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

Morozensky, L.

Pancheva, E.

Volkovich, Z.

Bulgarian Digital Mathematics Library at IMI-BAS

Pliska Stud. Math. Bulgar. 18 (2007), 315–326STUDIA MATHEMATICABULGARICAUPPER AND LOWER BOUNDS FOR RUIN PROBABILITYE. Pancheva1 Z. Volkovich L. MorozenskyIn this note we discuss upper and lower bound for the ruin probability inan insurance model with very heavy-tailed claims and interarrival times.1. BackgroundsThe framework of our study is set by a given Bernoulli point process (Bpp)N = {(Tk, Xk) : k ≥ 1} on the time-state space S = (0,∞) × (0,∞). Bydefinition (cf. Balkema and Pancheva 1996) N is simple in time (Tk 6=Tj a.s. fork 6= j), its mean measure is finite on compact subsets of S and all restrictions ofN to slices over disjoint time intervals are independent. We assume that:a) the sequences {Tk} and {Xk} are independent and defined on the sameprobability space;b) the state points {Xk} are independent and identically distributed randomvariables (iid rv’s) on (0,∞) with common distribution function (df) Fwhich is asymptotically continuous at infinity;c) the time points {Tk} are increasing to infinity, i.e. 0 < T1 < T2 < · · · ,Tk →∞ a.s.1Correspondence author.2000 Mathematics Subject Classification:Key words: compound extremal processes; α-stable approximation; ruin probability316 E. Pancheva, Z. Volkovich, L. MorozenskyThe main problem in the Extreme Value Theory is the asymptotic of theextremal process {∨kXk : Tk ≤ t} =N(t)∨k=1Xk, associated with N , for t→∞. Herethe maximum operation between rv’s is denoted by “∨” and N(t) := max{k :Tk ≤ t} is the counting process of N . The method usually used is to chooseproper time-space changes ζn = (τn(t), un(x)) of S (i.e. strictly increasing andcontinuous in both components) such that for n → ∞ and t > 0 the weakconvergence(1) Y˜n(t) := {∨ku−1n (Xk) : τ−1n (Tk) ≤ t} =⇒ Y˜ (t)to a non-degenerate extremal process holds. (For weak convergence of extremalprocesses consult e.g. Balkema and Pancheva 1996.)In fact, the classical Extreme Value Theory deals with Bpp’s {(tk, Xk) : k ≥1} with deterministic time points tk, 0 < t1 < t2 < · · · , tk →∞. One investigatesthe weak convergence to a non-degenerate extremal process(2) Yn(t) := {∨ku−1n (Xk) : tk ≤ τn(t)} =⇒ Y (t)under the assumption that the norming sequence {ζn} is regular. The later meansthat for all s > 0 and for n→∞ there exist point-wiselimn→∞u−1n ◦ u[ns](x) = U s(x)limn→∞τ−1n ◦ τ[ns](t) = σs(t)and (σs(t), U s(x)) is a time-space change. As usual “◦” means the compositionand [s] the integer part of s. The family L = {(σs(t), U s(x)) : s > 0} forms acontinuous one-parameter group w.r.t. composition.Let us denote the (deterministic) counting function k(t) = max{k : tk ≤ t},and put kn(t) := k(τn(t)), kn := kn(1). The df of the limit extremal process in(2) we denote by g(t, x) := P(Y (t) < x), and set G(x) := g(1, x). Then necessaryand sufficient conditions for convergence (2) are the following1. F kn(un(x))w−→G(x), n→∞2.kn(t)kn−→ λ(t), n→∞, t > 0.Upper and Lower Bounds for Ruin Probability 317The regularity of the norming sequence {ζn} has some important consequences(cf. Pancheva 1998). First of all, the limit extremal process Y (t) is self-similarw.r.t. L, i.e.Us ◦ Y (t)d= Y ◦ σs(t), ∀s > 0 .Furthermore:0.k[ns]kn−→ sa, n→∞, for some a > 0 and all s > 0;1’. the limit df G is max-stable in the sense that(3) Gs(x) = G(L−1s (x)) ∀s > 0, Ls := U a√s;2’. the intensity function λ(t) is continuous.Thus, under conditions 1. and 2. and the regularity of the norming sequence,the limit extremal process Y (t) is stochastically continuous with df g(t, x) =Gλ(t)(x) and the process Y ◦ λ−1(t) is max-stable in the sense of (3).Let us come back to the point process N with the random time points Tk.The Functional Transfer Theorem (FTT) in this framework gives conditions onN for the weak convergence (1) and determines the explicit form of the limitdf f(t, x) := P(Y˜ (t) < x). In other words, the weak convergence (2) in theframework with non-random time points can be transfer to the framework of Nif some additional condition on the point process N is met. In our case this iscondition d) below.Denote by M([0,∞)) the space of all strictly increasing, cadlac functionsy : [0,∞) → [0,∞), y(0) = 0, y(t) → ∞ as t → ∞. We assume additionally toa)–c) the following conditiond) θn(s) := τ−1n (T[skn]) =⇒ T (s)where T : [0,∞) → [0,∞) is a random time change, i.e. stochastically continuousprocess with sample paths in M([0,∞)). Let us set Nn(t) := N(τn(t)). In viewof condition d) the sequence318 E. Pancheva, Z. Volkovich, L. MorozenskyΛn(t) :=Nn(t)kn=1knmax{k : Tk ≤ τn(t)}= sup{s > 0 : τ−1n (T[skn]) ≤ t}= sup{s > 0 : θn(s) ≤ t}is weakly convergent to the inverse process of T (s). Let us denote it by Λ andlet Qt(s) = P (Λ(t) < s).Now we are ready to state a general FTT for maxima of iid rv’s on (0,∞).Theorem (FTT): Let N = {(Tk, Xk) : k ≥ 1} be a Bpp described byconditions a)–c). Assume further that there is a regular norming sequence ζn(t, x) =(τn(t), un(x)) of time-space changes of S such that for n→∞ and t > 0 conditions1., 2. and d) hold. Theni)Nn(t)knd−→Λ(t)ii) P(Nn(t)∨k=1u−1n (Xk) < x)w−→E[G(x)]Λ(t)Indeed, we have to show only ii). Observe that for n→∞Nn(t) = kn.Nn(t)kn∼ kn.Λ(t) ∼ kn(λ−1 ◦ Λ(t))In the last asymptotic relation we have used condition 2). Then by convergence(2)Y˜n(t) =Nn(t)∨k=1u−1n (Xk) =⇒ Y (λ−1 ◦ Λ(t))andP(Nn(t)∨k=1u−1n (Xk) < x) −→ f(t, x)∞∫0Gs(x)dQt(s) = E[G(x)]Λ(t)Let us apply these results to a particular insurance risk model.Upper and Lower Bounds for Ruin Probability 3192. Application to ruin probabilityThe insurance model, we are dealing with here, can be described by a particularBpp N = {(Tk, Xk) : k ≥ 1} wherea) the claim sizes {Xk} are positive iid random variables which df F has aregularly varying tail, i.e. 1−F ∈ RV−α. We consider the “very heavy tailcase” 0 < α < 1 when EX does not exist, briefly EX = ∞ ;b) the claims occur at times {Tk} where 0 < T1 < T2 < · · · < Tk → ∞a.s. We denote the inter-arrival times by Jk = Tk − Tk−1, k ≥ 1, T0 = 0and assume the random variables {Jk} positive iid with df H. Suppose1−H ∈ RV−β, 0 < β < 1;c) both sequences {Xk} and {Tk} are independent and defined on the sameprobability space.The point process N generates the following random processes we areinterested in.i) The counting process N(t) = max{k : Tk ≤ t}. It is a renewalprocess withN(t)t→ 0 as t → 0 for EJ∞. By the Stable CLT thereexists a normalizing sequence {b(n)}, b(n) > 0, such that[nt]∑k=1Jkb(n)converges weakly to a β- stable Levy process Sβ(t). One can chooseb(n) ∼ n1/βLJ(n), where LJ denotes a slowly varying function. Let usdetermine b˜(n) by the asymptotic relation b(b˜(n)) ∼ n as n→∞. Nowthe normalized counting processN(nt)b˜(n)is weakly convergent to thehitting time process E(t) = inf{s : Sβ(s) > t} of Sβ, see Meerschaertand Scheﬄer (2002). As inverse of Sβ, E(t) is β-selfsimilar.ii) The extremal claim process Y (t) = {∨Xk : Tk ≤ t} =N(t)∨k=1Xk. Inview of assumption a) there exist norming constants B(n) ∼ n1/αLX(n)such that[nt]∨k=1XkB(n)converges weakly to an extremal process Yα(t)with Frechet marginal df, i.e. P (Yα(t) < x)Φtα(x) = exp−tx−α.Consequently,Yn(t) :=N(nt)∨k=1XkB(b˜(n))=⇒ Yα(E(t)).320 E. Pancheva, Z. Volkovich, L. MorozenskyBelow we use theβα-selfsimilarity of the compound extremal processYα(E(t)) (see e.g. Pancheva et al. 2003).iii) The accumulated claim process S(t) =N(t)∑k=1Xk. Using the samenorming sequence as above we observe thatSn(t) :=N(nt)∑k=1XkB(b˜(n))=⇒ Zα(E(t)).Here Zα is an α-stable Levy process and the composition Zα(E(t)) isβα-selfsimilar.iv) The risk process R(t) = c(t) − S(t). Here u := c(0) is the initialcapital and c(t) denotes the premium income up to time t, hence it isan increasing curve. We assume c(t) right-continuous.Note, the extremal claim process Y (t) and the accumulated claim processS(t) need the same time-space changes ζn(t, x) =(nt,xB(b˜(n)))to achieveweak convergence to a proper limiting process. In fact, {ζn} makes theclaim sizes smaller and compensates this by increasing their number in theinterval [0, t]. Both processes Yn(t) and Sn(t) are generated by the pointprocess Nn ={(Tkn,XkB(b˜(n))): k ≥ 1}. With the latter we also associatethe sequence of risk processes Rn(t) =c(nt)B(b˜(n))− Sn(t). Let us assumeadditionally to a)–c) the conditiond)c(nt)B(b˜(n))w→ c0(t), c0 increasing curve with c0(0) > 0.Under conditions a)–d) the sequence Rn converges weakly to the risk process(cf Furrer et al. 1997) Rα,β(t) = c0(t)−Zα(E(t)) with initial capital u0 = c0(0).Using the Rα,β-approximation of the initial risk process R(t), when time andinitial capital increase with n, we next obtain upper (ψ¯) and lower(ψ) bound forthe ruin probability Ψ(c, t) : P (inf0≤s≤tR(s) < 0). Let Zα(1) and E(1) have df’sGα and Q, resp. Then we have:Upper and Lower Bounds for Ruin Probability 321ψ(c0, t) := P ( inf0≤s≤tRα,β(s) < 0)≤ P ( sup0≤s≤tZα(E(s)) > u0)≤ P (Zα(E(t)) > u0)=∞∫0Q¯((u0xtβα)α)dGα(x) =: ψ¯(c0, t)Here Q¯ = 1−Q. On the other handψ(c0, t) ≥ P (Yα(E(t)) > c0(t))=∞∫0Q¯((c0(t)xtβα)α)dΦα(x) =: ψ(c0, t)Here we have used the self-similarity of the processes Zα, Yα and E. Thus, finallywe getψ(c0, t) ≤ ψ(c0, t) ≤ ψ¯(c0, t)Remember, our initial insurance model was described by the point process Nwith the associated risk process R(t). We have denoted the corresponding ruinprobability by Ψ(c, t) with u = c(0). ThenΨ(c, t) = P inf0≤s≤t{c(s) −N(s)∑k=1Xk} < 0= P inf0≤s≤ tn c(ns)B(b˜(n)) −N(ns)∑k=1XkB(b˜(n)) < 0Now let initial capital u and time t increase with n → ∞ in such a way thatuB(b˜(n))= u0,tn= t0. We observe that under conditions a) – d) we mayapproximateΨ(c, t) ≈ ψ(c0, t0)322 E. Pancheva, Z. Volkovich, L. Morozenskyand consequently for u and t “large enough”(4) ψ(c0, t0) ≤ Ψ(c, t) ≤ ψ¯(c0, t0)3. ExamplesAssume that our model is characterized by α = 0.5, i.e. the df of Zα(1) isthe Levy df Gα(x) = 2(1− Φ(√1x)). Here Φ is the standard normal df.We suppose also that the random variable E(1) is Exp(1)-distributed, namelyQ(s) = 1 − e−s, s ≥ 0. Further, let us take the income curve c0 to be of thespecial form c0(t) = u0 + tβα c, c positive constant, that agrees with the self-similarity of the process Zα(E(t)). Now the upper bound depends on (u0, t0, β)and the lower bound depends on (u0, t0, β, c). We calculate the bounds ψ and ψ¯in two cases α > β = 0.25 and α < β = 0.75 by using MATLAB7. The resultsof 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, if we choose the income curve in the above special form, we maycalculate the ruin probability ψ(c0, t0) in the approximating model exactly, namelyψ(c0, t0) = P ( inf0≤s≤t0{u0 + sβα c− Zα(E(s))} < 0)= P ( inf0≤s≤t0{sβα (c− Zα(E(1)))} < −u0)= P ( inf0≤s≤t0{sβα (c− Zα(E(1)))} < −u0, c− Zα(E(1)) < 0)= P (tβα0 (c− Zα(E(1))) < −u0, c− Zα(E(1)) < 0)= P (Zα(E(1)) > c+u0tβα0)=∞∫0Q¯(c0(t0)xtβα0α)dGα(x)Below we give graphical results related to the computation of ψ(c0, t0) ,ψ(c0, t0) and ψ¯(c0, t0) in the 6 cases: c=0.1, c=1, c=10 when α = 0.5 andβ = 0.25 β = 0.75.Upper and Lower Bounds for Ruin Probability 3234. Graphics of ψ(c0, t0), ψ(c0, t0) and ψ¯(c0, t0)0.511.522468100.10.20.30.40.50.6time rangemoney in beginFigure 1: α = 0.5, β = 0.25, c = 0.10.511.522468100.10.20.30.40.50.6time rangemoney in beginFigure 2: α = 0.5, β = 0.25, c = 0.5324 E. Pancheva, Z. Volkovich, L. Morozensky0.511.522468100.10.20.30.40.50.6time rangemoney in beginFigure 3: α = 0.5, β = 0.25, c = 1.00.511.5224681000.20.40.60.8time rangemoney in beginFigure 4: α = 0.5, β = 0.75, c = 0.1Upper and Lower Bounds for Ruin Probability 3250.511.5224681000.20.40.60.8time rangemoney in beginFigure 5: α = 0.5, β = 0.75, c = 0.50.511.5224681000.20.40.60.8time rangemoney in beginFigure 6: α = 0.5, β = 0.75, c = 1.0326 E. Pancheva, Z. Volkovich, L. MorozenskyREFERE NCES[1] Balkema A., Pancheva E. Decomposition for Multivariate ExtremalProcesses. Commun. Statist.-Theory Meth., 25(4) (1996), 737–758.[2] Furrer H., Michna Zb. Stable Levy Motion Approximation in CollectiveRisk Theory. Insurance: Mathem. and Econom., 20 (1997), 97–114.[3] Meerschaert M.M., Scheffler H-P. Limit Theorems for ContinuousTime Random Walks. (2002), Submitted.[4] Pancheva E., Kolkovska E., Yordanova P. Random Time-ChangedExtremal Processes. Comun. Tecnica No I-03-14/07-08-2003 PE/CIMAT(2003).[5] Pancheva E. Self-similar Extremal Processes. J. Math. Sci., 92(3) (1998),3911–3920.E. PanchevaInstitute of Mathematics and Informatics,the Bulgarian Academy of Sciences, 1113 Sofia, Bulgariae-mail: pancheva@math.bas.bgZ. VolkovichSoftware Engineering Department,ORT Braude College of Engineering, Karmiel 21982, IsraelAffiliate Professor, Department of Mathematics and Statistics,the University of Maryland, Baltimore County, USAe-mail: vlvolkov@ort.org.ilL. MorozenskySoftware Engineering Department,ORT Braude College of Engineering, Karmiel 21982, Israele-mail: leonatm@bezeqint.net

Upper and Lower Bounds for Ruin Probability

In this note we discuss upper and lower bound for the ruin probabil-ity in an insurance model with very heavy-tailed claims and interarrival times

Upper and Lower Bounds for Ruin Probability

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