptied at unit rate. The evolution of the trunk load veries the so-called Bene equation (cf. Bene (1963), uillemin-mazumdar (1993), Roberts et al. (1991)), i.e., it is the reected process, in the sense of Skohorod, of the input stream. Except for the very particular case of an input ow with independent increments, cf. Bingham (1975)), very little is known about the load process. It is, in particular, important to know whether its law is absolutely continuous with respect to the Lebesgue measure. This is exactly where the Stochastic Calculus of Variations (SCV) is useful. This technique has been developed rst by P. Malliavin (1978) for the aussian probability spaces and adapted to some other probability spaces like the Poisson space (cf Bass-Cranston (1986), Bichteler-ravereaux-Jacod (1983), Bismut (1983), Bouleau-Hirsch (1986), Decreusefond (1994), Nualart-Vives (1988a), Privault (1994)). Our main motivation in this work is to introduce the Malliavin Calculus in this setting and prove some simple but non-trivial 1

2 2 results about the trunk load process which is the solution of the Bene equation. In the second section, we recall briey the SCV for the classical Wiener space and construct its counterpart for a compound Poisson process. There is a minor dierence from the current literature on Poisson space (cf. Bouleau-Hirsch (1986)), namely, we construct the Sobolev spaces with respect to the weak graph topology, under which the calculations of the third section are easier. We then give a sucient condition for the existence of the density of the law of a real-valued random variable dened on the product of Wiener and Poisson spaces. The third section shows how to interpret the Bene equation as a reection problem in the sense of Skohorod (cf. Chaleyat-Maurel et al. (198)). Afterwards, we give a sucient condition for the solution of a rather general reection problem to be in the space of Sobolev dierentiable random variables, improving some results of Nualart-Vives (1988b). Finally we give an application of these results to a semi-martingale which is representative for the class of processes encountered in the engineering applications of queues. 2. Preliminaries 2.1. Brownian Motion. Let be the canonical Wiener space = C ([; 1];R), i.e., the space of continuous functions on [; 1], null at time. Denote by (B t ) t2[;1] the canonical process dened by B t (!) =!(t) and by (F t ) t2[;1] its ltration. is equipped with the Wiener measure P which makes (B t ) t2[;1] a standard Brownian motion. Let H be the Hilbert space L 2 ([; 1]), denote by C 1 P (Rn ) the set of C 1 functions f, from R n to R with polynomial growth (as well as their derivatives). A smooth functional is a random variable F :! R of the form F = f (B t1 ; : : : ; B tn ) with f 2 C 1 P (Rn ); t 1 < < t n 1: The class of smooth functionals will be denoted by S. Denition 2.1. The derivative of a smooth functional F = f (B t1 ; : : : ; B tn ), f 2 C 1 P (Rn ) is dened by D t F (!) := nx i f (B t1 ; : : : ; B tn )1 [;ti ] (t):

3 Remark 2.1. Interpreting DF := (D t F ) t2[;1] as an element of L 2 (; H), we have for any h in H < DF; h > H = d d F (B : + : where (B : + R : h(s)ds)(t) := B t + R t h(s)ds. h(s)ds) = Due to the quasi-invariance of P, D is closable in L p, (p 1) and the completion of S with respect to the semi-norm kfk 2;1 :=kfk 2 +kdfk L 2 (;H) is denoted by D 2;1. We denote also by D the extension of the derivative operator to D 2;1. The formal adjoint of D is denoted by. The domain Dom L 2 (; H) of is dened in the following way : u is in Dom if and only if u belongs to L 2 () and E 1 D s F u s ds = E [F (u)] ; for any F in D 2;1. Note that the latter equation is the innite dimensional analogue of the integration by parts formula and it is essential to prove the following result : Theorem 2.1 (Bouleau-Hirsch (1986)). Let F :! R be in D 2;1. The law of F is absolutely continuous with respect to the Lebesgue measure as soon as < DF; DF > H > a:e:. Remark 2.2. As it will appear below, we will be interested in the Sobolev dierentiability, of functionals of the form sup s2[;1] X s, where X is a process in D 2;1 for all t. Such a problem has been already studied by Nualart-Vives (1988b) for a continuous process X and the main results are the following : if the random variable M := sup s2[;1] X s is square integrable, if X s belongs to D 2;1 for all s, if the H valued process fdx s ; s 2 [; 1]g has a continuous version with E h i sup s2[;1] k DX s k 2 < 1 and if P(X s = X t ) = for all s; t 2 Q \ [; 1], s 6= t, then M belongs to D 2;1. Moreover, if there exists 2 L 2 (; H) such that (inf t:xt=m < DX t ; > > ) almost surely then M has an absolutely continuous law with respect to the Lebesgue measure Compound Poisson Process. In this section, will denote the space of locally nite RR jump measures on [; 1] [; 1]. For any in, we consider N t () := (ds; dz) which is the canonical point process associated to. (F t ) t2[;1] denotes the canonical ltration generated by the ];t][;1] sample-paths and P is the unique probability measure on for which N is the compound Poisson process whose dual predictable projection (i.e., L vy : 3

4 4 measure) is := 1 [;1] (z)dz ds. Finally, we denote by P the predictable -eld on R + and we write P ~ := P B(R + ). We denote for short the stochastic integral of a P ~ measurable process Y (resp. Y 1 ];t] (s)) with respect to a random measure by Y (resp. Y t ). Let H be the space L 2 ([; 1] [; 1]) and let H 1 be the sub-space of bounded elements of H. We dene the space of L 2 -smooth functionals as Denition 2.2. F : 7?! R is an L 2 -smooth functional if for any l in L 2 (; H 1 ), there exists a square integrable random variable D l F such that E?2 jf ()? F ( l )? D l F ()j 2!?! ; where and v(s; z) = z l(s; u)du l ([; t] B) = 1[;t]B (s; z + v(s; z))(ds; dz): It is proved in Bass-Cranston (1986), as a consequence of the irsanov theorem, that, for any ~ P measurable l in L 2 (; H 1 ), we have (2.1) E [D l F ] = E [F l] ; where l denotes RR l d(? ). We denote by C 2 b (Rn ) the space of bounded, twice continuously dierentiable functions on R n with bounded derivatives and dene a cylindrical functional to be a random variable F :! R of the form (2.2) F := f (N t1 ; : : : ; N tn ) with f 2 C 1 b (Rn ), t 1 < < t n 1: The class of cylindrical functional will be denoted by S. Proposition For any t in [; 1], N t :?! N t () is an L 2 - smooth functional with (2.3) (2.4) D l N t () =? v(s; z)1 [;t] (s) (ds; dz) : 2. A cylindrical functional F := f (N t1 ; : : : ; N tn ); f 2 C 1 b (Rn ) is an L 2 - smooth functional and for any l 2 L 2 (; H 1 ), D l F =? nx i f (N t1 ; : : : ; N tn ): ti 1 v(s; z)(ds; dz):

5 Proof. 1. Since?1 (N t? N t ) is equal to the right hand side (2.3), it only remains to prove that v t belongs to L 2. From Jensen inequality, we get 5 (2.5) E jv t j 2 = = E E v 2 t " t 1 z E t 1 z z E klk 2 H : l(; s; u)du 2 dz ds l 2 (; s; u)du dz ds # 2. For the sake of notational simplicity, we give the proof of (2.4) only in dimension 1. Let F be as described in (2.2) with n = 1 and let l be in H. Denoting by F l the right member of (2.4), we have?1 jf ( l )? F ()? F l j =?1 jf (N t )? f (N t)? f (N t )(N t? N t )j 2kf k 1 t 1 jv(s; z)j(ds; dz): From (2.5), the setf?1 (F ( l )? F ()? F l )g > is bounded in L 2 () and it converges to almost surely, consequently the convergence takes place also in L 2 sense. From Theorem (2.2) and inequality (2.5), for any F in S, there exists a random variable DF in L 2 (; H) dened by E D l F 2 = E < DF; l > 2 H, for any l in L 2 (; H). A sequence (F n ) of elements of S is said to converge to F for the topology if and only if F n converges in L 2 to F and DF n converges weakly in L 2 (; H). We denote by D 2;1 the completion of S with respect to this topology and the limit of DF n by D F. Lemma 2.3. Let F and be two L 2 smooth functionals and l be a predictable element of L 2 (; H), we have (2.6) " E [D l F ] =?E [F D l ]? E F l d(? ) ];1][;1] #

6 6 Proof. The transformation T l : 7?! l is invertible and T?1 () = T l?l (), hence by the irsanov theorem, E [(F ()? F ( l ))] = E [()F ()]? E (?l )F ():E( l d(? )) = E [(()? (?l ))F ()]? E F ()(?l ):(E( l d(? ))? 1) ; where E is the Dol ans-dade exponential, i.e., for any semi-martingale, Q E() t = e t st (1 + s)e?s is the process which satises Y t = 1 + R t Y s? d s. We then obtain (2.6) by dividing the above expectations by and letting going to. Proposition 2.4. The limit D F does not depend on the choice of the sequence (F n ) n S. Proof. Let (F n ) n be a sequence of functionals from S. Suppose that, for any l in L 2 (; H) and in L 2 (), F n L 2????! n!+1 and E [ < DF n; l > H ]????! n!+1 E [ < ; l > H] We have to prove that =. By a density argument, it is sucient to consider bounded random variables and deterministic processes l. Then, hypothesis on the sequence (F n ) and lemma above imply that is a.s. null. From proposition (2.4), D is a well dened extension of the linear operator D from D 2;1 to L 2 (; H). Moreover, by a limiting argument, formula (2.1), still holds for F in D 2;1 : E [< DF; l >H] = E [F l] : (2.7) We can dene the adjoint of D, i.e., is the linear operator dened on Dom L 2 (; H) such that (2.7) is still veried. Clearly, coincides with on H and Theorem 2.5. Let ' be a twice dierentiable function on R with bounded derivatives and let F be in D 2;1. Then '(F ) belongs to D 2;1 and we have (2.8) D '(F ) = ' (F )D F :

7 Proof. Let (F n ) be a sequence from S converging to F in the -topology. We already know that D'(F n ) = ' (F n )DF n. E < ' (F )D F? ' (F n )DF n ; l > H = E < ' (F )(D F? DF n ); l > H + E < DF n (' (F )? ' (F n )); l > H k' k 1 E [j < D F? DF n ; l > H j] + k' k 1 E [jf? F n j < jdf n j; jlj > H ] ; hence by the boundedness of the sequence (DF n ) in L 2 (; H), we see that ' (F )D F is the weak limit of D('(F n )) and then '(F ) is in D 2;1, thus (2.8) is satised Mixing both processes. Let B be a Brownian motion independent of the compound Poisson process N. We work on := B N equipped with the product probability P := P B P N and for any! := (! 1 ;! 2 ) 2, let B t (!) := w 1 (t); N t (!) :=! 2 (t) and we F t := fb s ; N s ; s tg. In the sequel, notations referring to the Brownian motion B (resp. to the compound Poisson process N) will be indexed by B (resp. by N). Denition 2.3. A functional F :! R is said to belong to D 2;1 square integrable, ( F (:;! 2 ) :! 1 7?! F (! 1 ;! 2 ) 2 D B 2;1 ; PN a:e: ; F (! 1 ; :) :! 2 7?! F (! 1 ;! 2 ) 2 D ;N 2;1 ; P B a:e: ; 7 if it is and E jd B F j 2 + jd N F j2 < +1. For such a functional, we dene D by DF := (D B F (! 1 ;! 2 ); D N F (! 1 ;! 2 )): Lemma 2.6. For any functional F in D 2;1, h(! 1 ; s) in Dom B and l(! 2 ; s; z) in Dom N, we have E [< DF; (h; l) >] = E (2.9) F ( B h + N l) Proof. Let F (! 1 ;! 2 ) = F B (! 1 )F N (! 2 ) be in D 2;1. By the denition of D, we have DF = (F N (! 2 )D B F B (! 1 ); F B (! 1 )D N F N (! 2 )). Therefore E [< DF; (h; l) >] = E F N (! 2 ) < D B F B (! 1 ); h > + E F B (! 1 ) < D N F N (! 2 ); l > = E [F N (! 2 )]E < D B F B (! 1 ); h > + E [F B (! 1 )]E < D N F N (! 2 ); l > = E [F N (! 2 )]E F B (! 1 ) B h + E [F B (! 1 )]E F N (! 2 ) N l = E F B (! 1 )F N (! 2 )( B h + N l) = E F ( B h + N l) :

8 8 Hence (2.9) holds for nite sums of tensor products and the proof follows by a density argument. The following is the main result about the absolute continuity : Theorem 2.7. Let F :! R be in D 2;1 pair (h; l) 2 Dom B Dom N such that and suppose that there exists a (2.1) < D B F; h > H B + < D N F; l > HN > P? almost surely. Then the law of F is absolutely continuous with respect to the Lebesgue measure on R. Proof. Let us consider ' a C 1 (R;R)-function with compact support and assume that F belongs to D 2;1, then '(F ) 2 D 2;1 and D'(F ) =? ' (F )D B F; ' (F )D N F Let (h; l) as above. We have < D'(F ); (h; l) > HB H N = ' (F )? < D B F; h > H B + < D N F; l > H N : Taking expectations on both sides and applying the integration by parts of Lemma (2.9) yield to E ' (F )? < D B F; h > H B (2.11) + < D N F; l > H N = E '(F )( B h + N l) k ' k 1 E B h + N l C k ' k 1 : By a limiting argument, formula (2.11) holds for functions ' dened by ' (x) = 1 A (x) and '(?1) =, for any bounded borel subset A of R. We then obtain je [1 A (F )]j CjAj; where E denotes the expectation associated to the measure dp := (< D B F; h > H B + < D N F; l > H N )dp. The latter formula shows that the P -law of F is absolutely continuous with respect to the Lebesgue measure and under condition (2.1), P is equivalent to P, hence the P-law of F is absolutely continuous with respect to the Lebesgue measure.

9 9 3. The Bene equation Consider an innite trunk being emptied at unit rate. Let S t be the total amount of work sent by sources up to time t and W t the load at time t. We assume without loss of generality that all processes are cadlag. Intuitively, using the conservation principle, we set that W satises the equation (3.1) W t = S t? t + L t ; where L t is the cumulative idle time of the server up to time t. Remark 3.1. The process L t can not always be written R t 1 fg (W s )ds, since, for processes like the Brownian Motion, the term R t 1 fg (W s )ds is almost surely null whereas there exists a non-trivial process, namely, the local time at, which measures the time spent by the Brownian Motion at zero. Hence, it is better to work in a more general setting to cover the case where the points are not met by the process under question. For, we introduce the notion of Reection Problem dened as in Chaleyat-Maurel et al. (198) : Denition 3.1 (RP(X)). Let (X t ) be a cadlag process with X. The pair (W; L) is said to be the solution of the reection problem associated with X if and only if 1. W t = X t + L t, 8 t. 2. W is a non-negative cadlag process. 3. L is a cadlag non-decreasing process, null at time, such that (3.2) t W s? dlc s = ; 8 t ;P? almost surely and L s = 2W s where L s := L s? L s? with L s? is the continuous part of L. the left-hand limit of L at s and Lc Theorem 3.1 (cf. Chaleyat-Maurel et al. (198)). There is a unique solution to the problem of reection associated with X. Moreover, if the process sup st X? s is continuous, L is continuous and the solution is given by L t = sup st X? s ; W t = X t + L t ; where X? s := sup(?x s; ). Remark 3.2. A sucient condition for sup st X? s to be continuous is that X has positive jumps. Note that this hypothesis is in conformity with the nature of our queueing problem and hence, from now on, we will assume that X has positive jumps. We see that the process W which appears in (3.1) is the rst component of the solution of RP(X) with X t := S t?t. This shows

10 1 that the Bene Equation (3.1) has a unique solution which is (3.3) W t := X t + sup(?x s ): st 3.1. First consequence. Using (3.3), we can derive in a very simple way another well known form of the Bene equation, see Borovkov (1976). Proposition 3.2. Let X t := S t? t and (W; L) be the solution of RP(X). For any f in C 2 (R), b (3.4) f (X t? X s )dl s E [f (W t )] = E [f (X t )] + E t R Proof. By the change of variables formula, e ilt t = 1 + i eils dl s, but L t = W t?x t and the measure dl s (!) has a support included in fs; W s (!) = g, hence e ilt = 1 + i t e i(ws?xs) dl s = 1 + i t e?ixs dl s : Multiplying this equation by e ixt and taking the expectations of both sides yields = E h i e ixt + ie t e i(xt?xs) dl s ; E h e iwt i then, by a density argument, we get (3.4) Absolute continuity of the solution of (3.3). Recall that our aim is to apply Theorem (2.1) to the load process W. For this we need to prove that the load process belongs to D 2;1. Theorem 3.3. Suppose that the modied input stream X t := S t? t satises the following conditions 1. X, X s 8s. 2. The random variable L 1 := sup s2[;1] (?X s ) is square integrable. 3. X t is in D 2;1 for all t 2 [; 1] and " E # sup k D B X t k 2 H B + sup k D N X t k 2 < 1 L 2 (;H N ) t2[;1] t2[;1] Then W 1 and L 1 are in D 2;1. For the proof we need the following : Lemma 3.4. If X belongs to D 2;1 then so does jxj and DjXj = DX(1 fx>g? 1 fx<g ):

11 Proof of Lemma (3.4) -. Let X n = p X 2 + 1=n, then (X n ) n converges a.s. to jxj as n goes to +1 and from Theorem (2.8), X n belongs to D 2;1 and DX n = X p X 2 + 1=n DX = X p X 2 + 1=n DX 1 fx6=g: Now, we see that kd B X n k L 2 (;H B ) and kd N X n k L 2 (;H N ) are bounded uniformly with respect to n, hence there exists a weakly convergent subsequence (DX nk ) k in L 2 (; H B ) L 2 (; H N ). Since DX nk converges almost surely to DX(1 fx>g? 1 fx<g ), it follows that jxj belongs to D 2;1 with DjXj = DX(1 fx>g? 1 fx<g ). Proof of Theorem (3.3) -. We only have to show that L 1 = sup s1 (?X s ) belongs to D 2;1. Let us enumerate [; 1] \ Q by ft 1 ; : : : ; t n ; : : : g and dene M n = sup(?x t1 ; : : : ;?X tn ). Lemma (3.4) and the relation M n = sup(m n?1 ;?X M n?1? X tn + jm n?1 + X tn j tn ) = ; 2 imply that M n belongs to D 2;1 and from Lemma (3.4), Note that DM n =? nx i=1 supe kd B M n k 2 H B n DX ti 1 fxti >X tj ;8j6=ig: " # E sup kd B X t k 2 ; H B t2[;1] sup kd N M n k 2 L 2 (;H N sup kd N X ) t k 2 ; L 2 (;H N ) n t2[;1] hence there exists a L 2 (; H B ) L 2 (; H N ) weakly convergent subsequence (M nk ) k. Since, this sequence converges almost surely to L 1 and is uniformly integrable, the convergence holds in the L 2 sense hence L 1 belongs to D 2;1 and so does W 1. We are now ready to give the main criterion for the absolute continuity of the law of the solution of (3.3) Theorem 3.5. With the hypothesis of Theorem (3.3), the law of W 1 is absolutely continuous with respect to the Lebesgue measure as soon as there exists (h; l) 2 H B H N such that (3.5) P-almost surely. < DX 1 ; (h; l) >? sup t:x t=inf s X s (< DX t ; (h; l) >) > ; 11

12 12 Proof. iven (h; l) 2 H B H N, we need to verify (2.1) for W 1. Let = f< DW 1 ; (h; l) >= g, we have, using the subsequence of the previous proof, = = = = < DW 1 ; (h; l) > dp < DX 1 ; (h; l) > dp + < DL 1 ; (h; l) > dp < DX 1 ; (h; l) > dp + lim < DM nj ; (h; l) > dp j < DX 1 ; (h; l) > dp + lim inf < DM nj ; (h; l) > dp j < DX 1 ; (h; l) > dp + inf < D(?X t ); (h; l) > dp t:x t=inf s X s (< DX 1 ; (h; l) >? sup t:x t=inf s X s < DX t ; (h; l) >)dp and then by (3.5), P() =. Example 3.3. Let the input stream S t be given by S t = f (t; N t ) where f is a twice dierentiable function such that 1. E f (1; N 1 ) 2 < f y is strictly positive non-decreasing in both variables. 3. f t (1; z) < 1 for all z in R+. Then, the law of the load W 1 has a density with respect to the Lebesgue measure. In fact, we have D N X t = D N (S t? t) = f y (t; N t)d N N t; thus the law of W 1 has a density provided that there exists l 2 H N such that P a.s. f y (1; N 1) v d? sup f y (t; N t) t:x t=inf s X s ];t]e v d > : The left hand side of the previous line is greater than f (1; N 1 ) RR 1 T v(s; z)d where T is the last time that X t reaches its inmum. For the elements of ft = 1g, there exists a time interval ]1? ; 1] such that for any t in this interval, f (t; N t ) = f (t; N 1 ). This equality and X t X 1 imply that f (t; N 1 )? t f (1; N 1 )? 1. This turns out to be impossible since f t (1; N 1) is assumed to be less than 1. Hence, T is almost surely strictly less than 1 and then the law of W 1 is absolutely continuous with respect to the Lebesgue measure.

13 Acknowledgment : The authors would like to thank J. Bertoin (University of Paris VI) who indicated them Bingham (1975) and D. Nualart (University of Barcelona) for valuable comments. 13

14 14 References [1] R.F. Bass and M. Cranston, The Malliavin calculus for pure jump processes and applications to local time, The Annals of Probability 14 (1986) [2] V.E. Bene, eneral stochastic processes in the theory of queues, Addison-Wesley, [3] K. Bichteler and J. Jacod, Calcul de Malliavin pour les diusions avec sauts : existence d'une densit dans le cas unidimensionnel, in: J. Az ma & M. Yor, eds., S minaire de probabilit s XVII, 1983, pp [4] N.H. Bingham, Fluctuation theory in continuous time, Advances on Applied Probability 7 (1975), [5] J.-M. Bismut, Calcul des variations stochastiques et processus de sauts, eitschrift f r Warhscheinlichkeits Theory 63 (1983), [6] A.A. Borovkov, Stochastic processes in queueing theory, Springer-Verlag, [7] N. Bouleau and F. Hirsch, Dirichlet Forms and Analysis on Wiener Space, de ruyter Studies in Mathematics, [8] M. Chaleyat-Maurel, N. El Karoui, and B. Marchal, R exion discontinue et syst mes stochastiques, The Annals of Probability 8 (198) [9] F. uillemin and R. Mazumdar, The Bene equation for the distribution of excursions of general storage., To be published, April [1] P. Malliavin, Stochastic calculus of variations and hypoelliptic operators, in: K.Ito, ed., Proc. Intern. Symp. S.D.E. Kyoto, [11] D. Nualart and J. Vives, Anticipative calculus for the Poisson process based on the Fock space, in: J. Az ma & M. Yor, eds., S minaire de probabilit s XXIV ( Springer- Verlag, 1988), pp [12] D. Nualart and J. Vives, Continuit absolue de la loi du maximum d'un processus continu, C.R. Acad. Sci. Paris 37 (1988), [13] N. Privault, Chaotic and variational calculus for the Poisson processes., Ph.D. thesis, University of Paris VI, [14] Q. Ren and H. Kobayashi, A diusion approximation analysis of an ATM statistical multiplexer with multiple types of trac, Conference Record lobecom'93, vol. 2, IEEE, 1993, pp [15] J.W. Roberts et al., Information Technologies and Sciences COST 224 report. Published by the Commission of the European Communities, L. D. and A.S. U. 46, rue Barrault Paris Cedex 13, FRANCE.