Upper and lower bounds for ruin probability E. Pancheva,Z.Volkovich and L.Morozensky 3 Institute of Mathematics and Informatics, the Bulgarian Academy of Sciences, 3 Sofia, Bulgaria pancheva@math.bas.bg Software Engineering Department, ORT Braude College of Engineering, Karmiel 9, Israel Affiliate Professor, Department of Mathematics and Statistics, the University of Maryland, Baltimore County, USA vlvolkov@ort.org.il 3 Software Engineering Department, ORT Braude College of Engineering, Karmiel 9, Israel leonatm@bezeqint.net Abstract In this note we discuss upper and lower bound for the ruin probability in an insurance model with very heavy-tailed claims and interarrival times. Key Words and Phrases: compound extremal processes; α-stable approximation; ruin probability Correspondence author
Backgrounds The framework of our study is set by a given Bernoulli point process (Bpp) N = {(T k,x k ):k } on the time-state space S =(, ) (, ). By definition (cf. Balkema and Pancheva 99) N is simple in time (T k T j a.s. for k j), its mean measure is finite on compact subsets of S and all restrictions of N to slices over disjoint time intervals are independent. We assume that: a) the sequences {T k } and {X k } are independent and defined on the same probability space; b) the state points {X k } are independent and identically distributed random variables (iid rv s) on (, ) with common distribution function (df) F which is asymptotically continuous at infinity; c) the time points {T k } are increasing to infinity, i.e. <T <T <..., T k a.s. The main problem in the Extreme Value Theory is the asymptotic of the extremal process { X k : T k t} = N(t) X k, associated with N,fort. k k= Here the maximum operation between rv s is denoted by andn(t) := max{k : T k t} is the counting process of N. The method usually used is to choose proper time-space changes ζ n =(τ n (t),u n (x)) of S (i.e. strictly increasing and continuous in both components) such that for n and t> the weak convergence Ỹ n (t) :={ u n (X k):τn (T k) t} = Ỹ (t) () k to a non-degenerate extremal process holds. (For weak convergence of extremal processes consult e.g. Balkema and Pancheva 99.) In fact, the classical Extreme Value Theory deals with Bpp s {(t k,x k ): k } with deterministic time points t k,<t <t <..., t k. One investigates the weak convergence to a non-degenerate extremal process Y n (t) :={ k u n (X k ):t k τ n (t)} = Y (t) () under the assumption that the norming sequence {ζ n } is regular. The later means that for all s > and for n there exist point-wise lim n u n u [ns] (x) = U s (x) lim τ n n τ [ns](t) =σ s (t) and (σ s (t), U s (x)) is a time-space change. As usual means the composition and [s] the integer part of s. The family L = {(σ s (t), U s (x)) : s>} forms a continuous one-parameter group w.r.t. composition. Let us denote the (deterministic) counting function k(t) =max{k : t k t}, and put k n (t) :=k(τ n (t)), k n := k n (). The df of the limit extremal process
in() wedenotebyg(t, x) := P(Y (t) < x), and set G(x) := g(,x). Then necessary and sufficient conditions for convergence () are the following. F kn (u n (x)) w G(x), n. kn(t) k n λ(t), n, t >. The regularity of the norming sequence {ζ n } has some important consequences (cf. Pancheva 99). First of all, the limit extremal process Y (t) is self-similar w.r.t. L, i.e. U s Y (t) d = Y σ s (t), s >. Furthermore:. k [ns] k n s a,n,forsomea>andalls>;. the limit df G is max-stable in the sense that G s (x) =G(L s (x)) s >, L s := U a s; (3). the intensity function λ(t) is continuous. Thus, under conditions. and. and the regularity of the norming sequence, the limit extremal process Y (t) is stochastically continuous with df g(t, x) =G λ(t) (x) and the process Y λ (t) is max-stable in the sense of (3). Let us come back to the point process N with the random time points T k. The Functional Transfer Theorem (FTT) in this framework gives conditions on N for the weak convergence () and determines the explicit form of the limit df f(t, x) :=P(Ỹ (t) <x). In other words, the weak convergence () in the framework with non-random time points can be transfer to the framework of N if some additional condition on the point process N is met. In our case this is condition d) below. Denote by M([, )) the space of all strictly increasing, cadlac functions y :[, ) [, ), y() =, y(t) as t. We assume additionally to a) - c) the following condition d) θ n (s) :=τn (T [skn]) = T (s) where T :[, ) [, ) is a random time change, i.e. stochastically continuous process with sample paths in M([, )). Let us set N n (t) :=N(τ n (t)). In view of condition d) the sequence Λ n (t) := N n(t) = max{k : T k τ n (t)} k n k n = sup{s >:τn (T [skn]) t} = sup{s >:θ n (s) t} is weakly convergent to the inverse process of T (s). Let us denote it by Λ and let Q t (s) =P (Λ(t) <s). 3
Now we are ready to state a general FTT for maxima of iid rv s on (, ). Theorem (FTT): Let N = {(T k,x k ):k } be a Bpp described by conditions a) - c). Assume further that there is a regular norming sequence ζ n (t, x) =(τ n (t),u n (x)) of time-space changes of S such that for n and t> conditions.,. and d) hold. Then i) Nn(t) k n ii) P( Nn(t) d Λ(t) k= u n (X k) <x) w E[G(x)] Λ(t) Indeed, we have to show only ii). Observe that for n N n (t) =k n. N n(t) k n.λ(t) k n (λ Λ(t)) k n In the last asymptotic relation we have used condition ). Then by convergence () Ỹ n (t) = Nn(t) (X k )= Y (λ Λ(t)) and k= u n P( Nn(t) k= u n (X k) <x) f(t, x) = G s (x)dq t (s) =E[G(x)] Λ(t) Let us apply these results to a particular insurance risk model. Application to ruin probability The insurance model, we are dealing with here, can be described by a particular Bpp N = {(T k,x k ):k } where a) the claim sizes {X k } are positive iid random variables which df F has a regularly varying tail, i.e. F RV α. We consider the very heavy tail case <α<whenex does not exist, briefly EX = ; b) the claims occur at times {T k } where <T <T <... < T k a.s. We denote the inter-arrival times by J k = T k T k, k, T =and assume the random variables {J k } positive iid with df H. Suppose H RV β, <β<; c) both sequences {X k } and {T k } are independent and defined on the same probability space. The point process N generates the following random processes we are interested in.
i) The counting process N(t) =max{k : T k t}. It is a renewal process with N(t) t ast forej =. By the Stable CLT there exists a normalizing sequence {b(n)}, b(n) >, such that [nt] J k k= b(n) converges weakly to a β- stable Levy process S β (t). One can choose b(n) n /β L J (n), where L J denotes a slowly varying function. Let us determine b(n) by the asymptotic relation b( b(n)) n as n. Now the normalized counting process N(nt) is b(n) weakly convergent to the hitting time process E(t) =inf{s : S β (s) >t} of S β, see Meerschaert and Scheffler (). As inverse of S β, E(t) isβ-selfsimilar. ii) The extremal claim process Y (t) ={ X k : T k t} = N(t) X k. In view k= of assumption a) there exist norming constants B(n) n /α L X (n) such that [nt] k= X k B(n) converges weakly to an extremal process Y α(t) with Frechet marginal df, i.e. P (Y α (t) <x)=φ t α(x) =exp tx α.consequently, Y n (t) := N(nt) k= X k B( b(n)) = Y α(e(t)). Below we use the β α - selfsimilarity of the compound extremal process Y α(e(t)) (see e.g. Pancheva et al. 3). iii) The accumulated claim process S(t) = N(t) k= X k. Using the same norming sequence as above we observe that S n (t) := N(nt) k= X k B( b(n)) = Z α(e(t)). Here Z α is an α-stable Levy process and the composition Z α (E(t)) is β α - selfsimilar. iv) The risk process R(t) =c(t) S(t). Here u := c() is the initial capital and c(t) denotes the premium income up to time t, hence it is an increasing curve. We assume c(t) right-continuous. Note, the extremal claim process Y (t) and the accumulated claim process S(t) need the same time-space changes ζ n (t, x) =(nt, x B( b(n)) )toachieveweak convergence to a proper limiting process. In fact, {ζ n } makes the claim sizes smaller and compensates this by increasing their number in the interval [,t]. Both processes Y n (t) and S n (t) are generated by the point process N n = {( T k n, X k ) : k }. B( b(n)) risk processes R n (t) = the condition d) c(nt) B( b(n)) With the latter we also associate the sequence of c(nt) B( b(n)) S n(t). Let us assume additionally to a) - c) w c (t), c increasing curve with c () >. 5
Under conditions a) - d) the sequence R n converges weakly to the risk process (cf Furrer et al. 997) R α,β (t) =c (t) Z α (E(t)) with initial capital u = c (). Using the R α,β - approximation of the initial risk process R(t), when time and initial capital increase with n, we next obtain upper ( ψ) and lower(ψ) bound for the ruin probability Ψ(c, t) :=P (inf s t R(s) < ). Let Z α () and E() have df s G α and Q, resp. Then we have : ψ(c,t) := P ( inf α,β(s) < ) s t P (sup s t Z α (E(s)) >u ) P (Z α (E(t)) >u ) ) α = Q(( u )dg xt β α (x) =: ψ(c,t) α Here Q = Q. On the other hand ψ(c,t) P (Y α (E(t)) >c (t)) ( ) α c (t) = Q( )dφ α (x) =:ψ(c,t) xt β α Here we have used the self-similarity of the processes Z α, Y α and E. Thus, finally we get ψ(c,t) ψ(c,t) ψ(c,t) Remember, our initial insurance model was described by the point process N with the associated risk process R(t). We have denoted the corresponding ruin probability by Ψ(c, t) withu = c(). Then N(s) Ψ(c, t) =P ( inf {c(s) X k } < ) s t k= = P ( inf { c(ns) N(ns) s t n B( b(n)) X k B( b(n)) } < ) k= Now let initial capital u and time t increase with n in such a way that u = u t B( b(n)), n = t. We observe that under conditions a) - d) we may approximate Ψ(c, t) ψ(c,t ) and consequently for u and t large enough ψ(c,t ) Ψ(c, t) ψ(c,t ) ()
3 Examples Assume that our model is characterized by α =, i.e. the df of Z α () is the Levy df G α (x) =( Φ( x )). Here Φ is the standard normal df. We suppose also that the random variable E() is Exp()-distributed, namely Q(s) = e s, s. Further, let us take the income curve c to be of the special form c (t) =u + t β α c, c positive constant, that agrees with the selfsimilarity of the process Z α (E(t)). Now the upper bound depends on (u,t,β) and the lower bound depends on (u,t,β,c). We calculate the bounds ψ and ψ in two cases α > β = and α < β =.75 by using MATLAB7. The results of the calculations show clearly that in case β>α, when large claims arrive often, the bounds of the ruin probability are larger than in the case β<α, even in small time interval. Note,ifwechoosetheincomecurveintheabovespecialform,wemay calculate the ruin probability ψ(c,t ) in the approximating model exactly, namely ψ(c,t ) = P ( inf {u + s β α c Zα (E(s))} < ) s t = P ( inf {s β α (c Zα (E()))} < u ) s t = P ( inf {s β α (c Zα (E()))} < u, s t c Z α (E()) < ) = P (t β α (c Z α (E())) < u, c Z α (E()) < ) = P (Z α (E()) >c+ u ) t β α ( ) α c (t ) = Q( )dg xt β α (x) α Below we give graphical results related to the computation of ψ(c,t ), ψ(c,t )and ψ(c,t ) in the cases: c=., c=, c= when α = and β = β =.75. 7
Graphics of ψ(c,t ),ψ(c,t ) and ψ(c,t )...3...5 Figure : α =,β =,c=....3...5 Figure : α =,β =,c=
...3...5 Figure 3: α =,β =,c=......5 Figure : α =,β =.75,c=. 9
.....5 Figure 5: α =,β =.75,c=.....5 Figure : α =,β =.75,c=.
References [] Balkema A., Pancheva E. (99) Decomposition for Multivariate Extremal Processes. Commun. Statist.-Theory Meth., Vol. 5, Nr.. 737 75. [] Furrer H., Michna Zb. (997) Stable Levy Motion Approximation in Collective Risk Theory. Insurance : Mathem. and Econom., Vol.. 97. [3] Meerschaert M.M., Scheffler H-P. () Limit Theorems for Continuous Time Random Walks. Submitted. [] Pancheva E., Kolkovska E., Yordanova P. (3) Random Time-Changed Extremal Processes. Comun. Tecnica No I-3-/7--3 PE/CIMAT. [5] Pancheva E. (99) Self-similar Extremal Processes. J. Math. Sci., Vol. 9, Nr. 3. 39 39.