Laplace Transform Of The Time Of Ruin For A Perturbed Risk Process Driven By A Subordinator

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1 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 Laplace Tranform Of The Time Of Ruin For A Perturbed Rik Proce Driven By A Subordinator Moncef Elghribi and Ezeddine Haouala Abtract In thi paper, we contruct, by Bochner ubordination, a new model (an extenion of the Sparre-Anderen model with invetment that i perturbed by diffuion). For thi rik proce, we derive a general integro-differential equation for the Laplace tranform of the time of ruin with poitive urplu initial via the elementary propertie of the claical conditional expectation. The pecial cae, for different inter-arrival time ditribution, are given in ome detail. We alo deduce a comparion for Laplace tranform of the time of ruin, for different inter-arrival time ditribution. Keyword: Ruin theory; Bochner Subordination; Laplace tranform; Integro-differential equation. Introduction The claical rik model perturbed by a diffuion wa introduced by Gerber (97) and ha been further tudied by many author during the lat few year; uch a Dufrene and Gerber (99), Furrer and Schmidli (994), Gerber and Landry (998), Wang and Wu (), Tai and Willmot (a), Li and Garrido (5) and the reference therein. The purpoe of the paper are twice. At firt time, we conider a generalization of the perturbed rik model of Li and Garrido (5). We ubtitute the Brownian motion by a ubordinated proce in the ene of Bochner by unpecified ubordinator. Thi ubtitution yield a new model given by equation (3.6) (ee Section 3 for more detail). At econd time, for the above rik proce, we derive an integro-differential equation for the Laplace tranform of the time of ruin with poitive urplu initial via the elementary propertie of the claical conditional expectation. The organization of thi paper i a follow. The next Département de Mathématique, Faculté de Science de Tuni, 9 Tuni, Tuniia. Fax: Elmoncef.Elghribi@ft.rnu.tn, ezdine.haouala@ft.rnu.tn Département de Mathématique, Faculté de Science de Tuni, 9 Tuni, Tuniia. Fax: ezdine.haouala@ft.rnu.tn Section tart with a brief decription of the claical rik model, and ome attention i payed to the quantitie: the Laplace tranform of the time of ruin, the elementary propertie of the claical conditional expectation and the infiniteimal generator. In Section 3, we contruct a new model, Sparre-Anderen model with invetment perturbed by diffuion under ubordination, and we end thi ection by a particular cae of ubordination which how the coincidence of the reult in the Sparre- Anderen model with invetment perturbed by diffuion (.) and in our model (3.6). In Section 4, for the new model, we tart by deriving an integro-differential equation for the Laplace tranform of the time of ruin with poitive urplu initial under very general condition regarding the claim ize, the claim arrival and the return from invetment, via the elementary propertie of the claical conditional expectation, and we end thi ection by giving everal example of the integro-differential equation atified by the Laplace tranform of the time of ruin, for different inter-arrival time ditribution. Finally, in Section 5, we preent a comparion of Laplace tranform of the time of ruin, for different inter-arrival time ditribution. Model Decription and Notation We will aume that all procee and random variable are defined on a filtered complete probability pace (Ω, F, P, (F t ) t ). The filtration (F t ) t i right continuou and all the tochatic procee to be defined in thi paper are adapted. Conider a time-continuou Sparre-Anderen urplu proce perturbed by a diffuion N(t) U(t) = u + ct ξ k + λb(t), t, (.) k= where u i the initial capital and c > i the incoming premium rate. The claim ize (ξ k ) k N are poitive i.i.d. random variable with common probability ditribution function F ξ and denity function f ξ, repreenting the k-th claim amount, with finite mean µ = E[ξ ], and variance σ = V ar(ξ ) <. (Advance online publication: November 9)

2 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 The ordinary renewal proce {N(t), t } denote the number of claim up to time t, with N(t) = up{n : T n t}, t, with, by convention, up =, where the (T n ) n N denote the claim time, with T = < T < T <... The inter-arrival time: τ = T, τ k = T k T k, k =, 3,... are i.i.d. ditributed with finite mean. Finally, {B(t) : t } i a tandard Wiener proce that i independent of the compound ordinary renewal proce S(t) = N(t) k= ξ k and the diperion parameter λ >. Further aume that the equence ξ k and τ k are independent of each other. Next, we denote by f τ the denity for the time in between claim (τ k ) k N that i atifie an ordinary differential equation of order n and with contant coefficient, formally denoted by L( d dt )f τ (t) =, (.) with L denoting the formal adjoint of the linear operator L. In general, the linear operator L i defined by with the adjoint L, L( d dt )f τ (t) = L ( d dt )f τ (t) = n j= α j d j dt j f τ (t), (.3) n () j d j α j dt j f τ (t). (.4) j= If the invetment price i modeled by geometric Brownian motion with drift a and volatility σ, then the equation of the urplu model i expreible a: U(t) = u + ct + a N(t) k= U() d + σ U() db() ξ k + λb(t), t, (.5) where B(.) i a tandard Brownian motion. Remark : If N(t) i Poion ditributed, the proce (.) i a compound Poion proce and refer to the claical Cramér-Lundberg model perturbed by a diffuion (ee e.g. Furrer and Schmidli 994: [7]). But if N(t) i renewal proce, then the proce (.) i called the Sparre-Anderen model perturbed by a diffuion (ee e.g. Li and Garrido 5: [9]) and conequently (.5) i referred to a the Sparre-Anderen model perturbed by a diffuion with invetment. Now define T u = inf{t ; U(t) < U() = u} (, otherwie), to be the time of ruin of (.) and Ψ = P(T u < U() = u) = P[ inf{t ; U(t) < U() = u}], (.6) to be the ultimate ruin probability with an initial urplu u. Next, for define Φ = E(e Tu {Tu< } U() = u), (.7) where {.} i the uual indicator function, to be the Laplace tranform of the time of ruin with an initial urplu u. The loading of ecurity i defined by r = c µe[n(t)]. If r >, then the activity i known a profitable. Indeed, the Law of Large Number enure that, in thi cae, the proce U(t) + almot urely (a..) a t +, and conequently Ψ. If r <, then U(t) a.. a t +. Generally, we will make the aumption that the activity i profitable. Recall that the tranition operator K t of a Markov proce U(t) i given by: K t f = E(f(U(t)) U() = u), and the infiniteimal generator of {K t, t > }, i the linear operator A defined by: Af(x) = lim tf(x)f(x) K t t for all real-valued, bounded, Borel meaurable function f defined on a metric pace S. The domain of A i denoted by D A. Uing Itô formula we find that the infiniteimal generator of (.5) A = ( σ u + λ ) d d + (au + c) du du. (.8) Paulen and Gjeing (997) (ee []) introduce a relationhip between the infiniteimal generator of the rik proce and the two quantitie of ruin (probability of ruin, Laplace tranform of the time of ruin). Actually, for example, they how that a function q that atifie the equation Aq = q with ome boundary condition, i the Laplace tranform of the time of ruin. The following theorem i an adapted form of their theorem to the Cramér-Lundberg cae with no invetment. Our Theorem 4. (ee Section 4) i baed on the theorem given below. Theorem : (See []) Aume that on the event {T u = }, U t a.. a t. Then with the above notation we have the following. )Aume that g i a bounded and twice continuouly differentiable function on u with a bounded firt derivative there, where we at u = mean the right-hand derivative. If g olve Ag = on u >, together with the boundary condition: (Advance online publication: November 9)

3 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 a) g = on u <, b) g() = if λ >, c) lim u g =, then g = P(T u < U() = u). ) Aume that q, i a bounded and twice continuouly differentiable function on u with a bounded firt derivative there, where we at u = mean the righthand derivative. If q olve Aq = q on u >, together with the boundary condition: a) q = on u <, b) q () = if λ >, c) lim u q =, then q = E(e Tu {Tu< } U() = u). 3 Model Contruction 3. Subordination of Brownian motion For the following claical notion, we refer the reader to [] and [3]. Let (E, E) be a meaurable pace and let m be a σ-finite poitive meaure on (E, E). Let B = (B t ) t be a Brownian motion on R. The aociated emigroup P = (P t ) t i defined by P t = g t f for t > ; f L (λ) and g t (x) = Πt exp{ x } i the function of Gau on R. t The aociated L (m)- generator (or generator) M i defined by Mf(x) = lim tf(x)f(x) P t t =, where i the Laplacian operator, on it domain D(M) which i the et of all function f L (m) for which thi limit exit in L (m). Let Z = (Z t ) t a unpecified Bochner ubordinator, i.e. Z = (Z t ) t i a convolution emigroup of probability meaure on R uch that, for each t >, we have Z t ε and Z t i upported by [, [. The aociated Berntein function h i defined by it Laplace tranform LZ t (r) = exp(th(r)) for all r, t >. It i known that h admit the repreentation h(r) = kr + ( exp(r)) ν(d), r >, (3.) where k and ν i a meaure on ], [ verifying ( ) ν(d) <. Moreover, k and ν are uniquely determined, they are called parameter of the Berntein function of Z. Then Y = (Y t ) t, where Y t = B Zt i called the ubordinated proce of the Brownian motion B t by mean of the ubordinator Z t. The aociated emigroup P Z = (Pt Z ) t be the (Bochner) ubordinated emigroup of P by mean Z i given by: P Z t = P Z t (d) for every t >. The aociated generator i denoted by M Z on it domain D(M Z ). Moreover, it i known that D(M) D(M Z ) and M Z u = kmu + where k and ν are given in (3.). (P t u u) ν(dt), u D(M), (3.) 3. Example of ubordinator Cae : Dirac ubordinator Let ε = (ε t ) t the dirac ubordinator. Then in the cae, the ubordinated proce Y of the Brownian motion B by mean of the dirac ubordinator ε coincide to the original Brownian motion i.e. ( Y t = B εt = B t for every t > ). Conequently P ε = P and where i the Laplacian operator. M ε = M =, (3.3) Cae : One-ided table ubordinator Let η α fractional power be the one-ided table ubordinator of order α ], [, i.e. the unique convolution emigroup η α := ηt> α on [, [ uch that for each t >, the Laplace tranform L(ηt α )(x) = exp(tx α ), x >. It i well known that the aociated Berntein function h(r) = r α admit the repreentation h(r) = α Γ( α) ( exp(r)) d. (3.4) α+ In the cae, the ubordinated proce Y of the Brownian motion B by mean of η α i expreible a: Y t = B ηα t, for each t. The aociated emigroup P ηα = (P ηα t ) t> be the (Bochner) ubordinated emigroup of P by mean η α (i.e. P ηα t = P η α t (d) for every t > ). Hence, (ee [3] for more detail) the infiniteimal generator M η α of Y t on it domain D(M η α) i expreible a: M η α = α c ( ) α, c >, (3.5) where i the cloure of the Laplacian operator. 3.3 The model Now, we contruct, by a Bochner ubordination, a new model in the following way: we take the model (.) and we ubtitute the Brownian motion: B t by the ubordinated proce of B t in the ene of Bochner by mean of (Advance online publication: November 9)

4 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 Z t that i denoted by Y t. In thi way the model (.), (ee Section ), i expreible a: N(t) U(t) = u + ct ξ k + λy (t), t, (3.6) k= where u i the initial capital and c > i the incoming premium rate. The claim ize (ξ k ) k N are poitive i.i.d. random variable with common probability ditribution function F ξ and denity function f ξ, repreenting the k-th claim amount, with finite mean µ = E[ξ ], and variance σ = V ar(ξ ) <. The ordinary renewal proce {N(t), t } denote the number of claim up to time t, with N(t) = up{n : T n t}, t, with, by convention, up =, where the (T n ) n N denote the claim time, with T = < T < T <... The inter-arrival time: τ = T, τ k = T k T k, k =, 3,... are i.i.d. ditributed with finite mean. Finally, {Y (t) = B Zt : t } i a ubordinated proce of the tandard Wiener proce {B(t) : t } in the ene of Bochner by mean of Z that i independent of the compound ordinary renewal proce S(t) = N(t) k= ξ k and the diperion parameter λ >. Further aume that the equence ξ k and τ k are independent of each other. In thi paper, a particular cae i conidered, namely the invetment price i modeled by geometric Brownian motion with drift a and volatility σ, then the equation of the urplu model i given by: U(t) = u + ct + a N(t) k= U() d + σ U() db() ξ k + λy (t), t, (3.7) where B(.) i a tandard Brownian motion and Y (.) i ubordinated proce of B(.) in the ene of Bochner by mean of ubordinator Z(.). Hence, by the equation (3.) and Itô formula, the infiniteimal generator of (3.7) i expreible a: A Z = ( σ u + kλ ) d du +(au+c) d du where I denoted the identity operator. (P t I) ν(dt), (3.8) Since the ruin may occur only at the claim time, T k, the urplu proce (3.6) may be dicredited. The dicrete verion N(T k ) U k = U(T k ) = u + ct k ξ k + λy (T k ), (3.9) k= i a dicrete time Markov proce (Markov chain). The proce U k may be written immediately after the payment of the kth claim ξ k. U k = V U k τ k ξ k, (3.) where V U k τ k repreent the worth of a portfolio that reult from inveting the capital U k (immediately after the payment of the k claim) and the premium collected over the time τ k, into a riky aet. Recall that for the dicrete time Markov proce {(U k ) k N U() = u}, on the et of all real-valued, bounded, Borel-meaurable function ϕ, define the tranition operator P ϕ : R R, P h := E(h(U ) U() = u) = f τ (t)e(h(z u t x) U() = u)f X (x) dx dt, (3.) the generator of the time dicrete Markov proce i given by: A U ϕ = (P I)ϕ, where I denoted the identity operator, and D AU denoted the domain of the operator A U. Remark : For the preident trivial cae of ubordination (Dirac ubordinator), if the invetment price i modeled by geometric Brownian motion with drift a and volatility σ, then the equation of the urplu model i given by: U(t) = u + ct + a N(t) k= U() d + σ U() db() ξ k + λy (t), t, (3.) where B(.) i a tandard Brownian motion and Y (.) i a ubordinated proce of B(.) in the ene of Bochner by mean of ubordinator ε. Hence, by the equation (3.3) and the Itô formula, the infiniteimal generator of (3.) i expreible a: A ε = ( σ u + λ ) d d + (au + c) du du. (3.3) We remark that the equation (.5) and the equation (3.) poe the ame infiniteimal generator. Thi explain a coincidence between the Sparre-Anderen model perturbed by diffuion given by the equation (.) and the ubordinated Sparre-Anderen model perturbed by diffuion in the ene of Bochner by mean of ε given by the equation (3.6). 4 The integro-differential equation of the Laplace tranform of the time of ruin The claical approach in deriving equation atified by the Laplace tranform of the time of ruin i conditioning on the time of the firt claim and it ize, followed (Advance online publication: November 9)

5 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 by differentiation [Dickon and Hipp (998), Jun Cai (4) in the claical rik model without perturbation] and [Dufrene and Gerber (99), Furrer and Schmidli (994), Gerber and Landry (998), Wang and Wu (), Li and Garrido (5) and the reference therein in the claical rik model with perturbation]. In contrat, the uniform approach of thi paper (ee [6] for the claical rik model without perturbation) conit in deriving a general equation for the claical conditional expectation that relate to the Laplace tranform of the time of ruin via our Theorem 4. and Theorem 4. given below. Theorem 4. Let q D AZ, with P q (u, ) = q (u, ). If f τ atifie the ordinary differential equation with contant coefficient and L( d dt )f τ (t) = ) f (k) τ () =, for k =,..., n, ) lim x f (k) τ (x) =, for k =,..., n, then L (A Z I)P q (u, ) = f (n) τ () E[q (u, ξ )]. (4.) Proof. For, we take the ame technical of the proof of the theorem 3 in thei of Corina D. Contantinecu (ee page 3-33). Theorem 4. Aume that on the event {T u = }, U t a t. Aume that Φ i P invariant. Then the following axiom are equivalent: ) Any bounded function q D AU uch that q = e Tu g; atifie A U q = (P I)q =, together with the boundary condition for the function g: a) g = on u <, b) g() = if λ >, c) lim u g =, ) q i the Laplace tranform of the time of ruin, in other word q = Φ. Proof. Firt part ) = ). Let E u [q (U k )] := E u [q (U(T k ))] It i known that for any n = E(q (U(T k )) U() = u). n M n = q (U n ) A U q (U k ) i a martingale. Indeed k= n = q (U n ) (P I)q (U k ) k= E(M n+ F(U,..., U n )) = E(q (U n+ ) U,..., U n ) n (P I)q (U k ) k= = P q (U n ) P q (U n ) +q (U n ) (P I)q (U k ) n k= = M n. The aumption: A U q = implie that q (U k ) i a martingale, i.e. for any k, q = E u [q (U(T k ))] = E u [e T k g(u(t k ))]. The time of ruin T u i a topping time, thu q = E u [q (U(T u ))] = E u [e Tu g(u(t u ))] and moreover q = E u [q (U(T u T k ))] = E u [e (Tu T k) g(u(t u T k ))] = E u [e (Tu T k) g(u(t u T k )) {Tu<T k }] +E u [e (Tu T k) g(u(t u T k )) {T u>tk }] = E u [g(u(t u ))] E u [e Tu {Tu<T k }] +E u [g(u(t k ))] E u [e T k {Tu>T k }]. The reult thu follow by letting T k and uing the boundary condition for the function g, q = E u [e Tu {Tu< }] + = E u [e Tu {Tu< }] = Φ. When λ > the proce tarting from will immediately aume a negative value, hence the extra boundary condition q () = g() = in the cae. Thi prove part (i). Second part ) = ). Since the proce U k i a renewal proce and ince ruin cannot occur in the interval (, T ), where T repreent (Advance online publication: November 9)

6 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 the time of the firt claim, then the Laplace tranform of the time of ruin atifie the renewal equation, q = E(q (U ) U() = u) = P q. It i proved in the previou Theorem that P q atifie the equation for any q D AZ. Since, P q = q it follow that q atifie the equation. Since q i the Laplace tranform of the time of ruin, it alo atifie the boundary condition. Combining Theorem 4. with Theorem 4. above, we get that the Laplace tranform of the time of ruin atifie the following integro-differential equation: L (A Z I)Φ = f (n) τ () together with the boundary condition: a) lim u Φ =, b) Φ () = if λ >, c) (BC), Φ (u x)f ξ (x) dx, (4.) where (BC) tand for boundary condition and n repreent the degree of the ordinary differential equation atified by the denity of the inter-arrival time. The boundary condition (BC) may be derived from compatibility condition auming that the integro-differential equation and it derivative hold at zero. For intance, if the invetment i a geometric Brownian motion then the equation ha order n. 4. Application Many well-known equation are a particular form of the equation (4.). For intance, in the ubordinated Sparre-Anderen model with invetment perturbed by diffuion given by (3.7), the equation and their boundary condition can be derived for different inter-arrival time. Example : Erlang(, β) Conider that the urplu model (3.7) ha inter-arrival time τ k that are Erlang(, β), ditributed with the denity function f τ (t) = β t exp{βt}, t. Then for an Erlang(, β) ditribution, L( d dt ) = ( d dt + β), and L ( d dt ) = ( d dt + β). Hence, the equation (4.) i pecifically: (A Z + I + βi) Φ = f () τ () with the boundary condition: Φ (u x)f ξ (x) dx, (4.3) a) lim u Φ =, b) Φ () = if λ >, c) A Z A Z Φ ()(β +)A Z Φ ()+(β +) Φ () = β, d) the firt two derivative of the equation (4.3) evaluated at zero. The equation (4.3) i equivalent to A Z A Z Φ (β + )A Z Φ + (β + ) Φ u = β Φ (u x)f ξ (x) dx + β f ξ (x) dx, (4.4) where A Z = ( σ u + kλ ) d du +(au+c) d du and u (P t I) ν(dt) A Z A Z = ( σ u + kλ )A Z + (au + c)a Z (P t A Z A Z ) ν(dt) with (by taking account of derivation theorem under integral ign) and A ZΦ = ( σ u + kλ )Φ(3) +(σ u + au + c)φ () + aφ() (P t Φ () A ZΦ = ( σ u + kλ )Φ(4) Φ() ) ν(dt) +(σ u + au + c)φ (3) +(a + σ )Φ () (P t Φ () Φ() ) ν(dt). With the boundary condition obtained from the fact that the equation hold at zero and o do the firt two derivative of the equation. Example : Erlang(n, β) Conider that the urplu model (3.7) ha inter-arrival time τ k that are Erlang(n, β), ditributed with the denity function f τ (t) = βn Γ(n) tn exp{βt}, t and n N. Then for an Erlang(n, β) ditribution, L( d dt ) = ( d dt + β)n, and L ( d dt ) = ( d dt + β)n. Hence, the equation (4.) i (Advance online publication: November 9)

7 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 pecifically: n () k n! k!(n k)! Ak Z( + β) nk Φ = k= u β n Φ (u x)f ξ (x) dx + β n f ξ (x) dx, (4.5) with the boundary condition: a) lim u Φ =, b) Φ () = if λ >, c) n n! k= ()k k!(nk)! Ak Z ( + β)nk Φ () = β n, d) the firt n derivative of the equation (4.5) evaluated at zero. Example 3: Sum of two exponential Conider that the urplu model (3.7) ha inter-arrival time τ k that are um of two exponential, with the denity function f τ (t) = β β β β [exp{β t} exp{β t}], t and β β. In the cae of inter-arrival time ditributed a a mixture of exponential, the denity f τ atifie an ordinary differential equation of order, the linear operator L i L( d dt )f τ (t) = ( d dt + β )( d dt + β )f τ (t) =, and the adjoint linear operator L i expreible a: Aume that L ( d dt )f τ (t) = ( d dt + β )( d dt + β )f τ (t). u H := β β Φ (u x)f ξ (x) dx + β β f ξ (x) dx. Thu the equation (4.) i pecifically: H = A Z A Z Φ (β + β + )A Z Φ +[β β + (β + β ) + ]Φ, (4.6) with the boundary condition: a) lim u Φ =, b) Φ () = if λ >, c) A Z A Z Φ () (β + β + )A Z Φ () + [β β + (β + β ) + ]Φ () = β β, d) the firt two derivative of the equation (4.6) evaluated at zero. u u Thu, the explicit equation for the Laplace tranform of the time of ruin in cae on invetment in a geometric Brownian motion with inter-arrival time ditributed a a um of two exponential i a forth order integrodifferential equation: H = [ σ u + kλ ] Φ (4) + (σ u + kλ )(au + σ u + c)φ (3) + [(σ u + kλ ) (a + σ (β + β )u u)]φ () + (au + c)(σ u + au + c)φ () + (au + c)(a β β )Φ () (P t A Z Φ A Z Φ ) ν(dt) ( σ u + kλ ) (au + c) (P t Φ () Φ() ) ν(dt) (P t Φ () λ(β + β + ) Φ() ) ν(dt) (P t Φ Φ ) ν(dt) + [β β + (β + β ) + ]Φ, with the boundary condition obtained from the fact that the equation hold at zero and o do the firt two derivative of the equation. 5 Ordering of the Laplace tranform of the time of ruin Since the Laplace tranform of the time of ruin are olution of the newly introduced integro-differential equation, a new comparion of the Laplace tranform of the time of ruin when only the inter-arrival time ditribution are different i poible. Let U () (t) = u + ct + a N () (t) k= U () () d + σ U () ()db() ξ k + λy (t), t, (5.) be a ubordinated Cramér-Lundberg rik model with invetment perturbed by diffuion in the ene of Bochner in riky aet with a price which follow a geometric Brownian motion. The inter-arrival time {τ () k } k N are independent, exp(β) ditributed random variable. The claim arrival proce N () (t) i a Poion proce. The (Advance online publication: November 9)

8 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 Laplace tranform of the time of ruin for thi proce will be denoted by where Φ () T = E(e u {T () u < } U () () = u), T () u = inf{t ; U () (t) < U () () = u} (, otherwie), i the time of ruin of (5.). Let U () (t) = u + ct + a N () (t) k= U () () d + σ U () ()db() ξ k + λy (t), t, (5.) be a ubordinated Sparre-Anderen rik model with invetment perturbed by diffuion in the ene of Bochner in riky aet a in (5.), but with inter-arrival time {τ () k } k N are independent, Erlang(, β) ditributed random variable. The claim arrival proce N () (t) i a renewal proce. The correponding Laplace tranform of the time of ruin i where Φ () T = E(e u {T () u < } U () () = u), T () u = inf{t ; U () (t) < U () () = u} (, otherwie), i the time of ruin of (5.). The comparion of the urplu procee under thee different inter-arrival time ditribution may be achieved by a coupling of both procee derived from the common underlying Brownian motion. To be precie, one ue X(t) = X() exp{(a σ )t + σb t} [exp{(a σ [exp{(a σ +σ(b t B u )}] dy, (5.3) the explicit repreentation in term of the Brownian motion of the olution of the tochatic differential equation governing the invetment proce dx t = (ax t + c) dt + σx t db t + λ dy t, (5.4) thi can be verified uing Itô lemma. Lemma 5.If X(t) atifie the equation (5.3), then for any t, X(t) = X() exp{(a σ )(t ) + σ(b t B )} [exp{(a σ [exp{(a σ +σ(b t B u )}] dy. Proof. A X(t) atifie the equation (5.3) one ha X(t) = X() exp{(a σ )t + σb t} Then for any t, one ha [exp{(a σ [exp{(a σ +σ(b t B u )}] dy. X(t) = X()[exp{(a σ )(t + ) +σ(b t B + B )}] [exp{(a σ )[(t ) + ( u)] +σ[(b t B ) + (B B u )]}] du [exp{(a σ [exp{(a σ )[(t ) + ( u)] +σ[(b t B ) + (B B u )]}] dy [exp{(a σ +σ(b t B u )}] dy = X() exp{(a σ )(t ) + σ(b t B )} [exp{(a σ [exp{(a σ +σ(b t B u )}] dy. Propoition 5. For the procee U () and U () defined above, the Laplace tranform of the time of ruin have the following order: Φ Φ. (5.5) (Advance online publication: November 9)

9 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 Proof. In order to compare the two Laplace tranform of the time of ruin: Φ and Φ, one compare the two urplu procee U () and U () along each ample path of the Brownian motion. Both tart with the ame initial urplu u and have the ame underling Brownian motion. Let T () denote the time of the firt claim in the urplu proce U (). Then for any t T (), according to the equation (5.3) one ha U () (t) = u exp{(a σ )t + σb t} = U () (t). For t = T (), one ha U () (T () ) = u exp{(a σ + (c λσ) + λ T () [exp{(a σ [exp{(a σ +σ(b t B u )}] dy T () )T () + σb () T } [exp{(a σ () )(T u) +σ(b () T B u )}] du [exp{(a σ () )(T u) +σ(b () T B u )}] dy u exp{(a σ + (c λσ) + λ T () T () ξ () = U () (T () ). )T () + σb () T } [exp{(a σ () )(T u) B u )}] du +σ(b T () [exp{(a σ () )(T u) +σ(b () T B u )}] dy For T () t T (), according to lemma 5. U () (t) = U () (T () )[exp{(a σ T () )(t T () ) +σ(b t B T () )}] [exp{(a σ T () [exp{(a σ +σ(b t B u )}] dy )(t T () ) U () (T () )[exp{(a σ +σ(b t B () T }) T () = U () (t). T () [exp{(a σ [exp{(a σ +σ(b t B u )}] dy It follow by induction that U () (t) U () (t) for any t. Therefore, Φ Φ for any u. Propoition 5.3 For m < n, in the cae of Erlang(m, β), Erlang(n, β) rik procee Φ m Φ n. (5.6) Proof. Analogouly to the previou proof, it can be hown that Inductively, thi mean Φ m Φ n. Φ Φ Φ 3... Φ n, n N. Thu, for any m < n, the reult follow. Reference [] S. E. Anderen, On the collective theory of rik in cae of contagion between claim, Bull. Int. Math. Appl., V, pp , /57 [] J. Cai, Ruin probability and penalty function with tochatic rate of interet, Stochatic Proce. Appl., V, pp , /4, [3] D. Corina Contantinecu, Renewal Rik Procee with Stochatic Return on Invetment Thei on eptembre, 6. [4] F. Dufrene and H. Gerber, Rik theory for the compound Poion proce that i perturbed by diffuion, Inurance Mathematic and Economic, V, pp. 5-59, 99 [5] D. C. M. Dickon and C. Hipp, Ruin probabilitie for Erlang () rik procee, Inurance Mathematic and Economic, V, pp. 5-6, 998 (Advance online publication: November 9)

10 IAENG International Journal of Applied Mathematic, 39:4, IJAM_39_4_5 [6] M. Elghribi and E. Haouala, Laplace tranform of the time of ruin in the Sparre Anderen model with invetment, Math. Sci. Re. J., V, pp , /7 [7] H. Furrer and J. Schmidli, Exponential inequalitie for ruin probabilitie of rik procee perturbed by diffuion, Inurance Mathematic and Economic, V5, pp. 3-36, 994 [8] H. U. Gerber, An extenion of the renewal equation and it application in the collective theory of rik, Skandinavik Aktuarietidkrift, V, pp. 5-, 97 [9] S. Li and J. Garrido, On the Gerber-Shiu function in a Sparre-Anderen rik model perturbed by diffuion, Scandinavian Acturial Journal, V, pp. 6-86, 5 [] N. Jacob, Peudo Differential Operator and Markov Procee, nd Edition, Imperial College Pre, London, 3 [] H. Meng, C. Zhang and R. Wu, The expectation of aggregate dicound dividendd for a Sparre Anderen rik proce perturbed by diffuion, Applied Stochatic Model In Buine and Indutry, V, pp. 9-3, 5/ [] J. Paulen and H. Gjeing, Ruin theory with Stochatic Return on invetment, Advanced Applied Probability, V9, pp , 6/97 [3] K.I. Sato, Lévy Procee and infinitely Diviible Ditributio, nd Edition,Cambridge Univerity Pre 999 [4] E. Thomann and E. Waymire, Contingent claim on aet with converion cot, Journal of Statitical Planning and Inference, V3, pp , 3, [5] C.C.L. Tai and G.E. Willmot, A generalized defective renewal equation for the urplu proce perturbed by diffuion, Inurance Mathematic and Economic, V3, pp. 5-66, 4/ [6] G. Wang and R. Wu, Some ditribution for claical rik procee that i perturbed by diffuion, Inurance Mathematic and Economic, V6, pp. 5-4, 5/ (Advance online publication: November 9)