RUIN PROBABILITIES FOR RISK PROCESSES WITH NON-STATIONARY ARRIVALS AND SUBEXPONENTIAL CLAIMS. 1. Introduction

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1 RUIN PROBABILITIES FOR RISK PROCESSES WITH NON-STATIONARY ARRIVALS AND SUBEXPONENTIAL CLAIMS LINGJIONG ZHU Absrac. In his paper, we obain he finie-horizon and infinie-horizon ruin probabiliy asympoics for risk processes wih claims of subexponenial ails for non-saionary arrival processes ha saisfy a large deviaion principle. As a resul, he arrival process can be dependen, non-saionary and nonrenewal. We give hree examples of non-saionary and non-renewal poin processes: Hawkes process, Cox process wih sho noise inensiy and selfcorrecing poin process. We also show some aggregae claims resuls for hese hree examples. Le us consider a classical risk model. Inroducion N.) U = u + p C i, where C i are i.i.d. claims disribued as an R + -valued random variable C, p > is he premium rae, u > is he iniial reserve and N is a simple poin process. We are ineresed in he case when C i have heavy ails. A disribuion funcion B is subexponenial, i.e. B S if PC + C 2 > x).2) lim = 2, x PC > x) where C, C 2 are i.i.d. random variables wih disribuion funcion B. Le us denoe Bx) := PC x) and le us assume ha E[C ] < and define B x) := By)dy, where F x) = F x) is he complemen of any disribuion E[C] x funcion F x). In he paper, he noaion fx) gx) means lim x fx) gx) =. Goldie and Resnick [9] showed ha if B S and saisfies some smoohness condiions, hen B belongs o he maximum domain of aracion of eiher he Freche disribuion or he Gumbel disribuion. In he former case, B is regularly varying, i.e. Bx) = Lx)/x α+, for some α > and we wrie i as B R α ), α >. We assume ha B S and eiher B R α ) or B G, i.e. he maximum domain of aracion of Gumbel disribuion. G includes Weibull and lognormal disribuions. Dae: 6 April 23. Revised: 6 April Mahemaics Subjec Classificaion. 9B3; 6G55; 6F. Key words and phrases. Risk processes, ruin probabiliies, subexponenial disribuions, nonsaionary processes, Hawkes processes, sho noise processes, self-correcing poin processes.

2 2 LINGJIONG ZHU T i = τ i τ i is he lengh of he ime inerval beween wo consecuive arrival imes of he poin process τ i and τ i. τ i sands for he ih arrival ime of he poin process. If T i are i.i.d., wih mean E[T ], hen N is a renewal process and assume he usual ne profi condiion.3) ρ := E[C ] pe[t ] <, hen, i is well known ha see Teugels and Veraverbeke [6], Veraverbeke [9] and Embrechs and Veraverbeke [8]),.4) lim ψu) B u) = ρ ρ, where ψu) := Pτ u < ) is he infinie-horizon ruin probabiliy, where.5) τ u := inf{ > : U }. The exensions when N is no a renewal process has been sudied in Asmussen e al. [4] when he auhors consider a risk process wih regeneraive srucures or a saionary and ergodic process saisfying cerain condiions. See also Araman and Glynn [], Schlegel [3] and Zwar e al. [23]. Bu in general, for a simple poin process N, we may no have a regeneraive srucure and i may no be saionary and ergodic as assumed in Asmussen e al. [4]. For example, none of he examples ha we will inroduce laer in Secion 3 are saionary or have a regeneraive srucure. In his paper, we poin ou ha he classical infinie-horizon ruin probabiliy esimae.4) and also finie-horizon ruin probabiliy esimae sill hold as long as here exiss a large deviaion principle for N / ), which is he main resul of his paper, i.e. Theorem 3 and Theorem 7 in Secion 2.. The inuiion behind i is ha if he arrival imes deviae away from is mean wih an exponenially small probabiliy, i will be dominaed by he subexponenial disribuions of he claim sizes. Our proof is essenially based on checking he condiions proposed in Asmussen e al. [4]. In Secion 2.2, we review some known resuls abou esimaes of aggregae claims when N is no necessarily renewal and show ha a condiion is saisfied given he large deviaion principle of N / ). Finally, in Secion 3, we give hree examples of non-renewal processes, i.e. Hawkes process which answers a quesion of Sabile and Torrisi [5]), Cox process wih sho noise inensiy which reproves a resul ha is known, see Asmussen and Albrecher [2]), and self-correcing poin process for which our resuls apply. 2. Risk Process wih Non-Renewal Arrivals and Regularly Varying Claims 2.. Ruin Probabilies. Before we proceed, recall ha a sequence P n ) n N of probabiliy measures on a opological space X saisfies he large deviaion principle LDP) wih rae funcion I : X R if I is non-negaive, lower semiconinuous and for any measurable se A, we have 2.) inf Ix) lim inf x Ao n n log P na) lim sup n n log P na) inf Ix). x A

3 RISK PROCESSES WITH SUBEXPONENTIAL CLAIMS 3 Here, A o is he inerior of A and A is is closure. See Dembo and Zeiouni [7] or Varadhan [8] for general background regarding large deviaions and he applicaions. Also Varadhan [7] has an excellen survey aricle on his subjec. Asmussen e al. [4] proved ha.4) holds if we have Lemma and Lemma 2. So our main ask here is o prove Lemma and Lemma 2 under following assumpions. Noice ha Lemma holds if T i ) i is a saionary and ergodic sequence by using ergodic heorem). And ha is he only place Asmussen e al. [4] used he saionariy and ergodiciy assumpion. Tha is why as long as we can prove Lemma we can drop he saionariy and ergodiciy assumpion. The following is he main assumpion for he asympoic resuls of ruin probabiliies ha we are going o esablish in his paper. Assumpion. i) N / ) saisfies a large deviaion principle wih rae funcion I ) such ha Ix) = if and only if x = µ. ii) Ix) is increasing on [µ, ) and decreasing on [, µ]. iii) The ne profi condiion is saisfied, 2.2) ρ := µe[c ] p <. iv) There exiss some θ > such ha E[e θ P n Ti ] < for any n N. Lemma. Under Assumpion, for any fixed ɛ, ɛ >, here exiss a consan M > such ha { n ) }) p 2.3) P p T i n µ + ɛ + M > ɛ. n= Proof. Replacing ɛ by pɛ and M by pm in he above equaion, i is sufficien o prove ha 2.4) lim sup P { n n= ) }) T i > n µ + ɛ + M = Observe ha {N n} = { n T i > } for any n N and R + and also for any fixed µ < µ, here exiss some δ > such ha Iµ ) δ > and for sufficienly large, 2.5) PN / < µ ) e [Iµ ) δ ], where we used fac ha Iµ ) > and I ) is decreasing on [, µ] from Assumpion. Also for any N N, 2.6) lim sup n P n<n ) ) T i > n µ + ɛ + M =.

4 4 LINGJIONG ZHU Togeher, ake N N sufficienly large, { n ) }) 2.7) lim sup P T i > n µ + ɛ + M n= n ) ) lim sup P T i > n µ + ɛ + M = lim sup = lim sup lim sup lim sup n= n P n N P n N P n= n N T i > n µ + ɛ Nnµ +ɛ)+m ) ) + M ) nµ + ɛ) + M n nµ + ɛ) + M Nnµ +ɛ)+m nµ + ɛ) + M µ ) + µɛ e nµ +ɛ)+m)[i µ +µɛ ) δ ] =. Lemma 2. Under Assumpion, Psup n {npµ ɛ) p n 2.8) lim T i} u) =, B u) for any sufficienly small ɛ >. Proof. Noice ha 2.9) P sup n { } ) n n npµ ɛ) p T i u P n= = P n> p u µ ɛ P n> p u µ ɛ n> u p µ ɛ P T i n n n µ ɛ p T i n µ ɛ p ) ) u p T i n µ ɛ ) ) p N n µ ɛ p ) n n e µ ɛ p )[I µ ɛ p ) ) δ ], n> p u µ ɛ ) ) ) u p which is exponenially small in u as u. Since B u) is subexponenial, we have he desired resul. We have he following asympoic esimaes for infinie-horizon ruin probabiliies.

5 RISK PROCESSES WITH SUBEXPONENTIAL CLAIMS 5 Theorem 3. Under Assumpion, we have 2.) lim ψu) B u) = ρ ρ. Proof. I is a direc resul of Lemma, Lemma 2 and Theorem 3.. in Asmussen e al. [4]. Remark 4. In Theorem 3, we can replace he large deviaion assumpion of N / ) by a large deviaion assumpion of n n T i ). Bu usually, if N / ) saisfies a large deviaion principle wih rae funcion Ix) if and only if n n T i ) saisfies a large deviaion principle wih rae funcion xi/x). The reason we chose o assume he large deviaion for N / ) in Assumpion is because when N is no renewal, he iner-occurrence imes are no i.i.d. and i is usually easier and more naural o esablish he large deviaion for N / ), which is a leas in he case of our hree examples, Hawkes process, Cox process wih sho noise inensiy and self-correcing poin process. Nex, le us consider he finie-horizon ruin probabiliies. Le eu) := E[C u C > u] be he mean excess funcion and 2.) ψu, z) := Pτ u z), z >, be he finie-horizon ruin probabiliy. Remark 5. i) Regularly Varying Disribuions) If Bu) = Lu) u, α, ), i.e. α+ B R α ), hen, eu) u α. ii) Lognormal Disribuions) If Bu) = log u µ)/σ 2π e x2 /2 dx, hen, B G and B S and eu) σ2 u log u µ. iii) Weibull Disribuions) If Bu) = e uα, where α, ), hen, B G and B S and eu) u α α. Gx+y) Lemma 6. For any y <, lim x x G S. = uniformly for y [, y ] for any Lemma 6 can be found in Chaper X of Asmussen and Albrecher [2]. We have he following asympoic esimaes for finie-horizon ruin probabiliies. Theorem 7. Under Assumpion and furher assume ha B S, we have, for any T >, i) If B R α ), [ ψu, eu)t ) 2.2) lim = ρ + ρ) T ) ] α. B u) ρ α ii) If B G, ψu, eu)t ) 2.3) lim = ρ [ ] e ρ)t. B u) ρ Proof. The proof is based on he ideas in Asmussen e al. [4] wih some modificaions. When N is a renewal process, Asmussen and Klüppelberg [3] proved boh

6 6 LINGJIONG ZHU i) and ii). Now if N saisfies Assumpion, hen, by Lemma, { n } ) n 2.4) ψu, eu)t ) = P sup C i p T i > u n eu)t ɛ )P sup n eu)t { n C i n Now, in boh cases i) and ii), we know ha ex) ) } p µ + ɛ > u + M). R x By)dy Bx). Since boh Bx) and ex+y) B belong o S, Lemma 6 implies ha lim x ex) = uniformly for y [, y ] for any y <. Therefore, for any ɛ, ), we have eu) eu + M) ɛ ) for any sufficienly large u and hus we ge 2.5) ψu, eu)t ) ɛ )P sup n eu+m)t ɛ ) { n ) } p C i n µ + ɛ > u + M). Now assume B R α ). We have by he corresponding resul for renewal N in Asmussen and Klüppelberg [3] and Lemma 6, 2.6) lim inf ψu, eu)t ) B u) = lim inf ψu, eu)t ) B u + M) ρ ɛ ɛ [ ) + ρɛ )T ɛ )/α) α], ρ ɛ where ρ ɛ := E[C] p µ +ɛ. Since i holds for any ɛ, ɛ, ɛ >, we proved he lower bound. The case for B G is similar. Now, le us prove he upper bound. Choose ɛ > small enough ha p µ ɛ > E[C ], 2.7) lim sup = lim sup lim sup ψu, eu)t ) B u) Psup n eu)t { n C i npµ ɛ) + npµ ɛ) n T i} > u) B u) PX ɛ u) + Y ɛ > u), B u) where X ɛ u) := sup n eu) { n C i npµ ɛ)} and Y ɛ := sup n {npµ ɛ) n T PY i}. By Lemma 2, we have lim ɛ>u) = and by he resuls for he B u) renewal case Asmussen and Klüppelberg [3]), for B R α ), 2.8) PX ɛ u) > u) ρ ɛ ρ ɛ [ + ρɛ )T/α) α] B u), where ρ ɛ := E[C] pµ ɛ. Le us recall he Proposiion.9. of Chaper X in Asmussen and Albrecher [2] which says ha for any disribuions A, A 2 on R +, if we have A i x) a i Gx) for some G S and some consans a +a 2 >, hen, A A 2 x) a + a 2 )Gx). In our case Gx) = B x) S and A, A 2 are he disribuions of X ɛ u) and Y ɛ wih a > and a 2 =. Noice ha X ɛ u) and Y ɛ may be negaive. To save he argumen, we can simply use he fac ha X ɛ u) max{x ɛ u), }

7 RISK PROCESSES WITH SUBEXPONENTIAL CLAIMS 7 and Y ɛ max{y ɛ, } hen apply i o max{x ɛ u), } and max{y ɛ, } insead. Also, in our case, X ɛ u) depends on u, bu he proof of Proposiion.9. Chaper X in Asmusseen and Albrecher [2] sill works. Hence, we ge 2.9) lim sup ψu, eu)t ) B u) ρ ɛ ρ ɛ [ + ρɛ )T/α) α]. Since i holds for any ɛ, we proved he upper bound. similar. The case for B G is 2.2. Aggregae Claims. Le A := N C i be he aggregae claims up o ime, where as before we assume here ha C i are i.i.d. posiive random variables Consider he following assumpions. Assumpion 2. i) E[N ] < for any and E[N ] as. N ii) E[N ], as. iii) There exis ɛ, δ > such ha 2.2) PN > k) + ɛ) k, as. k>+δ)e[n ] Klüppelberg and Mikosch [2] proved ha under Assumpion 2, for fixed ime, we have 2.2) PA E[A ] > x) E[N ]PC x), uniformly for x γe[n ] for any γ >. Remark 8. Indeed, Klüppelberg and Mikosch [2] proved a slighly sronger resul which says 2.2) holds assuming ha he claim sizes C i are i.i.d. wih a disribuion funcion F ERV α, β) for some < α β <, where ERV denoes he space of exended regular varying funcions. I is usually easy o check i) and also under he assumpions in Theorem 3, N µ and N / ) saisfies a large deviaion principle wih rae funcion Ix) which is nonzero if and only if x µ. Therefore, if we assume we could prove ha E[N ] µ as, hen ii) is saisfied. Moreover iii) can be replaced by iii ) For any µ >, c µ := inf Ix) x µ x >. Assume iii ), we can find some < δ < δ such ha for any sufficienly large, 2.22) PN > k) + ɛ) k PN > k) + ɛ) k k>+δ)e[n ] k>+δ )µ k>+δ )µ k>+δ )µ k>+δ )µ e Ik/) ɛ ) + ɛ) k e Ik/) k ɛ +δ )µ )k + ɛ) k e c +δ )µ ɛ +δ )µ )k + ɛ) k. If we pick up ɛ > small enough such ha ɛ +δ )µ < c +δ )µ, hen, we can pick up ɛ > small enough so ha c +δ )µ ɛ +δ )µ > log + ɛ) and herefore by leing, iii) is saisfied.

8 8 LINGJIONG ZHU 3. Examples of Non-Renewal Arrival Processes 3.. Example : Hawkes Process. Hawkes process is a simple poin process ha has self-exciing propery, clusering effec and long memory. I was firs inroduced by Hawkes [] and has been widely applied in finance, seismology, neuroscience, DNA modelling and many oher fields. A simple poin process N is a linear Hawkes process if i has inensiy 3.) λ = + τ< h τ), h ) : [, ), ) is inegrable and h L <. We also assume ha N sars wih empy pas hisory, i.e. N, ] =. By our definiion, he Hawkes process is non-saionary and is in general even non-markovian unless h ) is an exponenial funcion). Also, i does no have a regeneraive srucure. Thus, he condiions in Asmussen e al. [2] do no apply here. Noice ha i is well known ha, see for example Daley and Vere-Jones [6]) N 3.2) lim = µ :=, h L and Bordenave and Torrisi [5] proved he a large deviaion principle for N / ), i.e. Lemma 9. Therefore, i is naural ha we can apply he resuls in our paper o sudy he ruin probabiliies wih subexponenial claims when he arrival process is a non-saionary linear Hawkes process. Lemma 9 Bordenave and Torrisi [5]). N / ) saisfies a large deviaion principle wih rae funcion, { ) x log x + x h L + if x [, ) 3.3) Ix) = x +x h L + oherwise Remark. Indeed, in Bordenave and Torrisi [5], hey expressed he rae funcion I ) in an alernaive way, which is less explici. The expression of he rae funcion in Lemma 9 was firs poined ou in Zhu [22]. Lemma. E[N] h L as. Proof. Taking expecaion of he indeniy λ = + h s)nds), we ge 3.4) E[λ ] = + h s)e[λ s ]ds + h L which implies ha for any, sup s E[λ s ] uniformly in. Nex, le H) := hs)ds and [ ] 3.5) E[N ] = E λ s ds h L = + = + s u = + E[N ] h L h L hs u)de[n u ]ds hs u)dsde[n u ] sup E[λ s ]ds, s H u)de[n u ],. and herefore E[λ ]

9 RISK PROCESSES WITH SUBEXPONENTIAL CLAIMS 9 which implies ha 3.6) E[N ] = h L and 3.7) lim sup H u)e[λ u ]du h L H u)e[λ u ]du, = h L lim sup lim sup H u)du Hu)du =, since H) = hs)ds as. h. L Assume he ne profi condiion p > E[C] If C i have ligh ails, hen Sabile and Torrisi [5] obained he asympoics for he infinie-horizon ruin probabiliy ψu) and he finie-horizon ruin probabiliy φu, uz) for any z >. As poined ou in Sabile and Torrisi [5] he case when C i are heavy-ailed is open and now we have he ools o handle he case. Proposiion 2. Assume he ne profi condiion p > E[C] i) Infinie-Horizon) ψu) 3.8) lim B u) = E[C ] p h L ) E[C ]. 3.9) ii) Finie-Horizon) For any T >, h. L lim = ψu, uz) B u) E[C ] p h L ) E[C ] E[C ] p h L ) E[C ] [ + [ e p h L ) E[C ] p h L ) E[C ] p h L ) p h L ) T ] ) ) ] α T α if B R α ). if B G iii) Aggregae Claims) For fixed ime, 3.) PA E[A ] > x) E[N ]PC x), uniformly for x γe[n ] for any γ >. Proof. To prove i) and ii), by Theorem 3 and Theorem 7, i is enough o check he condiions in Assumpion. i) and ii) of Assumpion 2 can be verified by he large deviaions resul in Lemma 9 and he properies of he rae funcion. iii) of Assumpion is he assumpion of he Proposiion 2. To check iv) of Assumpion, noice ha by he definiion of Hawkes process, N sochasically dominaes N, a homogenous Poisson process wih parameer >. Bu Ti corresponding o N are i.i.d. exponenially disribued wih parameer and hey sochasically dominae T i, he lengh of ime inerval beween wo consecuive arrivals of a Hawkes process. Bu we know ha exponenially disribuion has exponenial ails and hus for θ > small enough, E[e θ P n Ti ] E[e θ P n T i ] = E[e θt ] n < for any n N. Thus iv) of Assumpion holds. Now, o prove iii), i is enough o check i), ii) and iii ) of Assumpion 2. In he proof of Lemma

10 LINGJIONG ZHU [ ] h uniformly in and hus E[N L ] = E λ sds < and i) of Assumpion 2 is verified. ii) of Assumpion 2 is a resul, we showed ha E[λ ] h L of Lemma and law of large numbers of N / and finally iii ) of Assumpion 2 can be verified by easily checking he rae funcion in Lemma Example 2: Cox Process wih Sho Noise Inensiy. We consider a Cox process N wih inensiy λ ha follows a sho noise process 3.) λ = ) + g τ ) ), τ ) < where τ ) are he arrival imes of an exernal homogenous Poisson process wih inensiy γ. Here, g ) : R + R + is inegrable, i.e. g)d < and ) is a posiive, coninuous, deerminisic funcion such ha ) as. The ruin probabiliies for heavy-ailed claims wih arrival process being a sho noise Cox process is known in he lieraure, e.g. see he book by Asmussen and Albrecher [2]. Bu he echniques in he lieraure use he very specific feaures of sho noise Cox process and he proofs are much longer. Our proof essenially only needs he large deviaion resul for N / ) which is very easy o esablish. Since N ) is a Poisson process wih inensiy γ, by he definiion of λ, i is easy o see ha 3.2) N + γ g L, as. I is no clear o he auhor if he large deviaion resul for N / ) is known in he lieraure. For he sake of compleeness, le us esablish he large deviaion principle here. Lemma 3. N / ) saisfies a large deviaion principle wih rae funcion, { { } supθ R θx e θ ) γe eθ ) g L ) if x [, ) 3.3) Ix) =. + oherwise Proof. For any θ R, we have 3.4) E[e θn ] = E [e eθ ) R λsds] = e eθ ) R s)ds E [e eθ ) R R ] s gs u)n ) du)ds = e eθ ) R s)ds E [e R [ R ] u eθ )gs u)ds]n ) du) Therefore, we have = e eθ ) R s)ds e γ R e R u e θ )gs u)ds )du = e eθ ) R s)ds e γ R R u e e θ )gs)ds )du = e eθ ) R s)ds e γ R e R u e θ )gs)ds )du. 3.5) lim log E[eθN ] = e θ ) + γe e θ ) g L ).

11 RISK PROCESSES WITH SUBEXPONENTIAL CLAIMS By Gärner-Ellis heorem, we conclude ha N / ) saisfies a large deviaion principle wih rae funcion { } 3.6) Ix) = sup θ R θx e θ ) γe eθ ) g L ). Now, if x <, hen for any θ <, θx e θ ) γe eθ ) g L ) θx if we le θ. Hence, Ix) = + for x <. Lemma 4. E[N] + γ g L as. Proof. Observe ha [ 3.7) E[N ] = E = = = ] λ s ds [ s)ds + E s)ds + γ s)ds + γ s s s ] gs u)n ) du)ds gs u)duds gu)duds, which implies ha E[N] + γ g L as. Proposiion 5. Assume he ne profi condiion p > E[C] + γ g L ). i) Infinie-Horizon) ψu) 3.8) lim B u) = + γ g L)E[C ] p + γ g L )E[C ]. ii) Finie-Horizon) For any T >, 3.9) lim = ψu, uz) B u) +γ g L )E[C ] p +γ g L )E[C ] +γ g L )E[C ] p +γ g L )E[C ] [ ) ) ] α + +γ g L )E[C] T p α if B R α ) [ ]. e p +γ g L )E[C ])T/p if B G iii) Aggregae Claims) For fixed ime, 3.2) PA E[A ] > x) E[N ]PC x), uniformly for x γe[n ] for any γ >. Proof. To prove i) and ii), by Theorem 3 and Theorem 7, i is enough o check he condiions in Assumpion. i) and ii) of Assumpion 2 can be verified by he large deviaions resul in Lemma 3 and he properies of he rae funcion. iii) of Assumpion is he assumpion of he Proposiion 5. To check iv) of Assumpion, noice ha by he definiion of Hawkes process, N sochasically dominaes N, an homogenous Poisson process wih parameer := max ). Bu Ti are i.i.d. exponenially disribued wih parameer and corresponding o N hey sochasically dominae T i, he lengh of ime inerval beween wo consecuive arrivals of a Hawkes process. Bu we know ha exponenially disribuion has

12 2 LINGJIONG ZHU exponenial ails and hus for θ > small enough, E[e θ P n Ti ] E[e θ P n T i ] = θt E[e ] n < for any n N. Thus iv) of Assumpion holds. Now, o prove iii), i is enough o check i), ii) and iii ) of Assumpion 2. I is easy o see ha ha E[λ ] = ) + γ [ ] gs)ds < for any > and hus E[N ] = E λ sds < and i) of Assumpion 2 is verified. ii) of Assumpion 2 is a resul of Lemma 4 and law of large numbers of N / and finally iii ) of Assumpion 2 can be verified by easily checking he rae funcion in Lemma Example 3: Self-Correcing Poin Process. A self-correcing poin process, also known as he sress-release model, is a simple poin process N wih empy hisory, i.e. N, ] = such ha i admis he F -inensiy 3.2) λ := λz ), and Z := N. The rae funcion λ ) : R R + is coninuous and increasing such ha 3.22) < λ = lim λz) < < lim λz) = z z + λ+ <. Noice ha in he definiion of inensiy in 3.2), we used N insead of N. Tha is crucial o guaranee ha he inensiy λ for he self-correcing poin process is F -predicable. The model was firs inroduced by Isham and Wesco [] as an example of a process ha auomaically correcs a deviaion from is mean. Laer, i was sudied as a model in seismology. The sress builds up a he linear rae in our model and releases by he amoun a ih jump. Vere-Jones [2] discussed an insurance inerpreaion. Under hese assumpions, i is well known ha N as See for example Proposiion 4.3 in Zheng [2]). Recenly, Sen and Zhu [4] proved he following large deviaion resul. Lemma 6 Sen and Zhu [4]). N / ) saisfies a large deviaion principle wih rae funcion Λ x) if x > if x = 3.23) Ix) = Λ +, x) if x < + oherwise where 3.24) Λ ± x) = log x λ ± ) x + λ ± x, x. Lemma 7. E[N] as. [ ] Proof. E[N ] = E λz s)ds. Zheng [2] proved ha here exiss a unique invarian measure πdz) for he Markov process Z. By ergodic heorem, we have 3.25) λz s )ds λz)πdz), as. We know ha Z = N has he generaor 3.26) Afz) = f + λz)fz ) fz)), z

13 RISK PROCESSES WITH SUBEXPONENTIAL CLAIMS 3 and we have Azπ = which implies ha λz)πdz) = and hus λz s)ds a.s. as. Since λ λ ) λ +, by bounded convergence heorem, we conclude ha E[N] as. Proposiion 8. Assume he ne profi condiion p > E[C]. i) Infinie-Horizon) ψu) 3.27) lim B u) = E[C ] p E[C ]. ii) Finie-Horizon) For any T >, 3.28) [ ) ) ] α ψu, uz) E[C ] p E[C lim = ] + E[C] T p α if B R α ) B u) E[C ] [ ]. e p E[C ])T/p if B G p E[C ] iii) Aggregae Claims) For fixed ime, 3.29) PA E[A ] > x) E[N ]PC x), uniformly for x γe[n ] for any γ >. Proof. To prove i) and ii), by Theorem 3 and Theorem 7, i is enough o check he condiions in Assumpion. i) and ii) of Assumpion 2 can be verified by he large deviaions resul in Lemma 6 and he properies of he rae funcion. iii) of Assumpion is he assumpion of he Proposiion 8. To check iv) of Assumpion, noice ha by he definiion of Hawkes process, N sochasically dominaes N λ, an homogenous Poisson process wih parameer λ. Bu Ti λ corresponding o N λ are i.i.d. exponenially disribued wih parameer λ and hey sochasically dominae T i, he lengh of ime inerval beween wo consecuive arrivals of a Hawkes process. Bu we know ha exponenially disribuion has exponenial ails and hus for θ > small enough, E[e θ P n Ti ] E[e θ P n T λ i ] = λ θt E[e ] n < for any n N. Thus iv) of Assumpion holds. Now, o prove iii), i is enough o check i), ii) and iii ) of Assumpion [ 2. I is easy o see ha ] ha λ λ + < for any > and hus E[N ] = E λ sds λ + < and i) of Assumpion 2 is verified. ii) of Assumpion 2 is a resul of Lemma 7 and law of large numbers of N / and finally iii ) of Assumpion 2 can be verified by easily checking he rae funcion in Lemma 6. Acknowledgemens The auhor is suppored by NSF gran DMS-947, DARPA gran and Mac- Cracken Fellowship a New York Universiy. References [] Araman, V. and P. W. Glynn. 26) Tail asympoics for he maximum of perurbed random walk. Ann. Appl. Probab. 6, [2] Asmussen, S. and H. Albrecher. Ruin Probabiliies, 2nd ediion, World Scienific, 2, Singapore. [3] Asmussen, S. and C., Klüppelberg. 996). Large deviaions resuls for subexponenial ails wih applicaions o insurance risk. Sochasic Process. Appl. 64, [4] Asmussen, S., Schmidli, H. and V. Schmid. 999). Tail approximaions for non-sandard risk and queueing processes wih subexponenial ails. Adv. Appl. Probab. 3,

14 4 LINGJIONG ZHU [5] Bordenave, C. and G. L., Torrisi. 27). Large deviaions of Poisson cluser processes. Sochasic Models. 23, [6] Daley, D. J. and D. Vere-Jones. An Inroducion o he Theory of Poin Processes, Volume I and II, 2nd Ediion. Springer-Verlag, New York, 23. [7] Dembo, A. and O. Zeiouni, Large Deviaions Techniques and Applicaions, 2nd Ediion, Springer, 998 [8] Embrechs, P. and N. Veraverbeke. 982). Esimaes for he probabiliy of ruin wih special emphasis on he possibiliy of large claims. Insurance: Mahemaics and Economics., [9] Goldie, C. M. and S. Resnick. 988). Disribuions ha are boh subexponenial and in he domain of aracion of an exreme value disribuion. Adv. Appl. Probab. 2, [] Hawkes, A. G. 97). Specra of some self-exciing and muually exciing poin processes. Biomerika. 58, [] Isham, V. and M. Wesco. 979). A self-correcing poin process. Soch. Proc. Appl. 8, [2] Klüppelberg, C. and T. Mikosch. 997). Large deviaions of heavy-ailed random sums wih applicaions o insurance and finance. Journal of Applied Probabiliy. 34, [3] Schelgel, S. 998). Ruin probabiliies in perurbed risk models. Insurance: Mahemaics and Economics. 22, [4] Sen, S. and L. Zhu, Large deviaions for self-correcing poin processes. Preprin, 23. [5] Sabile, G. and G. L. Torrisi. 2). Risk processes wih non-saionary Hawkes arrivals. Mehodol. Compu. Appl. Prob. 2, [6] Teugels, J. L. and Veraverbeke, N. 973). Cramér-ype esimaes for he probabiliy of ruin. C.O.R.E. Discussion Paper 736. [7] Varadhan, S. R. S. 28). Large deviaions. Annals of Probabiliy. 36, [8] Varadhan, S. R. S. Large Deviaions and Applicaions, SIAM, Philadelphia, 984. [9] Veraverbeke, N. 977). Asympoic behavior of Wiener-Hopf facors of a random walk. Sochasic Process. Appl. 5, [2] Vere-Jones, D. 988). On he variance properies of sress release models. Aus. J. Sais. 3A, [2] X. Zheng. 99). Ergodic heorems for sress release processes. Soch. Proc. Appl. 37, [22] Zhu, L. Large deviaions for Markovian nonlinear Hawkes processes. Preprin, 2. arxiv: [23] Zwar B., Bors, S. and K. Dȩbicki. 25). Subexponenial asympoics of hybrid fluid and ruin models. Ann. Appl. Probab. 5, Couran Insiue of Mahemaical Sciences New York Universiy 25 Mercer Sree New York, NY-2 Unied Saes of America address: ling@cims.nyu.edu