aa r X i v : . [ m a t h . P R ] J a n The Local Time of the Classical Risk Process ∗ F. Cortes, J.A. Le´on, and J. Villa
Abstract.
In this paper we give an explicit expression for the localtime of the classical risk process and associate it with the density ofan occupational measure. To do so, we approximate the local time bya suitable sequence of absolutely continuous random fields. Also, asan application, we analyze the mean of the times s ∈ [0 , T ] such that0 ≤ X s ≤ X s + ε for some given ε >
1. Introduction and main results
Henceforth, X = { X t , t ≥ } represents the classical risk process. Moreprecisely, X t = x + ct − N t X k =1 R k , t ≥ , where x ≥ c > N = { N t , t ≥ } is an homogeneous Poisson process with rate α and { R k , k ∈ N } is a sequence of i.i.d non-negative random variables, which isindependent of N . N t is interpreted as the number of claims arrivals duringtime t and R k as the amount of the k -th claim. We suppose that R hasfinite mean and it is an absolutely continuous random variable with respectto the Lebesgue measure.The risk process has been studied extensively because it is often used todescribe the capital of an insurance company. Indeed, among the propertiesof X considered by several authors, we can metion that the local time of X has been analyzed by Kolkovska et al. [ ], the double Laplace transformof an occupation measure of X has been obtained by Chiu and Yin [ ], orthat the probability of ruin has been one of the most important goals of Mathematics Subject Classification.
Key words and phrases.
Classical risk process, crossing process, local time, occupationmeasure, Tanaka-like formula. ∗ Partially supported by the CONACyT grant 45684-F, and by the UAA grants PIM05-3 and PIM 08-2. the risk theory (see, for example, Asmussen [ ], Grandell [ ], Rolski et al.[ ] and the references therein to get an idea of the analysis realized in thissubject). In this paper we are interested in continuing the development ofthe local time L of X and its applications as an occupational density inorder to improve the understanding of X .Note that X is a L´evy process due to P Nk =1 R k being a compound Poissonprocess. Thus, we can apply different criteria for general L´evy processes toguarantee the existence of L . For example, we can use the Hawkes’ result [ ]when R k is exponential distributed (see also [ ] and references therein forrelated works). However, we cannot obtain in general the form of L via thisresults. Moreover, in the literature there exist different characterizations ofthe local time (see Fitzsimmons and Port [ ] and the references therein). Forinstance, the local time have been introduced in [ ] (resp. [ ]) as an L (Ω)-derivative (resp. derivative in probability) of some occupation measure.Nevertheless, in [
6, 7 ], it is not analyzed some properties of the involvedlocal time using this “approximation of L ”.The purpose of this paper is to associate the local time of X with thecrossing process when L is interpreted as a density of the occupational mea-sure (see Theorem 1.c) below). The relation between the local time and thecrossing process was conjectured by L´evy [ ] for the Brownian motion case(i.e., whe X is a Wiener process). In this article we use the ideas of theproof of Tanaka’s formula for the Brownian motion (see Chung [ ], Chapter7) to obtain a sequence of absolutely continuous random fields (in time) thatconverges with probability 1 (w.p.1 for short) to L t ( x ) = 1 c ( 12 1 { x } ( X t ) + 1 ( x, ∞ ) ( X t ) −
12 1 { x } ( x ) − ( x, ∞ ) ( x ) − X
0) = 1 , wich imply that only a finite number of summands in (1.1) are different thanzero.In the following result we not only relate L to the number of crossingswith certain level, but also to the occupation measure(1.3) Y t ( A ) = Z t A ( X s ) ds, t ≥ A ∈ B ( R ) , where B ( R ) is the Borel σ -algrebra of R . Toward this end, we need thefollowing: HE LOCAL TIME OF THE CLASSICAL RISK PROCESS 3
Definition 1.
We say that there exists a crossing with the level x ∈ R at time s ∈ (0 , + ∞ ) if for all open interval I such that s ∈ I , x is aninterior point of { X t : t ∈ I } . That is, x ∈ ( X ( I )) ◦ . Moreover, the numberof crossings with the level x in the interval (0 , t ) is denoted by C t ( x ) . C isknown as the crossing process of X . Observe that if x ∈ R is a crossing point at time s , then X is continuous attime s and X s = x .Now we can state the main result of the paper. Theorem 1.
Let t > and x ∈ R . Then, the random field L defined in(1.1) has the following properties: a) L t ( x ) ≥ and L · ( x ) is not decreasing w.p.1. b) L t ( x ) = c (cid:0) { X t } ( x ) − { X } ( x ) + C t ( x ) (cid:1) w.p.1. c) For every bounded and Borel measurable function g : R → R , wehave (1.4) Z t g ( X s ) ds = Z R g ( y ) L t ( y ) dy w.p. . Note that Statement b) implies that the number of crossings C of X introduced in Definition 1 satisfies C t ( x ) = 1 ( −∞ ,X t ) ( x ) − ( −∞ ,X ) ( x ) + X x ∈ R . Also note that, from (1.4) and Statement a), therandom field L can be interpreted as an occupation density relative to theLebesgue measure on R . Hence, L in (1.1) is called the local time and theexpression L t ( x ) = 1 c ( 12 1 { X t } ( x ) −
12 1 { X } ( x ) + 1 ( −∞ ,X t ) ( x ) − ( −∞ ,X ) ( x )+ Z (0 ,t ] f ( x, X s ) dX s ) , is known as Tanaka-like formula for L t ( x ). Here f ( x, X s ) = ( ( Xs,Xs − ) ( x )∆ X s , ∆ X s = 0 , , ∆ X s = 0 . On the other hand, relation (1.4) can be extended to some occupationalresults. Indeed, as an example, we can state the following, which leads usto get some average of the pathwise behavior of X . Theorem 2.
Let g : R × R −→ R be a bounded and Borel measurablefunction. Then for each ε > , (1.5) E [ Z t g ( X s , X s + ε − X s ) ds ] = Z R E [ g ( x, X ε − x )] E [ L t ( x )] dx. F. CORTES, J.A. LE ´ON, AND J. VILLA
An application of this theorem is to answer the question:
What is theaverage in time that the capital of an insurance company is positive, andbigger than itself after twelve months?.
The paper is organized as follows. In Section 2 we provide the toolneeded to prove Theorem 1. In particular, we approximate the local timeby a sequence of suitable random fields. The proof of Theorem 1 is given inSection 3. Finally, in Section 4, we show Theorem 2 and answer the abovequestion in the case that the claim R has exponential distribution.
2. Main tool
In this section we provide the needed tool to show that Theorem 1 holds.In particular, we construct the announced sequence converging to the localtime L .In the remaining of this paper, T i denotes the i -th jump time of N , with T = 0. It is known that T i has gamma distribution with parameters ( i, α ), i ≥ . We will use the following technical resul in the proofs of this section.
Lemma 1.
Let x ∈ R , s > , Ω ( s ) = { ∆ X s = 0 } , and Ω = { X s − = x, ∆ X s = 0 for some s > }∪{ X s = x, ∆ X s = 0 for some s > } . Then, P (Ω ( s )) = 0 and P (Ω ) = 0 . Proof.
By the law of total probability P (Ω ( s )) = ∞ X k =0 P ( N s = k ) P (Ω ( s ) | N s = k ) . Notice that P (Ω ( s ) | N s = k ) = P (∆ X s = 0 | N s = k ) = P ( T k = s ) = 0 . On the other hand, let ν ∈ N and define˜Ω ν = { X s − = x, ∆ X s = 0 for some 0 < s < ν }∪{ X s = x, ∆ X s = 0 for some 0 < s < ν } . For k = 0 , P ( ˜Ω ν | N ν = 0) = P ( ∅| N ν = 0) = 0 , HE LOCAL TIME OF THE CLASSICAL RISK PROCESS 5 and for k ≥ P ( ˜Ω ν | N ν = k ) ≤ P ( X T j − = x for some j ∈ { , ..., k } | N ν = k )+ P ( X T j = x for some j ∈ { , ..., k } | N ν = k ) ≤ k X j =1 ( P ( X T j − = x | N ν = k ) + P ( X T j = x | N ν = k )) . For j = 1 we get P ( X T − = x | N ν = k ) = P ( T = ( x − x ) c − | N ν = k ) = 0 ,P ( X T = x | N ν = k ) = P ( cT − R = x − x | N ν = k ) = 0 , this is because T and R are independent and absolutely continuous randomvariables. Let P ( ·| N ν = k ) = P ∗ ( · ). When j > P ∗ ( X T j − = x ) = Z R P ∗ ( X T j − = x | X T j − − = y ) P ∗ ( X T j − − ∈ dy )= Z R P ∗ ( R j − = y − ( x − ( T j − T j − ) c )) P ∗ ( X T j − − ∈ dy )= 0and P ∗ ( X T j = x ) = Z R P ∗ ( X T j = x | X T j − = y ) P ∗ ( X T j − ∈ dy )= Z R P ∗ (( T j − T j − ) c − R j = x − y ) P ∗ ( X T j − ∈ dy )= 0 . Here we have used the fact that R j − has an absolutely continuous distri-bution. Finally notice that P (Ω ) ≤ P ∞ ν =1 P ( ˜Ω ν ) = 0 . (cid:3) Now we ap-proximate the local time L by a sequence of suitable random fields, whichallows us to see that Theorem 1.a) is true. Toward this end, let x ∈ R arbitrary and fixed. For each n ∈ N define ϕ x,n : R → R by ϕ x,n ( y ) = , y < x − /n, ( n ( y − x ) + 1) / , x − /n ≤ y ≤ x + 1 /n, , x + 1 /n < y .Notice that lim n →∞ ϕ x,n ( y ) = , y < x, / , y = x, , y > x, = 12 1 { x } ( y ) + 1 ( x, + ∞ ) ( y ) , (2.1) F. CORTES, J.A. LE ´ON, AND J. VILLA and ϕ ′ x,n ( y ) = , y < x − /n,n/ , x − /n < y < x + 1 /n, , x + 1 /n < y. For each n ∈ N , we define the random field L nt ( x ) = 1 c ( ϕ x,n ( X t ) − ϕ x,n ( X ) − X s ≤ t { ϕ x,n ( X s ) − ϕ x,n ( X s − ) − ϕ ′ x,n ( X s − )∆ X s } ) , where ∆ X t = X t − X t − . As in (1.1), we have by (1.2) that L n is well-defined.Before proving that { L n , n ∈ N } is the sequence that we are looking for,we need to approximate the fuction ϕ x,n by a sequence of smooth functions.To do so, setΩ ′ = ( { X s − = x ± /m = X s , for some s > , m ∈ N }∪{ N s < + ∞ , for all s > } c ∪ Ω ) c . Since, by Lemma 1, P ( X s − = x ± /m = X s , for some s > , m ∈ N ) ≤ ∞ X m =1 P ( X s − = x ± /m = X s , for some s >
0) = 0 , we have P (Ω ′ ) = 1 . Let ψ : R → R a symmetric function in C ∞ ( R ) with compact supporton [ − ,
1] and Z − ψ ( y ) dy = 1 . Define the sequence ( ψ m ) by ψ m ( y ) = mψ ( my ) , y ∈ R , and ϕ mx,n ( y ) = ( ψ m ∗ ϕ x,n )( y ) = Z R ϕ x,n ( y − z ) ψ m ( z ) dz. Since ψ m ∈ C ∞ ( R ), then ϕ mx,n ∈ C ∞ ( R ) and moreover (cid:0) ϕ mx,n (cid:1) m converges uniformly on compacts to ϕ x,n , (2.2) (( ϕ mx,n ) ′ ) m converges pointwise, except on x ± /n , to ( ϕ x,n ) ′ . (2.3)Let us use the notation L n,mt ( x ) = 1 c Z (0 ,t ] ( ϕ mx,n ) ′ ( X s − ) dX s . HE LOCAL TIME OF THE CLASSICAL RISK PROCESS 7
Then, by the change of variable theorem for the Lebesgue-Stieltjes integral,we have cL n,mt ( x ) = ϕ mx,n ( X t ) − ϕ mx,n ( x ) − X
Let n ∈ N . Then, lim m →∞ L n,mt ( x ) = L nt ( x ) , for all t > , w . p . . Proof.
For ω ∈ Ω ′ we have that (1.2) and (2.2) implylim m →∞ ( ϕ mx,n ( X t ) − ϕ mx,n ( x ) − X
Now we are ready to state the properties of { L n , n ∈ N } that we use inSection 3. Proposition 2.
The sequence { L n , n ∈ N } fulfill: a) L nt ( x ) = R t ϕ ′ x,n ( X s − ) ds + c P , w.p.1. b) lim n →∞ L nt ( x ) = L t ( x ) , for all t > , w.p.1. F. CORTES, J.A. LE ´ON, AND J. VILLA
Proof.
We first deal with Statement a). Fix t ≥ ω ∈ Ω ′ ∩ Ω ( t ) c . Then there is k ∈ N such that N t ( ω ) = k . Thus Z t ( ϕ mx,n ) ′ ( X s − ) dX s = k X i =1 Z ( T i − ,T i ] ( ϕ mx,n ) ′ ( X s − ) dX s + Z ( T k ,t ] ( ϕ mx,n ) ′ ( X s − ) dX s = k X i =1 Z ( T i − ,T i ] (cid:0) ϕ mx,n (cid:1) ′ ( X s − ) cds + Z [ T k ,t ) (cid:0) ϕ mx,n (cid:1) ′ ( X s ) cds + k X i =1 (cid:0) ϕ mx,n (cid:1) ′ ( X T i − )( X T i − X T i − )= c Z t ( ϕ mx,n ) ′ ( X s − ) ds + k X i =1 (cid:0) ϕ mx,n (cid:1) ′ ( X T i − )( X T i − X T i − ) . Notice that on each ( T i − , T i ] and ( T k , t ], there is at most one s such that X s − = x ± /n . Hence, by (2.3), we havelim m →∞ ( ϕ mx,n ) ′ ( X s − ) = ( ϕ x,n ) ′ ( X s − ) , λ - a.s. Therefore, by the dominated convergence theorem, we deducelim m →∞ Z t ( ϕ mx,n ) ′ ( X s − ) dX s = c Z t ( ϕ x,n ) ′ ( X s − ) ds + k X i =1 ( ϕ x,n ) ′ ( X T i − )∆ X T i , a.s. Consequently, the fact that L nt ( x ) and the right-hand side of last equalityare c`adl`ag processes implies that Statement a) holds.Now we consider Statement b) in order to finish the proof of the propo-sition. Let ω ∈ Ω ′ and t ≥
0. Then there exist k ∈ N such that N t ( ω ) = k .Hencelim n →∞ ( ϕ x,n ( X t ( ω )) − ϕ x,n ( x ) − k X i =1 { ϕ x,n ( X T i ( ω )) − ϕ x,n ( X T i − ( ω )) } )= 12 1 { x } ( X t ( ω )) + 1 ( x, ∞ ) ( X t ( ω )) −
12 1 { x } ( x ) − ( x, ∞ ) ( x ) − k X i =1 {
12 1 { x } ( X T i ( ω )) + 1 ( x, ∞ ) ( X T i ( ω )) −
12 1 { x } ( X T i − ( ω )) − ( x, ∞ ) ( X T i − ( ω )) } . HE LOCAL TIME OF THE CLASSICAL RISK PROCESS 9
On the other hand ω ∈ { X s − = x = X s , for some s > } c implies X T i − ( w ) = x, i = 0 , , ..., k. Here there exist a finite number of indexes i such that X T i ( w ) ≤ x < X T i − ( w ) . For large enough n we have X T i − ( ω ) / ∈ ( x − /n, x + 1 /n ) , i = 0 , , ..., k. Therefore lim n →∞ X
3. Proof of Theorem 1
The purpose of this section is to give the proof of Theorem 1. This proofwill be divided into three steps. It is worth mentioning that the proof ofStatement a) gives us a sequence of absolutely continuous random fields thatconverges to L . Namely, the sequence { R t ϕ ′ x,n ( X s − ) ds, n ∈ N } . From part a) and b) of Propo-sition 2 we have L t ( x ) = lim n →∞ L nt ( x ) = lim n →∞ Z t ϕ ′ x,n ( X s − ) ds. which yields that ( L n · ( x )) is non-negative and increasing. Since X s ≤ X s − we have L t ( x ) = 1 c ( 12 1 { X t } ( x ) −
12 1 { x } ( x ) + 1 ( −∞ ,X t ) ( x ) − ( −∞ ,x ) ( x )+ X x , X t < x and C t ( x ) = n. Let c , ..., c n thecrossing times with the level x . Then, by hypothesis, there exist jumpingtimes s ∈ (0 , c ),..., s n +1 ∈ ( c n , t ) such that x ∈ ( X s i , X s i − ). Hence1 ( −∞ ,X t ) ( x ) − ( −∞ ,x ] ( x ) + X
From (3.2) and (3.3) we have Z t g ( X s ) ds = 1 c Z R g ( x ) k X i =1 ( X Ti ,X Ti − ] ( x ) dx + 1 c Z R g ( x ) k X i =1 hh X Ti − ,X Ti ii ( x ) dx + 1 c Z R g ( x )1 ( X Tk ,X t ] ( x ) dx = 1 c Z R g ( x ) k X i =1 ( X Ti ,X Ti − ] ( x ) dx + 1 c Z R g ( x )1 hh X T ,X Tk ii ( x ) dx + 1 c Z R g ( x )1 ( X Tk ,X t ] ( x ) dx = 1 c Z R (1 hh X ,X t ii + k X i =1 ( X Ti ,X Ti − ] )( x ) g ( x ) dx = Z R c (1 ( −∞ ,X t ] − ( −∞ ,X ] + k X i =1 ( X Ti ,X Ti − ] )( x ) g ( x ) dx. Thus, the proof is complete by (3.1).
4. An occupation measure result By F ( · , t ) we denote the distribution of P N t k =1 R k [ N t > , and by f ( · , t )the density of F ( · , t ) , when it exists. In order to use Theorem 2 we needan expression for E [ L t ( x )], which is given in [ ] (Proposition 1). Namely, if f ∈ L ( R × [0 , t ]), then(4.1) E [ L t ( x )] = Z t [(( x − x ) /c ) ∨ ∧ t f ( x + cs − x, s ) ds. Consider the measurable set∆ = [0 , ∞ ) × [0 , ∞ ) ∈ B ( R ) . Then, from Theorem 2 and (4.1), we get E [ Z t ∆ ( X s , X s + ε − X s ) ds ]= Z R E [1 ∆ ( x, X ε − x )] Z t [(( x − x ) /c ) ∨ ∧ t f ( x + cs − x, s ) dsdx = Z ∞ P ( x ≤ X ε ) Z t [(( x − x ) /c ) ∨ ∧ t f ( x + cs − x, s ) dsdx = Z ∞ P ( N ε X k =1 R k ≤ cε ) Z t [(( x − x ) /c ) ∨ ∧ t f ( x + cs − x, s ) dsdx = Z ∞ F ( cε, ε ) Z t [(( x − x ) /c ) ∨ ∧ t f ( x + cs − x, s ) dsdx. Now assume that R has exponential distribution with parameter β ,then the density of P N t k =1 R k [ N t > is f ( x, t ) = e − αt − βx ∞ X n =1 ( βαt ) n x n − n !( n − ! (0 , ∞ ) ( x ) , t > . Hence, in this case, E [ Z t ∆ ( X s , X s + ε − X s ) ds ]= Z ∞ "Z cε e − αε − βy ∞ X n =1 ( βαε ) n y n − n !( n − dy + e − αε × Z t [(( x − x ) /c ) ∨ ∧ t e − αs e − β ( x + cs − x ) ∞ X k =1 ( βαs ) k ( x + cs − x ) k − k !( k − dsdx = Z x "Z cε e − αε − βy ∞ X n =1 ( βαε ) n y n − n !( n − dy + e − αε × Z t e − αs e − β ( x + cs − x ) ∞ X k =1 ( βαs ) k ( x + cs − x ) k − k !( k − dsdx + Z x + ctx "Z cε e − αε − βy ∞ X n =1 ( βαε ) n y n − n !( n − dy + e − αε × Z t ( x − x ) /c e − αs e − β ( x + cs − x ) ∞ X k =1 ( βαs ) k ( x + cs − x ) k − k !( k − dsdx. For example, under the conditions x = 4 , α = 1 , β = 1 , c = 1 . , t = 1 , HE LOCAL TIME OF THE CLASSICAL RISK PROCESS 13 with ε = 12 and considering five iterations on the sums we get E [ Z ∆ ( X s , X s +12 − X s ) ds ] ≈ Z "Z . e − − y X n =1 (12) n y n − n !( n − dy + e − × Z e − (2 . s − x X k =1 s k (4 + (1 . s − x ) k − k !( k − dsdx + Z . "Z . e − − y X n =1 (12) n y n − n !( n − dy + e − × Z x − / (1 . e − (2 . s − x X k =1 s k (4 + (1 . s − x ) k − k !( k − dsdx = 7 . × − . Note that this value may help the insurance company to decide if it investspart of its wealth in another assets.
We will use the monotone class theorem(see, for example, Ethier and Kurtz [ ], Theorem 4.2) to show that the resultholds. Set H = { ψ : R −→ R , ψ is measurable, bounded and satisfies (1.5) } . It is not difficult to see that H is a real linear space and, by Theorem 1, wehave Z R E [ L t ( x )] dx = E [ Z R L t ( x ) dx ] = E [ Z t R ( X s ) ds ] = t. It means, 1 R ∈ H . Moreover H is closed under monotone convergence: Let( ψ n ) ⊂ H , such that 0 ≤ ψ n ↑ ψ , ψ bounded, then ψ is measurable and E [ Z t ψ ( X s , X s + ε − X s ) ds ] = lim n →∞ E [ Z t ψ n ( X s , X s + ε − X s ) ds ]= lim n →∞ Z R E [ ψ n ( x, X ε − x )] E [ L t ( x )] dx = Z R E [ ψ ( x, X ε − x )] E [ L t ( x )] dx, which gives that ψ ∈ H .Now we use the notation K = { ψ : R −→ R , ψ ( · , ·· ) = 1 A ( · )1 B ( ·· ) , A, B ∈ B ( R ) } . Then the family K is closed under multiplication and K ⊂ H . In fact, byTheorem 1 we obtain E [ Z t A ( X s )1 B ( X s + ε − X s ) ds ]= Z t E [1 A ( X s )] E [1 B ( X s + ε − X s )] ds = Z t E [1 A ( X s )] E [1 B ( εc − N s + ε X k = N s +1 R k )] ds = Z t E [1 A ( X s )] E [1 B ( εc − N ε X k =1 R k )] ds = Z t E [1 A ( X s )] E [1 B ( X ε − x )] ds = E [1 B ( X ε − x )] Z t E [1 A ( X s )] ds = E [1 B ( X ε − x )] E [ Z R A ( x ) L t ( x ) dx ]= Z R E [1 A ( x )1 B ( X ε − x )] E [ L t ( x )] dx. Finally, the Dynkin monotone class theorem yields that the proof is finished.
Acknowledgement.
The last two authors would like to thank Cinvestav-IPN and Universidad Aut´onoma de Aguascalientes for their hospitality dur-ing the realization of this work.
References [1] S. Asmussen (2000). Ruin Probabilities, World Scientific Publishing Co., Singapure.[2] J. Bertoin (1996). L´evy Processes, Cambridge University Press.[3] S.N. Chiu, C. Yin (2002).
On occupation times for a risk process with reserve-dependent premium , Stochastic Models, (2), 245-255.[4] K.L. Chung, R.J. Williams (1990). Introduction to Stochastic Integration, Birkh¨auser,Boston.[5] S.N. Ethier, T.G. Kurtz (1986). Markov Processes: Characterizations and Conver-gence, John Wiley & Sons, New York.[6] P.J. Fitzsimmons, S.C. Port (1990). Local times, occupation times, and the Lebesguemeasure of the range of a L´evy process . Seminar on Stochastic Processes, 1989 (SanDiego, CA, 1989), 59–73, Progr. Probab. , Birkh¨auser, Boston.[7] E.T. Kolkovska, J.A. L´opez-Mimbela, J. Villa (2005). Occupation measure and localtime of classical risk processes,
Insurance: Mathematics and Economics, (3), 573-584.[8] J. Grandell (1991). Aspects of Risk Theory, Springer-Verlag, New York. HE LOCAL TIME OF THE CLASSICAL RISK PROCESS 15 [9] J. Hawkes (1986).
Local times as stationary processes , K.D. Elworthy (Ed.), Fromlocal times to global geometry, Pitman Research Notes in Math. Vol. , Chicago111-120.[10] P. L´evy (1948). Processus Stochastiques et Mouvement Brownien, Gauthier-Villars,Paris.[11] T. Rolski, H. Schmidli, V. Schmidt, J. Teugels (1999). Stochastic Processes for In-surance and Finance, John Wiley & Sons, New York.
Universidad Aut´onoma de Aguascalientes, Departamento de Matem´aticasy F´ısica, Av. Universidad 940, C.P. 20100 Aguascalientes, Ags., Mexico
E-mail address : [email protected] Cinvestav-IPN, Departamento de Control Autom´atico, Apartado Postal14-740, 07000 M´exico D.F., Mexico
E-mail address : [email protected] Universidad Aut´onoma de Aguascalientes, Departamento de Matem´aticasy F´ısica, Av. Universidad 940, C.P. 20100 Aguascalientes, Ags., Mexico
E-mail address ::