SELF-SIMILAR PROCESSES IN COLLECTIVE RISK THEORY

Collective risk theory is concerned with random fluctuations of the total assets and the risk reserve of an insurance company. In this paper we consider self-similar, continuous processes with stationary increments for the renewal model in risk theory. We construct a risk model which shows a mechanism of long range dependence of claims. We approximate the risk process by a self-similar process with drift. The ruin probability within finite time is estimated for fractional Brownian motion with drift. A similar model is applicable in queueing systems, describing long range dependence in on/off processes and associated fluid models. The obtained results are useful in communication network models, as well as storage and inventory models.


Introduction
Consider a company which only writes ordinary insurance policies such as accident, disability, health and whole life.The policyholders pay premiums regularly, and at certain random times report claims to the company.A policyholder's premium, the gross risk premium, is a positive amount composed of two components.The net risk premium is the component calculated to cover the payments of claims on the aver- age.The security risk premium, or safety loading, is the component which protects the company from large deviations of claims from the average, and also allows an accumulation of capital.When a claim occurs, the company pays the policyholder a positive amount called the positive risk sum.
As a mathematical model for this situation, we shall assume that claims occur at 1Supported in part by the Swedish Research Council for Engineering Sciences grant 908911 TFR 91-747.jumps of a point process (N(t):t > 0).While most work in collective risk theory assumes that N(t) is a Poisson process, this restrictive assumption plays no role in our analysis.The successive risk sums (Yk:k ) are assumed to form a sequence which is stationary and strongly dependent, with E[Yk] # > 0. Furthermore, we shall assume that the initial risk reserve of the company is u > 0, and that the policyholders pay a gross risk premium of c > 0 per unit time.Thus, the risk process has the form: N(t) R(t) u + ct-E Yk" (1) k=l We define the ruin time T as the first time the company has a negative risk reserve: T inf{t > 0: R(t) < 0} if the set is nonempty, and T-otherwise.In order to avoid T < cx, a.s.we E(t) assume the net profit condition limt__.o> 0 holds.The principal problems of collective risk theory have been to calculate the ruin probability (u)= P{T < cx] R(0)= u}.Many of the results for these distributions are complicated expres- sions which have been obtained using analytical methods.For a comprehensive treat- ment of the theory up to 1955, the reader should consult Cramer [8].A more recent account of the theory is available in Chapter 7 of Takcs [44], Grandell [17], and As- mussen [3].By 1940, Hadwiger [19] was already comparing a discrete-time risk process with a diffusion.This can be viewed, though theoretically not comparable with the modern approach as the first treatment of diffusion approximations in risk theory.A more modern version, based on weak convergence, is due to Iglehart [20], Grandell [17], and Furrer, Michna and Weron [16].The basic premise is to let the number of claims grow in a unit time interval, and to make the claim sizes smaller in such a way that the risk process converges weakly to a self-similar process.The idea is to approximate the risk process with a self-similar process with drift.While the classical theory of risk processes requires independence of claims, this assumption can be drop- ped in our approach.A dependence of claims can guarantee that this model may be similar to a risk model with heavy-tailed claims.As an example of a risk process with such dependence, we construct a risk model in which claims appear in good and bad periods (e.g., good weather and bad weather).We assume that claims in bad per- iods are bigger than claims in good periods (e.g., the expected value of the claims in bad periods is bigger than the expected value of the claims in good periods).Under natural assumptions on the structure of the good and bad periods, we compute that the claims are strongly dependent.The use of the results obtained in this paper is motivated by communication net- work models, as well as storage and inventory models.Traffic on the data networks (e.g., Ethernet LANs) has characteristics substantially different from those of tradi- tional voice traffic.An important feature of data traffic lies in its dependence struct- ure; traditional models are based on assumptions of short-range dependence, while recent measurement and analysis of data traffic has produced strong indications of long-range dependence and self-similarity.Several empirical studies present statisti- cal evidence for existence of these non-standard dependence structures: see for exam- ple Heath, Resnick and Samorodnitsky [18]; Leland [24]; Leland and Wilson [25]; Leland, Taqqu, Willinger and Wilson [26, 27]; Norros [30-32]; Willinger, Taqqu,  Leland and Wilson [47]; Crovella and Bestavros [9]; and Cunha, Bestavros and  Crovella [10].In Norros [30-32], the cumulative traffic process (i.e., the amount of traffic arriving between 0 and t) is modeled with fractional Brownian motion (the self-similarity parameter H > 1/2).
To study the total traffic for all source-destination network pairs, Willinger, et al.   [47] first studied the asymptotic behavior of the integral of the covariance function for one such pair.Their methods (Laplace transform and Tauberian theorem) yield the interesting result that the superposition of a large number of suitably scaled, source-destination pairs is approximately a fractional Brownian motion.In Heath et   al. [18], one can find an explanation for the observed long range dependence and self- similarity in a simple on/off model applied in communication network models, as Well as storage and inventory models.
In applications involving source-destination network pairs, one defines the process with N and (Yk: k E ) as above.Consider the process V(t) suP0 < s < tS(s) as the workload process of a queueing system, with N representing the tirffe-rversed point process of arrivals of customers, and (Yk:k ) representing service time in reverse order.Then V(t) is the workload process.We assume that V is generated by a sta- tionary, marked point process M ((Tk, Uk): k G ), such that T k is the time of arri- val of the customer, and U k is his service time.By a standard procedure we can ex- tend M to a stationary, marked point process ((Tk, Uk): k Z), with doubly infinite time.
In this paper, in contrast to the classical, independent, identically distributed assumptions, we are interested in the case where (Yk:k G ) are strongly dependent.
In queueing systems, the relevance of such dependence assumptions is currently receiv- ing much attention as we mentioned above.As an example of a mechanism generat- ing such dependence, one can consider, the aforementioned alternating environment.The paper is organized as follow: Section 2 contains some preliminaries on weak convergence of stochastic processes in the Skorokhod topology and on self-similar pro- cesses; in Section 3, we define a sequence of risk processes and show that it converges weakly to a self-similar process with drift; Section 4 deals with the convergence of functionals of the risk process, showing that the finite-time passage probabilities con- verge; in Section 5, we briefly discuss our approximation when the claim arrival pro- cess is a renewal process; and in Section 6, we consider fractional Brownian motion as an example in risk theory.We give an approximation to the ruin probability when the initial capital is sufficiently large.

Preliminaries
In this section, we assemble those concepts and results from weak convergence theory as those apply to collective risk theory.Furthermore, we define a class II of process- es.
Denote by D the space of all cadlag (i.e., right-continuous with left-hand limits) functions on [0, oe) endowed with the Skorokhod topology (see Ethier and Kurtz  [4]).D[0, o)is a complete and separable metric space.All stochastic processes in this paper are assumed to be in D.
Definition 1: A sequence (X(n):n E N) of stochastic processes is said to converge weakly in the Skorokhod topology to a stochastic process X, if for every bounded con- tinuous functional f on D, it follows that: In this case, one writes X(')X.
One of the most useful results in weak convergence theory is the continuous mapp- ing theorem.Let h be a measurable mapping of S into another metric space S' with a-field f' of Borel sets.Each probability measure P on (S,b') induces on (S',f') a uni- que probability measure Ph-I(A)-P(h-1A) for A E b".Let D h be the set of dis- continuities of h.Then we have: Proposition 1: (Billingsley [7]) /f Pn and P are probability measures on (S,f)  such that Pn=P and P(Dh) O, then Pnh-l=:Ph-1.
In collective risk theory, we are interested in sums of a random number of ran- dom variables.We also need a general theorem of random change of time.Let I denote the identity function.
Proposition 2: Let (Bn:n e N) be processes in D[0, oc), B be a process with con- tinuous sample paths, and suppose that Bn==B.Let (Nn:n N) be a sequence of pro- cesses with nondecreasing sample paths starting from 0 such that Nn=I > O.
For each n N, B n and N n are assumed to be on the same probability space.Then: Bn(Nn)B(I). ( Proof: The process B has continuous sample paths so the assertion is an imme- diate application of the method used in Billingsley [7]. The concept of semi-stability was introduced by Lamperti [22].Mandelbrot and  Van Ness [29] call it self-similarity when appearing in conjunction with stationary in- crements as it does here.Definition 2: A process Z H possesses properties II (i.e., Z H II), if for some 0<H<I: 1.

2.
Z H has strictly stationary increments, that is the random function Uh(t ZH(t + h)-ZH(t), h > O, is strictly stationary.

3.
Z h is self-similar of order H (Hss), that is: in the sense of finite-dimensional distributions.EZH(t 0 and E ZH(t) " < cx3 for 7 < .

5.
Z H is a.s.continuous.
Examples of such a process are fractional Brownian motion [Gaussian process]   and the Rosenblatt process [non-Gaussian process] (see Mandelbrot and Van Ness [29]  and Rosenblatt [35]).If not stated otherwise explicitly, we make the following assumptions for the rest of the paper.

Weak Convergence of Risk Reserve Processes
We shall now construct a sequence of risk processes and show that it converges to a process Z H with drift where Z H E II.
The sequence (Q(n):n N) of risk processes is given as follows: for every n G N let u (n) > 0 denote the initial risk reserve, c(n)> 0 the premium rate, and N (n) the corresponding point process.The claim sizes are denoted by (yn): k e N).Then: We assume that the claims are of the form yn)_ 1 y k, where (Yk:k N) is a sta- tionary sequence with common distribution function F and mean # such that" [nt] (5) where (n)-hilL(n), and the function L is slowly varying at infinity.To construct such a sequence (see Taqqu [46]), let us take a stationary Gaussian sequence (Xk: Let G be a real-valued measurable function such that G(Xi) has mean 0 and finite variance.As mentioned above, We shall focus on values of H satisfying < H < 1.We assume E[XiX + k] for some slowly varying function L and some constant D > 0. H > arises when D < , where m, the Hermite rank of G, is the index of the first non-zero coefficient in the Hermit polynomials expansion of G.
Under these assumptions, = 1E[G(Xi)G(Xi + k)] , and the sequence (G(Xi): N) is so strongly depen- dent that the limit of may not be Gaussian.
Now we construct an insurance model, which as assumed above, produces strong- ly dependent claims.
We assume that we have good periods and bad periods when we observe arriving claims (e.g., periods ofood weather and periods of bad weather).These two periods G alternate.Let (T ,Tn,n N) be independent, identically distributed non-negative random variables representing good periods; similarly, let (SB, SBn, n N) be indepen- dent, identically distributed non-negative random variables representing bad periods.The T's are assumed independent of the S's, the common distribution of good periods is FG, and the distribution of bad periods is FB.We assume both F a and F B have finite means uG and UB, respectively, and we set u ua + UB" Consider the pure renewal sequence initiated by a good period (0, E = I(T + S), n N).The interarrival distribution is F a,F B and the mean interarrival time is u.This pure renewal process has stationary version (see Asmus-  sen [2]) [D, D + E 7= I(T + S), n N], where D is a delay random variable.How- ever, by defining the initial delay interval of length D this way, the interval does not decompose into a good and a bad period the way subsequent interarrival intervals do.
Consequently, we turn to an alternative construction of the stationary renewal pro- cess (see Heath et al. [18]).
Define three independent random variables B, To G and S0 G, which are independent of (S B, Tan San n G N) as follows" B is a Bernoulli random variable with values in {0, 1} and mass functions z G P{B -1} ---I P{B O} and (x > 0), where P{TGo > x}-f 1-Fa( )   ua ds 1 FGO (X), x p{SBo > x} F 1 uBrB(s).ds 1 FBo (x).One can verify that this delayed renewal sequence is stationary (see Heath et al. [18]).
We now define L(t) to be 1 if falls in a good period, and L(t)-0 if t is in a bad period.More precisely, the process (L(t), t >_ 0) is defined in terms of (Sn, n >_ O)   as follows: L(t) BI[o, Tao )(t + I a n=0 [Sn<-t<Sn+Tn+l)" (7) Proposition 3: The process (L(t), t > O) is strictly stationary and P{L(t) 1} EL(t) Proof: See Heath et al. [18], Proposition 2.1.Let (Yan,n _> 0) be independent, identically distributed random variables repre- senting claims appearing in good periods (e.g., Yn a describes a claim which may appear at the nth moment in a good period).Similarly, let (YBn,n >_ 0) be indepen- dent, identically distributed random variable representing claims appearing in bad periods (e.g, yB describes a claim which may appear at the nth moment in a bad period).We assume that a B (Yn,n > 0), (Yn,n > 0) and (L(t), t > 0) are independent, E[YGo] g and E[YBo]b (g < b), and the second moments of Y0 G and Y0 B exist.Then the claim Yn appearing at the nth moment (n > 0) is Yn L(n The sequence (Yn, n > 0) is stationary.
Proof: Let us notice that ( 11) We assumed that the good period dominates the bad period but one can approach the problem reversely, (e.g., the bad period can dominate the good period).
One can see the symmetry of this good and bad period characteristics in the covar- iance function (see Heath et al. [18]).This same argument can be used on/off models and associated fluid models.
Let us determine the limiting process of Z (n) given in (6).It is sufficient to study the convergence of the finite-dimensional distributions of process Z(n), when G Hm, where H m denotes the Hermite polynomial of order m.
() When m = 1, Z converges weakly to fractional Browman motion B H with parameter 1 / 2 < H-1-< 1.This limiting process is Gaussian, with zero mean and E IBH(t)-BH(s)I 2-t-sl 2H.The process is defined for 0<H<I.It is Brownian motion when H-1 / 2 .For a detailed treatment of BH, see Mandelbrot and   Van Ness [29], and Samorodnitsky and Taqqu [36].
When m-2, Z (n) converges weakly to the non-Gaussian process called the Rosenblatt process (see Rosenblatt [35]).
Partial results for m-3, where the limiting process is not Gaussian, are given in Taqqu [45].
We put Yk G(Xk) + # for k E We define the process (Q(t): t _> 0) by Q(t) u + ct-HZH(t), (12) where u and c are positive numbers, and (ZH(t)'t >_ 0) is a process endowed with pro- perties II(H).Here is some positive constant which will be specified in the next theorem.The following theorem shows that the sequence (Q(n):n N) converges weakly to the process in ( 12)" Theorem 3: sequence of point processes such that N(nl(t)-nt --0 in probability in the Skorokhod topology for some positive constant that Let the sequence (Yk:k E N) be as above and let (N(n):n ) be a (13) Assume also and lim u (n) u. (15 in the Skorokhod topology as Proof: Let us write the process Q()(t) in the following form" From the assumption in (13), we obtain in probability in the Skorokhod topology as n.From (13) and Proposition 2 we obtain that (')(t) in the Skorokhod topology as n--,cx.Because u(n)+t c (n)-)n converges to u / ct in probability in the Skorokhod topology, the proof is complete. [:]  The distribution of a risk process can be approximated by the distribution of process in (12).

The Convergence of Functionals of Risk Processes
Collective risk theory has mainly been concerned with functionals which represent the total assets of the insurance company at time t, namely Q(n), and with the ruin time T(n), which is defined as T (n) T(Q(n)), where T(x)-inf{t > 0: x(t) < 0} (17 if the set {t > 0:x(t) < 0} is not empty, and / cx otherwise.We need a theoretical result which permits us to approximate the finite-time ruin probability by the ruin probability of the corresponding process Q.The process given in (12) has continuous Sample paths.It is known that the convergence to a continuous function in the Sko- rokhod topology is equivalent to uniform convergence on compacts.Using Proposition 1, we get: Proposition 5: Let T be the functional defined in (17).If Q(n)=Q, with Q being given in (12), and Proof."Let x n converge to x in the Skorokhod topology, where x is a continuous trajectory of the process Q.Then x n tends to x uniformly on compacts.First assume that T(x)c.This means that x(t)> 0 for all t> 0 because we assumed that P{inf0<s<tQ(s)-O}-O for all t>0.Let N be such that, for sufficiently large n, xn(t-0 for all 0 _<t_< N. Letting Noc, we obtain that T(xn)---<x as Now let T(x)< cxz, and assume that T(xn)T and T > T(x)(more precisely there is such a subsequence of {T(xn)}).Then there is 6 > 0 such that x(T(x)+ 6) < 0 and T(x)+ 6 < T. Since xn(T(x + 5) < 0 for sufficiently large n, this is a con- tradiction.
Remark: Proposition 3 shows that the finite-time ruin probabilities converge" dkmP{T(Q()) <_ t} P{T(Q) <_ t}, or equivalently, n--,cx 0 < s < 0 < s < Hence, we can approximate the finite-time ruin probabilities by the probability of the first 0-downcrossing of the process in (12).However, it is not clear whether the convergence of the infinite-time ruin probabilities holds, i.e., whether: In Proposition 3, we proved much more (i.e., we proved that if Qn converges almost surely to Q then T(Q,)T(Q) a.s.).

A Renewal Type Model
To construct an example of risk processes which converge to a self-similar process, we have to check the conditions of Theorem 3. We consider the case where the occurrence of the claims is described by a renewal process N: N(t) max n" T k <_ k=l The inter-occurrence times (Tk:k E ) are assumed to be independent, positive ran- dom variables.We define N(n)(t)-N(nt). (20) Let B (B(t):t _> 0)denote standard one-dimensional Brownian motion.Then the following proposition holds: Proposition 6: Let (N(t): t > 0) be a renewal process with inter-occurrence times (Tk:k 5d), and assume that there exists a positive constant such that in the Skorokhod topology as n--,oc, where (n)-nl/2L(n) (L is slowly varying at infinity).Then, for 1/2 < H < Let us notice that if 0 < H < 1, then the above supremum is attained on bounded intervals.Because H > 1/2, the assumption in (21) implies that nH k__ l(Tk --) n--0 (25) in probability in the Skorokhod topology.In the second case (see ( 24)), we obtain ,kns en H E Tk> ns.k--1 For nu-Ansn H we have Thus, 1 sup nH 0 < u < ,kt-cn H 1 Finally, from(25) we have O < s < n g > e n 0 and the proof is complete.
Note that ( 21) is true in the ordinary renewal theory situation where (Tk) is a se- quence of independent, identically distributed random variables with mean and var- iance r2.In this case, we have N(nt)-nt =:er/3/2B(t).
6.An Example: Fractional Brownian Motion where 0 < H < 1, H is the parameter of self-similarity, C H is a real constant depend- ing on H, and B is a standard Brownian motion defined on the whole real line.
1 is Brown- The process B H possesses properties II.Observe that B H with H-Jan motion, and that in general, the process B H has stationary, but not independent increments.The fractional Brownian motion process B H is a very important generali- zation of Brownian motion because B H is the only Gaussian H-ss process with sta- tionary increments.The fractional Brownian motion has expectation El_BIt(t)] 0  Mandelbrot and Van Ness [29] and Samorodnitsky nd Taqqu [36].
We shall use the standard notation (I)(z) and (z) for the standard normal distri- bution and density functions, respectively.We recall the elementary relation: Let us define where u and c and A are positive constants.Recall that we assume 1 / 2 < H < 1.Our main aim is to find the ruin probability of the process in (30).We need bounds or limit theorems for the ruin probability of process B H because we do not know the exact form of this probability.This will be made by applying the easy consequence of the Normal Comparison Lemma (Slepian [42], see also Corollary 4.2.3 in Lead- better et al. [23]) and the continuity of the sample paths.Lemma 1: Let X 1 and X 2 be Gaussian continuous processes.Suppose that for We give another lemma: Lemma 2: Let B be a standard Brownian motion, and u >_ O, c >_ O. Then P{ si}fo(U + cs + B(s)) < 0} exp(-2uc). (32) Proof: See Grandell [17].
El Now we state a theorem which enables us to estimate the ruin probability of the process in (30) for an arbitrary amount of initial capital: Theorem 6: Let Q be the process given in (30).Then P{T(Q) < < I-4P (f f( + ct )_ ,kt)H + exp{-2uct I 1 ((u-ct (33)  pression is equal to which completes the proof.
[] An immediate consequence of Theorem 2 is the following result known for the Lvy processes.
The following theorem enables us to approximate the ruin probability of the pro- cess in (30) for a sufficiently large initial capital.
The numerator in (39) is clearly greater than or equal to the denominator.Thus it suffices to show that P{T(Q)<t} lim sup_.
The time horizon t equals 5.

Table 1 :
Comparison of the finite-time ruin probabilities of the fractional