ON THE DISTRIBUTIONS OF THE SUP AND INF OF THE CLASSICAL RISK PROCESS WITH EXPONENTIAL CLAIM*

Transcription

1 Communication on Stochatic Analyi Vol., No. (9) Serial Publication ON THE DISTRIBUTIONS OF THE SUP AND INF OF THE CLASSICAL RISK PROCESS WITH EXPONENTIAL CLAIM* JORGE A. LEÓN AND JOSÉ VILLA Abtract. The purpoe of thi article i to ue the double Laplace tranform of the occupation meaure of the claical rik proce X with exponential claim to deduce the ditribution of the random variable up{x : t} and inf{x : t}, for every t >. A a conequence, we alo get the ditribution of the time to ruin in finite time and the firt paage of a given level.. Introduction In thi paper, we deal with the claical rik proce with exponential claim defined on a complete probability pace (Ω, F, P). More preciely, let N t X t x + ct R k, t. (.) Here x i the initial capital, c > i the premium income per unit of time, N {N t, t } i an homogeneou Poion proce with rate λ and {R k, k,,...} i a equence of i.i.d. random variable independent of N. Henceforth we uppoe that R ha exponential ditribution with mean /r. Our goal i to calculate explicit expreion for the ditribution of the random variable up{x : t} and inf{x : t}, t >. Toward thi end, we apply the complex inverion theorem of the Laplace tranform (or Lerch theorem) to the double Laplace tranform of ome occupation meaure of X (ee Section below). A a conequence, we are alo able to give the ditribution of the firt paage of certain level x R of the proce X. It mean, the ditribution of S x inf{t > : X t x} k and T x inf{t > : X t < x}, becaue the right-continuity of the proce X yield { } {S x < t} upx > x t, x > x and {T x t} { } inf X < x, x < x. t Mathematic Subject Claification. Primary 6K; Secondary 6K99. Key word and phrae. Claical rik proce, Laplace tranform, ruin probability, time to ruin. * Partially upported by a abbatical year grant of CONACyT and PIM 8- of UAA. 69

2 7 JORGE A. LEÓN AND JOSÉ VILLA The tudy of the ditribution of the time to ruin T in finite time (i.e., for the cae x ) i extenive in the literature on rik theory due to it application in buine activitie. For intance, numerical procedure have been utilized by everal author in the analyi of T (ee, for example, Dickon and Water [6] or Seal []). Thi numerical approximation have been improved in everal work by deriving expreion for the mentioned ditribution (ee Amuen [, ], Dickon [4], Dickon et al. [5, 7], Drekic and Willmot [9], and Ignatov and Kaihev [], among other). In particular, the method ued in [5], [7] and [9] i baed on the complex inverion theorem, a we doe here. For x R, the proce S x have been analyzed by do Rei [8] and Gerber [] via a martingale method. The paper i organized a follow. In Section we relate the Laplace tranform of the function { P upx x } and P { } inf X x (.) to the double Laplace tranform of ome occupation meaure of X. Then, in Section we ue the complex inverion theorem to calculate the two probabilitie in (.).. Occupation Meaure Now we are intereted in the Laplace tranform of the occupation meaure Y x (t) t (x,+ ) (X )d and Y x (t) t (,x) (X )d, with x R and t >. So, we aume that the reader i familiar with the elementary propertie of the Laplace tranform a they are preented, for example, in Spiegel []. Throughout, the Laplace tranform of a meaurable function h : [, ) R i denoted by L(h). That i, L(h)() e t h(t)dt, for R uch that thi integral i convergent. The relation between the occupation meaure Y x and Y x, and the probabilitie in (.) i given by the following reult. Propoition.. Let X be the claical rik proce defined in (.) and x R. Then for each t >, { } { } {Y x (t) } upx x and {Y x (t) } inf X x. t t Proof. We firt oberve that {Y x (t) } { up t X x } i trivial. Now we ee the revere incluion. Let ω Ω be uch that t (x,+ ) (X (ω))d. (.)

3 CLASSICAL RISK PROCESS 7 If there i ome (, t) uch that X (ω) > x, then, by the right-continuity of X, there exit a non-empty open interval I (, t) uch that I and X (ω) > x, for all I. Conequently, t (x,+ ) (X (ω))d (x,+ ) (X (ω))d I >, I where I i the length of I. But thi i a contradiction to (.). Therefore X (ω) x for all t, which implie that ω alo belong to { up t X x }. We proceed imilarly for the remainder of the proof. In order to expre the double Laplace tranform of Y x and Y x, we need to introduce the following notation. Let be a poitive real number. The poitive and negative root of the quadratic equation are denoted by v + and v, repectively. [ e t E e αyx(t)] dt cv + (rc λ )v r Propoition.. Let and α be two poitive real number. Then { +α + α( c v+ ) e (x x)v v +α v+ α +α c and [ ] e t E e αy x (t) dt + αc +α ( c v+ )v +α α(v +α v+ α c ) }, x x,, x < x, c(v + + α c )(v +α v+ α c )e (x x )v+ { +α + α( c v ) e (x x)v+ v + +α v α +α c + αc +α ( c v )v + +α α(v+ +α v α c ) }, x x,, x x. c(v + α c )(v + +α v α c )e (x x )v Proof. For a < b and t > define the double Laplace tranform of T [a,b] (t) t [a,b](x r )dr a f(x ) e t E x [e ] αt [a,b](t) dt. Thi i a Feynman-Kac repreentation of the olution of equation { ( + α)f(x ), a < x Af(x ) + < b, f(x ), x < a or x > b, where A i the infiniteimal generator aociated to the emigroup of proce X. Solving thi equation and letting a and b, repectively, we are done. For detail ee the paper of Chiu and Yin [] (Corollary 4.). The following reult i a conequence of Propoition. and it will be ued in Section. Propoition.. Let X be the claical rik proce given by (.). Then, for every >, we have {, x x, L(P (Y x ( ) ))() e (x x )v+ (.), x < x,

4 7 JORGE A. LEÓN AND JOSÉ VILLA and lim α L(P (Y x ( ) ))() {, x x, λe (x x )v c(v + +r), x > x. (.) Proof. We firt deal with equality (.). By the dominated convergence theorem we can write [ e t E e αyx(t)] dt lim α ( e t + {Y x(t)} e t P (Y x (t) )dt + e t P (Y x (t) )dt. Therefore, from Propoition. we get {Y x(t)>} e t ) e αyx(t) dpdt {Y x(t)>} ( lim e αyx(t)) dpdt α e t P (Y x (t) )dt (.4) lim α { } +α + α( c v+ ) e (x x)v v +α v+ α +α, x x, c + αc +α ( c v+ )v +α α(v +α v+ α c ) c(v + + α c )(v +α v+ α c )e(x x )v+, x < x. Note that the definition of v+α implie that lim ( α v +α /α ), which lead u to ( ( lim α v α +α v+ α ) ) c. c Hence, the fact that v+α <, for all α >, together with (.4), yield that equality (.) i true for x x. On the other hand, for x < x, ( ) lim αc α + +α c v+ v+α α ( v+α ) v + α c c ( v + ) ( + α c v +α v + ) α c e (x x )v + ( ) lim c α + c v+ v +α +α ( v+α ) v + α c ( ) c v + (v α + c +α v + ) e (x x)v+ α c e (x x)v+. Thu (.) hold. Finally, in order to ee that (.) i atified, we only need to oberve that lim v + α ( +α /α ) ( and lim v + c α +α v α ) v + c + r, and proceed a in the beginning of thi proof.

5 CLASSICAL RISK PROCESS 7. The Ditribution of the Sup and Inf of X Within Finite Time The purpoe of thi ection i to apply Lerch theorem to the Laplace tranform obtained in Propoition. to calculate the ditribution of the up and inf of X (ee Propoition.). In order to tate the main reult of thi paper, we need to introduce the following notation. Note firt that ( + λ rc) + 4cr ( + λ + rc) 4λrc ( + λ + rc λrc)( + λ + rc + λrc) ( r )( r ), with r λrc λ rc and r λrc λ rc. Hence r < r <, and v + + λ rc + ( r )( r ), (.) c v + λ rc ( r )( r ). c (.) For ake of implicity, let u utilize the convention a x x c and b λ + rc. Theorem.. Let X be the claical rik proce (.) and t >. Then we have ( ) P up X > x (.) t ( e a(λ rc) e a r r + ) r e (t a)u in(a u r / u r / ) du, π r u for every x (x, x + ct), and ( ) ( e P inf X < x λe a(λ rc) a rr t b + (.4) r r ( )) + r π Im e (t a)u ai u r / u r / u ( u + b i u r / u r /)du, for every x < x. r Remark.. We make two remark. i) Note that ( ) ( ) P up X > x t for x x + ct, and P up X > x t for x < x. ii) Similarly we have P ( ) inf X < x for x > x. t In the following two ubection, we eparate the proof of (.) and (.4) for the convenience of the reader becaue both of them are long and tediou.

6 74 JORGE A. LEÓN AND JOSÉ VILLA.. Proof of equality (.). Let h M () e a a ( r )( r ), C\[r, r ], with ( r )( r ) ( r )( r ) / exp(iχ()), where χ() arg( r ) + arg( r ). Then, by (.), (.), the invere theorem of the Laplace tranform (ee for example []) and the fact that h M i an analytic function on C\[r, r ], we have, for σ large enough, P (Y x (t) ) πi σ+i πi ea(rc λ) πi e t ( e (x x)v σ i σ+i e tea(rc λ ( r )( r )) σ i σ+i σ i + e t h M ()d. Notice that we can ue the invere theorem becaue it i not difficult to ee that t P(X t x) i continuou, which follow from the fact that P(X t x). Let C(ρ, ε) C C be the following contour of integration: ) d d C C C B ρ C 4 C 5 C 6 r r ǫ C 7 σ ρ η(ǫ) C 9 C 8 C C C C A Now oberve that i a pole of order one and r, r are branch point of h M. Therefore, by the reidue theorem (ee []), σ+i e t h M ()d πie a r r lim lim e t h M ()d. (.5) σ i ρ ε C(ρ,ε)

7 CLASSICAL RISK PROCESS 75 Note that we only need to analyze the integral in the right-hand ide of (.5) on each arc C j, j {,...,}, in order to finih the proof. To do o, now we divide the proof in everal tep. Step. We begin our tudy on the arc C (ρ) and C (ρ). For C (ρ) we take the parametrization ρe iθ, θ θ π/, where ρ and θ are indicated in the following figure: C ρ θ r r θ ρ co θ σ In thi cae, it i eay to ee that arg(ρe iθ r ) < π and arg(ρe iθ r ) < π, which give χ(ρe iθ ) < π/. Since coθ, when θ [, π/], we have ( ) Re (ρe iθ r )(ρe iθ r ) ρe iθ r / ρe iθ r / co(χ(ρe iθ )) >. Uing thi and the fact that a >, we can conclude e are (ρe iθ r )(ρe iθ r ). (.6) Now note that t > a, due to x + ct > x, and that ρ coθ < σ. Thu, π e t e (t a)ρeiθ a (ρe iθ r )(ρe iθ r ) h M ()d ρie iθ dθ C (ρ) θ π θ π θ ρe iθ e (t a)ρ co θ are (ρe iθ r )(ρe iθ r ) dθ e (t a)σ dθ ( ) ( ) σ σ π e (t a)σ in e (t a)σ ρ ρ. Hence lim ρ C (ρ) et h M ()d. We can proceed in the ame way to ee that lim ρ C (ρ) et h M ()d i alo equal to zero.

8 76 JORGE A. LEÓN AND JOSÉ VILLA Note that an important point in thi analyi i the fact that t a > and a <. Thi will be alo important in the remaining of thi proof. Step. Now we conider the integral over C (ρ) and C (ρ). Over C (ρ), we conider the parametrization ρe iθ, π/ θ θ < π, a it i ilutrated in the next figure: C ρ θ r r θ A in the previou cae, the inequality (.6) i till true. So, taking into account that in θ θ/π, for θ [, π/], we conclude θ e t h M ()d e (t a)ρ co θ are (ρe iθ r )(ρe iθ r ) dθ C (ρ) π θ π e (t a)ρ co θ dθ From which we get Similarly, θ π θ π θ π e (t a)ρ co(θ+ π ) dθ e (t a)ρ in θ dθ e (t a)ρ(θ/π) dθ lim e t h M ()d. ρ C (ρ) lim e t h M ()d. ρ C (ρ) π ρ(t a).

9 CLASSICAL RISK PROCESS 77 and Step. Here we will how that lim ρ C (ρ) e t h M ()d lim e t h M ()d. ρ C (ρ) On C (ρ), we till ue the parametrization ρe iθ, with π < θ θ θ < π: C ρ θ θ r r θ Since x < x + ct, then t > a. Therefore we can take η > uch that ( t > + ( + η) /4) a. (.7) Moreover take ρ > uch that Notice that Hence ηρ > r r. (.8) π arg(ρeiθ r ) < θ and arg(ρe iθ r ) π. π 4 χ(ρeiθ ) < θ + π < θ. Since coθ i decreaing on [π/4, π], we have ( ) Re (ρe iθ r )(ρe iθ r ) ρe iθ r / ρe iθ r / coθ. (.9) On the other hand, the fact that r ρ coθ implie ρe iθ r ρ. (.)

10 78 JORGE A. LEÓN AND JOSÉ VILLA And uing (.8), we get ρe iθ r r r + ρ in (θ) r r + ρ (η + ) / ρ. (.) Therefore (.9), (.) and (.) yield ( ) Re (ρe iθ r )(ρe iθ r ) ρ / (η + ) /4 ρ / coθ (.) (η + ) /4 ρ coθ. From thi and (.7) we obtain e t h M ()d C (ρ) θ θ θ θ θ θ θ π θ π θ π θ π e (t a)ρ co θ are (ρe iθ r )(ρe iθ r ) dθ e (t a)ρ co θ a(η+)/4 ρ co θ dθ e (t a(+(η+)/4 ))ρco θ dθ e (t a(+(η+)/4 ))ρ co(θ+ π ) dθ e (t a(+(η+)/4 ))ρ in θ dθ Thi implie that θ π θ π e (t a(+(η+)/4 ))ρ θ π dθ π ρ(t a( + (η + ) /4 )). lim e t h M ()d. ρ C (ρ) Proceeding a the beginning of thi tep, we can conclude that we alo have e t h M ()d. lim ρ C (ρ) Step 4. Now we deal with the arc C 4 (ρ, ε) and C (ρ, ε). Here we conider the ame parametrization of previou tep. That i, ρe iθ, π/ θ θ θ4 4 π:

11 CLASSICAL RISK PROCESS 79 C 4 ρ θ θ 4 4 ρ η(ǫ) r r θ Notice that π arg(ρeiθ r ) θ and π arg(ρeiθ r ) θ. Thu, π/ χ(ρe iθ ) θ π. Moreover ince r, r ρ coθ then ρe iθ r ρ and ρe iθ r ρ, and uing the monotony of co on [π/, π], we have ( ) Re (ρe iθ r )(ρe iθ r ) ρe iθ r / ρe iθ r / co(χ(ρe iθ )) ρe iθ r / ρe iθ r / coθ ρ coθ. The above etimation i analogou to (.). Now the concluion follow a in Step. Step 5. Here we conider C 5 (ρ, ε) and C 9 (ρ, ε). Over C 5 (ρ, ε) we take the parametrization u + εi, ρ + η(ε) u r : C 5 ε ρ + η(ε) r r Notice that π arg(u + εi r ) π and π arg(u + εi r ) π,

12 8 JORGE A. LEÓN AND JOSÉ VILLA then π/ χ(u + εi) π. Since coine i negative and decreaing on [π/, π] we have, for ε < r r, Hence ( ) are (u + εi r )(u + εi r ) a u + εi r / u + εi r / ( co(χ(u + εi))) / a u r. e t h M () (u+εi r)(u+εi r ) e(t a)u are u + εi e(t a)r+/ a ρ r. r (.) By (.) we can apply the dominated convergence theorem and ince we have arg(u + εi r ) π and arg(u + εi r ) π, a ε, r lim e t h M ()d lim ε C ε 5(ρ,ε) ρ+η(ε) e (t a)(u+εi) a (u+εi r )(u+εi r ) u + εi r e (t a)u a u r / u r / e iπ ρ r e (t a)u+a u r / u r / ρ On the other hand, on C 9 (ρ, ε) we ue the parametrization u εi, r u ρ η(ε) : u u du. du du ρ + η(ε) r r ε C 9 Working a in previou cae, and noting that arg( u εi r ) π and arg( u εi r ) π, a ε,

13 CLASSICAL RISK PROCESS 8 we have Therefore, lim e t h M ()d ε C 9(ρ,ε) ρ η(ε) lim ε e (t a)( u εi) a ( u εi r)( u εi r ) u εi r r e (t a)u a u r / u r / e iπ ρ r ρ lim ρ ( lim ε u e (t a)u+a u r / u r / C 5(ρ,ε) u du. du e t h M ()d + lim e ε C t h M ()d 9(ρ,ε) ( du) ). Step 6. Now we deal with C 6 (ε). To do thi, we take the parametrization u + εi, r u r : C 6 r r Notice that for ε <, e t h M () e (t a)(u+εi) a (u+εi r )(u+εi r ) u + εi Since e(t a)u a u+εi r / u+εi r / co χ(u+εi) u + εi e(t a)r e a u+εi r / u+εi r / co χ(u+εi) r e(t a)r e a u+εi r / u+εi r / r ) / e (t a)r+a(+ r r. (.4) r arg(u + εi r ) π and arg(u + εi r ), a ε,

14 8 JORGE A. LEÓN AND JOSÉ VILLA then (u + εi r )(u + εi r ) u + εi r / u + εi r / exp(iχ(u + εi)) u r / u r / i, a ε. Due to (.4) we are able to apply the dominated convergence theorem: r lim e t h M ()d lim ε C ε 6(ε) r r r e (t a)(u+εi) a (u+εi r)(u+εi r ) u + εi e (t a)u a u r / u r / i Step 7. Now we uppoe that C 8 (ε) i defined by u εi, r u r : u du. du r r C 8 we can imitate Step 6 to we get r lim e t h M ()d ε C 8(ε) r e (t a)u+a u r / u r / i Step 8. Finally we deal with C 7 (ε). Here, we put r + εe iθ, π/ θ π/ : u du. C 7 ε θ r r Thu χ(εe iθ + r ) [ ] π, π [, ] π. Conequently, uing the fact that coθ, for θ [ π, π] [, π], we obtain ( ) Re εe iθ (εe iθ + r r ) ε / εe iθ + r r / coχ(εe iθ + r ). (.5)

15 CLASSICAL RISK PROCESS 8 Alo, for < ε < r /, we have r + εe iθ r + r ε co( θ) + ε r + ε r ε > r. (.6) The etimation (.5), (.6) and t > a yield e t h M ()d C 7(ε) εe (t a)r π εe(t a)r r εe(t a)r r π π π π π εe(t a)(r+ε) π. r Therefore lim ε C 7(ε) et h M ()d. e (t a)εe iθ a εe iθ (εe iθ +r r ) r + εe iθ dθ e (t a)εe iθ a εe iθ (εe iθ +r r ) dθ e (t a)εco( θ) are εe iθ (εe iθ +r r ) dθ Step 9. To finih the proof, we only need to take into account Step -8, together with (.5), and Propoition. and.... Proof of (.4). Let u define then Since K() + b + ( r )( r ), C\[r, r ]. b + r + r 4λrc r r b, ( + b) ( r )( r ), C. Thi implie that /K i analytic over C\[r, r ]. Moreover, it i not difficult to ee that, for ρ large enough, we have λrc, C C 4 C 5 C 7 C 9 C C, b, C C, K() (ρ + r ) / (ρ + r ) / in π 4, C C, ε + Re() r / Re() r / in π 4, C 6 C 8. A in the proof of (.) we have by (.) P (Y x (t) ) σ+i e tλe (x x)v πi c ( v + + r ) d λe a(λ rc) πi σ i σ+i σ i e t h m ()d, (.7)

16 84 JORGE A. LEÓN AND JOSÉ VILLA where h m () e a+a ( r )( r ). K() Finally, the reult follow from Propoition., (.7) and the proof of (.). Oberve that a < and t a >, together with (.7), allow u to copy, line by line, the proof of (.) to how that (.4) i alo true. Acknowledgment. The firt author would like to thank Univeridad Autónoma de Aguacaliente and the econd author appreciate the hopitality of Cinvetav- IPN and Univeridad Juárez Autónoma de Tabaco during the realization of thi work. Reference. Amuen, S.: Approximation for the probability of ruin within finite time, Scand. Act. J. (984) Amuen, S.: Ruin Probabilitie, World Scientific Publihing Co., Singapure,.. Chiu, S. N., Yin, C.: On occupation time for a rik proce with reerve-dependent premium, Stochatic Model 8() () Dickon, D. C. M.: Some finite time ruin problem, Reearch Paper Serie of Faculty of Economic & Commerce, The Univerity of Melbourne 5 (7). 5. Dickon, D. C. M., Hughe, B.D., Lianzeng, Z.: The denity of the time to ruin for a Sparre Anderen proce with Erlang arrival and exponential claim, Scan. Actuarial J. 5 (5) Dickon, D. C. M., Water, H. R.: The probability and everity of ruin in finite and in infinite time, Atin Bulletin () (99) Dickon, D. C. M., Willmot, G. E.: The denity of the time to ruin in the claical Poion rik model, Atin Bulletin 5() (5) do Rei, A. E.: How long i the urplu below zero?, Inurance: Mathematic and Economic (99) Drekic, S., Willmot, G. E.: On the denity and moment of the time of ruin with exponential claim, Atin Bulletin () () -.. Gerber, H. U.: When doe the urplu reach a given target?, Inurrace: Mathematic and Economic 9 (99) Ignatov, Z. G., Kaihev, V. K.: A finite-time ruin probability formula for continuou claim everitie, J. Appl. Prob. 4 (4) Seal, H. L.: Numerical calculation of the probability of ruin in the Poion/Exponential cae, Mitt. Verein Schweiz. Verich. Math. 7 (97) Spiegel, M. R.: Laplace Tranform, McGraw-Hill, New York, 988. Jorge A. León: Cinvetav-IPN, Departamento de Control Automático, Apartado Potal 4-74, México D.F., Mexico addre: jleon@ctrl.cinvetav.mx Joé Villa: Univeridad Autónoma de Aguacaliente, Depart. de Matemática y Fíica, Av. Univeridad 94, C.P. Aguacaliente, Ag., Mexico addre: jvilla@correo.uaa.mx