Interest-Bearing Surplus Model with Liquid Reserves

Transcription

1 Inteet-Beaing Suplu Model with Liquid Reeve Kitina P Sendova 1 and Yanyan Zang 2 Abtact: We conide a uin model whee the uplu poce of an inuance company i contucted o that pat of the cuent uplu i kept available at all time and the emaining pat i inveted The fome potion of the capital i called liquid eeve In thi pape, we tudy the expected dicounted penalty function at uin Fit, we deive an intego-diffeential equation atified by the Gebe-Shiu function Second, we apply Laplace tanfom to the equation and educe it to a fit ode linea diffeential equation fo the function in quetion Finally, we find an explicit fom of the Gebe-Shiu function by conideing exponential claim [Key wod: Gebe-Shiu function, inteet, liquid eeve] T INTRODUCTION he Office of Supeintendent of Financial Intitution (OSFI) in Canada ha eleaed Minimum Continuing Capital and Suplu Requiement (MCCSR), which ae in place to potect policyholde by enuing that inuance companie maintain adequate capital level while the emaining uplu of the inue may be inveted in a competitive global maketplace In thi pape, we intend to model thee equiement Howeve, it i not poible to implement MCCSR in thei full detail due to thei high complexity Intead, we apply ome baic idea fom MCCSR to contuct the uplu poce Peviouly, thi wa achieved in two pape by Cai et al 1 Depatment of Statitical and Actuaial Science, Univeity of Weten Ontaio, kendova@ tatuwoca 2 Depatment of Statitical and Actuaial Science, Univeity of Weten Ontaio, yzang4@uwoca The autho wih to thank an anonymou efeee fo numeou valuable uggetion that undoubtedly impoved the claity of the pape Suppot fom a gant fom the Natual Science and Engineeing Reeach Council of Canada i gatefully acknowledged by the fit autho 178 Jounal of Inuance Iue, 21, 33 (2): Copyight 21 by the Weten Rik and Inuance Aociation All ight eeved

2 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 179 (29a) and Cai et al (29b), who choe a thehold-type model Moe pecifically, the autho aume that the potion of the uplu that i below a peet level i liquid and the amount in exce of thi level i inveted unde a deteminitic inteet ate In the peent pape, intead of implementing a thehold, we pefe to invet a pecentage of the cuent uplu of the inue Thi appoach i moe in line with the MCCSR Ou objective i to tudy the expected value of the dicounted penalty function at uin unde a pecific uplu poce Thi function i alo known a the Gebe-Shiu dicounted penalty function, which we define by mu ( ) E{ e T wut ( ( ), UT ( ) )IT ( < ) U( ) u}, u It wa fit intoduced by Gebe and Shiu (1998) fo analyzing the claical compound Poion model and wa ubequently tudied unde numeou othe uin theoy model Hee T i the time of uin, which i denoted by T inf{t U(t) < }, i the foce of inteet, and w(x 1, x 2 ), x 1, x 2 >, i a nonnegative function of the uplu immediately befoe uin (U(T ) above) and the deficit at uin ( U(T) above) The function i known a penalty function Alo, I(E) i the indicato function of an event E The indicato aign a value of one when the event occu and zeo when the event doe not occu Initially, the function wa intended a a tool fo analyzing the expected dicounted penalty a a function of the uplu x 1 and the deficit x 2 It ha, though, a much boade meaning and eve to ecove a numbe of quantitie of pecial inteet in uin theoy Thee include the pobability of ultimate uin, the Laplace tanfom of the time of uin, the joint and maginal ditibution and moment of the uplu immediately befoe uin, and the deficit at uin The following lit povide moe detail how thee quantitie ae obtained: pobability of uin:, w(x 1, x 2 ) 1 fo all x 1, x 2 > ; (defective) joint and maginal moment of the uplu and deficit: k l, w(x 1, x 2 ) x 1x2, k, l nonnegative intege; l (defective) dicounted moment of the deficit: w(x 1, x 2 ) x 2 ; joint (defective) ditibution of the uplu and deficit:, w(x 1, x 2 ) I(x 1 x)i(x 2 y) I(x 1 x, x 2 y) fo all x 1, x 2 > ; maginal (defective) ditibution ae obtained by letting eithe x o y in the above;

3 18 SENDOVA AND ZANG (defective) ditibution of the claim cauing uin:, w(x 1, x 2 ) I(x 1 + x 2 z) fo all x 1, x 2 > ; tivaiate Laplace tanfom of the time of uin, the deficit, and the uplu: w(x 1, x 2 ) e x 1 zx 2 ; the maginal tanfom ae deived by etting any two of,, o z to Reeache attempt to expe quantitie of inteet, uch a ome of thoe lited above, in tem of known quantitie A potential appoach i to find an explicit expeion fo the Gebe-Shiu function and then invetigate it paticula cae In eveal intance, it i not poible to obtain uch an expeion Numeical method might then be ueful Altenatively, pecific cae of the expected dicounted penalty function might be eaie to analyze In thi pape, we complement the wok of Cai and Dickon (22), Cai (27), Yang et al (28), Cai et al (29a), and Cai et al (29b) whee inteet i alo incopoated into the paticula uplu pocee Thi aticle i tuctued a follow In Section 2, we intoduce the model we ae inteeted in tudying Then, in Section 3, we deive an intego-diffeential equation fo the Gebe-Shiu dicounted penalty function and apply Laplace tanfom to analyze thi function in Section 4 We then ty to obtain ome explicit expeion when claim ize ae exponentially ditibuted, in Section 5 In contat to peviou aticle conideing uin model with inteet, we manage to keep all paamete in the model without fixing them to a paticula value MODEL DESCRIPTION Aume that the claim amount {Y 1, Y 2, Y 3, } ae independent and identically ditibuted (iid) poitive andom vaiable with common cumulative ditibution function (cdf) F(y), y >, with F() and Laplace-Stieltjie tanfom y f ( ) e dfy ( ) Let the numbe-of-claim poce {N(t) t } be a homogeneou Poion poce with ate λ >, which i independent fom {Y 1, Y 2, Y 3, }, and {V 1, V 2, V 3, } be the time of epective claim occuence We uppoe that the inteclaim time {V 1, V 2 V 1, V 3 V 2, }, which ae iid exponential andom vaiable with mean 1/λ, ae alo independent fom the claim amount We alo define

4 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 181 Nt () the aggegate-claim poce St () Y i t to be the um of all i 1 claim up to time t, with the undetanding that S(t) if N(t) Finally, we aume the initial uplu to be u and the contant pemium ate to be c > λe{y 1 } We intend to keep pat of the diffeence between pemium and claim, α [,1], available at all time and invet the emaining pat with foce of inteet > The uplu poce {U(t) t } i then defined a Ut () αct [ S() t ] e t t u ( 1 α) c e Nt () d e V j + + Y j j 1 (21) Obeve that the foce of inteet depend on the choice of invetment of the inuance company The ik that i allowed by the MCCSR depend on the type of inuance poduct and the ating that the company want to maintain A fo the foce of inteet that appea in the Gebe-Shiu function, it i et when and if the company bankupt Alo, to obtain ome quantitie of inteet a pecial cae of the Gebe-Shiu function one need to fix to a paticula value o teat it a a vaiable of a Laplace tanfom It i then mathematically convenient to have the foce of inteet and a two epaate paamete INTEGRO-DIFFERENTIAL EQUATION FOR THE GERBER-SHIU FUNCTION In thi ection we deive an equation atified by the Gebe-Shiu function Since the function of inteet i found unde both the integal ign and the deivative ign, the identity that we obtain i called intego-diffeential equation A it i, numeical appoache may be ued fo olving it In ou late tudie, though, we conide two othe appoache to deducing the expected dicounted penalty function Theoem 31 Unde the uplu poce U(t), which i govened by identity (21), the Gebe-Shiu function m atifie the following intego-diffeential equation

5 182 SENDOVA AND ZANG m' ( u) u c u mu λ ( ) c u mu ( y ) dfy ( ) wuy (, u) dfy ( ) + u, (31) u Poof Auming that the time of the fit claim i t and it amount i y, the uplu U(t) may be peented a Ut () αct e t ( 1 α)c + u ( e t 1) y σ( tu ; ) y, whee σ( tu ; ) αct e t ( 1 α)c + u ( e t 1) Let Δ > be a ufficiently mall numbe and conide the inteval [,Δ) Denote by m(u E) the conditional expectation E{ e T wut ( ( ), UT ( ) )IT ( < ) U( ) ue, } whee E i an event Since the numbe of claim in [,Δ) ha a Poion ditibution with paamete λδ and the inteclaim time ditibution i the memoile exponential ditibution, then conditioning on the numbe of claim in [,Δ) and applying the Total Pobability Theoem, we may ewite the Gebe-Shiu dicounted penalty function a m(u) (1 λδ)e Δ m(u no claim in [,Δ)) + λδe Δ m(u one claim in [,Δ)) + o(δ) In the cae when thee ae no claim in [,Δ), the uplu at time Δ i σ(δ; u) When one claim of amount y occu in [,Δ) thee ae two poible cenaio: thi claim i malle than the accumulated capital and the uplu poce tat ove with a new initial uplu of σ(δ; u) y, o the claim amount lead to uin with penalty w[σ(δ; u), σ(δ; u) y] We may then apply the Total Pobability Theoem fo all poible amount y and obtain

6 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 183 mu ( ) ( 1 λδ)e Δ m[ σδu ( ; )] + (32) λδe Δ σδu ( ; ) m[ σδu ( ; ) y] dfy ( ) + w[ σδu ( ; ), σ( Δ; u) y] dfy ( ) + o( Δ) σδu ( ; ) By employing Taylo' expanion, m[ σδu ( ; )] i peented a m ( n) ( u) m[ σδu ( ; )] mu ( ) + m' ( u) [ σ( Δ; u) u] [ σδu ( ; ) u] n n! n 2 Then, ineting thi equation into (32) yield mu ( ) ( 1 λδ)e Δ m ( n) ( u) mu ( ) m' ( u) [ σ( Δ; u) u] [ σδu ( ; ) u] n + + n! n 2 λδe Δ σδu ( ; ) + m[ σδu ( ; ) y] dfy ( ) + w[ σδu ( ; ), σ( Δ; u) y] dfy ( ) σδu ( ; ) +o( Δ) We move all the tem to the left-hand ide and divide by Δ 1 e Δ + λδe Δ ( 1 λδ)e Δ [ σδu ( ; ) u] mu ( ) m' ( u) Δ Δ ( 1 λδ)e Δ m ( n) ( u) [ σδu ( ; ) u] n Δ n! n 2 λe Δ σδu ( ; ) m[ σδu ( ; ) y] dfy ( ) + w[ σδu ( ; ), σ( Δ; u) y] dfy ( ) σδu ( ; ) o ( Δ) Δ

7 184 SENDOVA AND ZANG We apply L Hôpital' ule to the above equation when Δ and the tem involving Δ become lim Δ 1 e Δ + λδe Δ Δ lim ( e + Δ λe Δ λδe Δ ) Δ λ +, limσ Δ ( Δ; u) lim Δ 1 α αcδ + ueδ + ( )c ( e Δ 1) u, [ σδu ( ; ) u] n αcδ ue Δ ( α)c e Δ n + + ( 1) u lim lim Δ Δ Δ Δ 1 α lim n αcδ + ueδ + ( )c ( e Δ 1) u Δ n 1 [ αc + ue Δ + ( 1 α)ce Δ ] Hence, equation (32) yield c + u, n 1, n 2, 3, 4 lim o( Δ) Δ Δ u ( )mu ( ) ( c + u)m' ( u) λ mu ( y) dfy ( ) λ wuy (, u) dfy ( ) u (33) which i the equied intego-diffeential equation fo the Gebe-Shiu function Obeve that the intego-diffeential equation atified by the Gebe- Shiu function doe not depend on the choice of α, whee α wa intoduced a the ate of the diffeence between pemium and claim to be kept available at all time Thi i a notable diffeence fom the uin model

8 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 185 popoed by Cai et al (29a) and Cai et al (29b), whee all expeion depend on the thehold level Cai and Dickon (22) dicu the cae when α i equal to, which i intepeted a inveting the whole amount of uplu with cetain foce of inteet, and the intego-diffeential equation (22) deived in thei pape i exactly the ame a the equation obtained in Theoem 31 In contat to peviou uin model that lead to intego-diffeential equation atified by the dicounted penalty function (ee, eg, Lin et al, 23; Mitic et al, 21), identity (31) deduced in Theoem 31 i linea but with non-contant coefficient It i exactly the tem c + u in (33) that lead to complication To obtain an explicit eult, one need to impoe additional aumption on the foce of inteet, the penalty function w, o the claim-ize ditibution Fo example, the mot geneal explicit olution to (31) appea in the pape by Cai and Dickon (22), whee i et to zeo (Thi eult i alo implemented in the model conideed by Cai et al, 29a, and Cai et al, 29b) Le geneal i the eult of Theoem 51 in Cai (27), whee along with the aumption the autho conide exponential claim He then deduce an explicit expeion of the Gebe-Shiu function The leat geneal eult appea in Yang et al (28), who make the peviou two aumption and uppoe futhe that the penalty function depend only on the deficit at uin In Section 4 of thi pape, we popoe an altenative expeion to the intego-diffeential equation (31), which i the fit ode non-homogeneou diffeential equation (41) atified by the Laplace tanfom of the expected dicounted penalty function Although thi type of equation i well tudied in the aea of odinay diffeential equation and may be olved explicitly, identity (41) poe ome difficultie A a eult, Theoem 41 may athe povide an altenative equation to be analyzed numeically Anothe contibution to the exiting liteatue i the explicit fom of the Gebe-Shiu function povided by Theoem 52 in Section 5, whee claim ae aumed to follow an exponential ditibution but no othe etiction ae impoed to the model paamete LAPLACE TRANSFORM OF THE GERBER-SHIU FUNCTION Cai and Dickon (22) peent equation (31) a a Voltea-type integal equation in thei attempt to olve it A a conequence, fo the explicit olution they need to pecify the value of the Gebe-Shiu function with no initial uplu They achieve thi in the dicount fee cae, ie, when In thi ection we attempt to tudy the behavio of the Gebe-Shiu function though it Laplace tanfom A long a continuou integable

9 186 SENDOVA AND ZANG function ae concened, the invee of thei Laplace tanfom i unique (ee Theoem 2-1 in Spiegel, 1965) A a eult, obtaining an explicit expeion fo the Laplace tanfom of uch a function yield an explicit expeion fo the function being tanfomed, a long a a way of inveting the Laplace tanfom i found Thi i why Laplace tanfom ae often chaacteized fo uin model dicued in the actuaial liteatue Reeache then eithe ty to invet the Laplace tanfom and deduce an expeion of the expected dicounted penalty function, o apply a numeical method fo the inveion A it will be een late, neithe of thee two appoache i poible unde the model tudied in thi pape Thu, one need to eot to numeical method available fo non-homogeneou linea odinay diffeential equation Denote by m ( ) e u mu ( ) du the Laplace tanfom of the Gebe- Shiu function We alo define ζ( u) wuy (, u) dfy ( ) and u ζ ( ) e u ζ( u) du Theoem 41 The Laplace tanfom of the Gebe-Shiu function atifie m' ( ) g ( )m ( ) + h ( ) (41) whee g ( ) [ c λ + λf ( ) ] and h ( ) Poof Fom Theoem 31, we have λ c ----ζ ( ) ----m ( ) u m' ( u) c + u mu λ ( ) c u mu ( y ) dfy + ζ( u), u Applying Laplace-Stieltjie tanfom to the above equation and eaanging it, we obtain

10 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 187 e u ( c + u)m' ( u) du ( λ + ) e u u mu ( ) du λ e u mu ( y) dfy ( ) du λ e u ζ( u) du We then ewite the above equation a c e u m' ( u) du + e u um' ( u) du ( )m ( ) λ e u mu ( y) dudf( y) λζ( ) y Pefoming integation by pat on the left-hand ide, we have c e u m' ( u) du + e u um' ( u) du c e u mu ( ) + e u mu ( ) du e u um( u) ( e u ue u )mu ( ) du + cm [ ( ) m( ) ] + [ m ( ) m' ( ) ] ( c )m ( ) cm( ) m' ( ) Then the equation become ( c )m ( ) cm( ) m' ( ) ( )m ( ) λ e t ( + y) mt ()ΔFy d ( ) λζ ( ) ( )m ( ) λm ( )f ( ) λζ ( ) Finally, a needed m' 1 λ c ( ) ---- [ c λ + λf ( ) ]m ( ) ζ ( ) ----m ( ),

11 188 SENDOVA AND ZANG Obeve that in contat to the claical compound Poion model, i not a oot to the function c + λf ( ) ( + ) even when, unle a well Thu, it i ufficient to conide only the cae > Notice that the highet deivative of the Laplace tanfom m in equation (41) i the fit deivative Alo, it contain a tem h that doe not involve m Thi make the equation non-homogeneou with epect to m In othe wod, we have deived a linea non-homogeneou fit ode diffeential equation atified by the Laplace tanfom of the Gebe-Shiu function We ae able to olve thi odinay diffeential equation though tandad technique (ee, fo intance, in Petovki, 1966, p 21) Namely, m ( ) m exp g( ξ) dξ + hx ( ) exp g( ξ) dξ x dx, (42) whee > i a contant uch that m m ( ) i known We aleady know that m ( ) convege to when convege to Thi i why we let and m o Next, we need to veify whethe g( ξ) dξ convege o not when g( ξ) dξ g( ξ) dξ [ cξ λ + λf ( ξ) ] dξ ξ c - 1 λ f ( ξ) ---- ( + λ + ) dξ ξ ξ c -ξ + λ logξ + λ -- f ( ξ) dξ ξ We find that the fit tem of the equation on the ight-hand ide convege to Then we need to check how the econd tem behave We begin by applying Theoem 1-11 in Spiegel (1965)

12 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 189 f ( ξ) dξ ξ + + e ξy + + Fy ( ) dydξ Fy ( ) e ξy dξdy + e y Fy ( ) dy y f ( ξ) Eithe of the two epeentation of the integal dξ ξ + e yfy ( ) dy ae difficult to tet fo convegence unle the claim-ize y cdf F i known A long a ou poblem i concened, though, it i ufficient to know that they convege eithe to a poitive finite value o to infinity A a eult, the whole integal g( ξ) dξ divege to and exp g( ξ) dξ convege to Thu, the fit tem on the ight-hand ide of the equation (42), namely, m exp g( ξ) dξ, i Thu, equation (42) educe to m ( ) hx ( ) exp g( ξ) dξ x (43) We now want to conide the econd pat of equation (42), namely, the ight-hand ide of (43) Howeve, it i difficult to claify it convegence popetie becaue the integal become complex afte expanding Moeove, it involve m(), which need to be pecified additionally Cai and Dickon (22) manage to do o only in the dicount-fee cae when Theefoe, it eem that numeical appoache to olving equation (41) might be moe appopiate EXPONENTIAL CLAIMS In thi ection, we deive an explicit expeion fo the Gebe-Shiu function when claim amount ae exponentially ditibuted with no othe etiction It i notewothy that the pape by Cai (27), Yang et al (28), Cai et al (29a), and Cai et al (29b) all conide the cae with exponential dx

13 19 SENDOVA AND ZANG claim amount but etict themelve to the dicount-fee cae, ie, when Moeove, Yang et al (28) impoe futhe that the penalty function depend olely on the deficit at uin We poceed by etating the intego-diffeential equation fo the Gebe-Shiu function a ( c + u)m' ( u) ( )mu ( ) λ[ Amu ( ( )) + ζ( u) ] (51) u whee Amu ( ( )) mu ( y) dfy ( ) and the claim amount {Y 1,Y 2, } ae independent, identically, and exponentially ditibuted with common pdf fy ( ) βe βy, y > Lemma 51 When claim ize ae exponentially ditibuted, then d Amu ( ( )) βmu ( ) βamu ( ( )) du Poof By a change of vaiable we have, u u Amu ( ( )) my ( )fu ( y) dy m( y)βe β( u y) dy Then we diffeentiate the equation with epect to u, u d d Amu ( ( )) βmu ( ) [ my ( )βe β( u y) dy] du du u βm( u) β my ( )βe β( u y) dy βm( u) βamu ( ( )), which complete the poof We now educe the intego-diffeential equation (51) to a econd ode non-homogeneou diffeential equation To thi aim, we fit diffeentiate equation (51) and it become ( c + u)m'' ( u) + ( + )m' ( u) λ d Amu ( ( )) + ζ' ( u) du (52) Then, we multiply equation (51) by β,

14 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 191 β( c + u)m'' ( u) β( λ + )mu ( ) λβ[ Amu ( ( )) + ζ( u) ] (53) By adding (52) and (53) togethe and employing Lemma 51, we obtain ( c + u)m'' ( u) + [ λ + β( c + u) ]m'( u) βmu ( ) λζ' [ ( u) + βζ( u) ],(54) a need We may now olve equation (54) explicitly Theoem 52 When the claim amount ae exponentially ditibuted with mean 1/β, the Gebe-Shiu function ha the fom mu ( ) [ κ 1 + C 1 ( βu βc ) ]y 1 ( βu βc ) + [ κ 2 + C 2 ( βu βc ) ]y 2 ( βu βc ), whee κ 1 and κ 2 ae abitay contant, y 1 ( x) Φ λ -- 1 λ , ; x, , 2, x Φ λ --, 1 λ ; x, λ , 2, (55) y 2 ( x) Ψ λ -- 1 λ , ; x λ, , 2, λ x Ψ λ --, 1 λ ; x, , 2, (56) a ( k) x k Φ( a, b; x) , and (57) b ( k) a ( k) aa ( + 1) ( a + k 1), k 1 k! Ψ( abx, ; ) Γ( 1 b) Γ( a b + 1) Φ abx Γ( b 1) (, ; ) Γ( a) x 1 b Φ( a b+ 12, bx ; ), (58) λ + x C 1 k 1 y 1 ( v)v e v dv, (59)

15 192 SENDOVA AND ZANG x 2 y 2 x C 2 k 2 y 2 ( v)v e v dv ( z)z e z z η ( v )y ( v )v e v dv dz, 2 + (51) whee κ 1 and κ 2 ae alo ome abitay contant, and η( x) λ ζ' 1 --x c βζ --x c - β β β (511) Poof We intend to make ome ubtitution in ode to educe equation (54) to a imple fom Let x be uch that u + c --x β and y(x) be uch that mu ( ) m 1 β --x c - yx ( ) A we know, m' ( u) βyx ( ) and m'' ( u) β 2 y'' ( x) Then equation (54) become βy'' ( x) [ λ x]βy' ( x) βy( x) 1 λ ζ' c --x βζ c --x - β β By dividing the above equation by β, it educe to xy'' ( x) x y' ( x) + -- yx ( ) hx ( ) (512) In ode to olve the above non-homogeneou diffeential equation, we have to olve the aociated homogeneou equation fit, that i, xy'' ( x) x y' ( x) + -- yx ( ) (513)

16 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 193 The olution to equation (513) i povided in Polyanin and Zaitev (23), p 22, and i decibed a the degeneate hypegeometic equation It i tated a whee κ 1, κ 2 ae ome abitay contant, which may be found unde bounday condition, and y 1 ( x) and y 2 ( x) ae pecified by (55) and (56), epectively Polyanin and Zaitev (23) alo tate that the geneal olution to the non-homogeneou diffeential equation i the um of the geneal olution to the coeponding homogeneou equation and any paticula olution to the non-homogeneou equation In ou cae, the paticula fom of the non-homogeneou equation (512) i y p ( x) C 1 ( x)y 1 ( x) + C 2 ( x)y 2 ( x) Now, we eplace yx ( ) by y p ( x) in equation (512), deducing Since we need only one paticula olution to the non-homogeneou diffeential equation, we may find paticula function fo C 1 and C 2 in ode to make the above equation hold One way of doing o without lo of geneality i by auming that and y h ( x) κ 1 y 1 ( x) + κ 2 y 2 ( x), xy 1 ( x)c 1 ''( x) 2xy 1 '( x) 1 λ x y1 ( x) C 1 '( x) + xy 2 ( x)c 2 ''( x) + 2xy 2 '( x) + 1 λ x y2 ( x) C 2 '( x) η( x) xy 1 ( x)c 1 ''( x) + 2xy 1 '( x) x y1 ( x) C 1 '( x) xy 2 ( x)c 2 ''( x) + 2xy 2 '( x) x y2 ( x) C 2 '( x) η( x) We ae able to olve the above fit ode linea diffeential equation in C 1 '( x) and C 2 '( x),

17 194 SENDOVA AND ZANG x 2xy 2 '( x) x y2 ( x) C 1 '( x) C exp dx xy 2 ( x) x C exp 2ln( y 2 ( x) ) 1 λ lnx + x+ l 2 k 1 y 1 ( v)v e v, whee C 1 '( x) pae though ome point ( x, C ), and l i ome contant calculated fom the integal Thu, equation (59) epeent the olution fo C 1 ( x) and imilaly, we may confim the olution of C 2 ( x), which i defined in (51) Theefoe, the geneal olution to the non-homogeneou diffeential equation (512) ha the fom a needed yx ( ) y h ( x) + y p ( x), CONCLUSIONS In thi pape we tudy the evolution of the uplu of an inuance company that invet a pecentage 1 α of it cuent uplu into a ikfee aet that povide a etun at a foce of inteet The emaining pecentage α of the company uplu i allocated to the o-called liquid eeve that do not yield any etun Nevethele, highe liquid eeve ae tied to a highe cedit ating Thi i the incentive fo the company to avoid etting α to a vey mall value, o in othe wod, to invet almot entiely it uplu Unde thi etup, it i natual to ak what i an appopiate balance between invetment and liquid eeve The anwe i povided by the value of α, which depend on the ik enitivity of the company Moe peciely, one need to know how high a pobability of ultimate uin o a deficit at uin the inue i willing to toleate Thee two quantitie and eveal othe meaue of the ikine of the buine ae pecial cae of the expected dicounted penalty function that i dicued in thi pape

18 INTEREST-BEARING SURPLUS MODEL WITH LIQUID RESERVES 195 Inteetingly, it appea fom ou analyi that whethe the inuance company keep cetain pecentage α > of it cuent uplu a liquid eeve and invet the emainde, o the company invet the entie uplu (α ), the expected dicounted penalty function atifie the ame intego-diffeential equation The dependence on α might be only elevant to the aocated initial condition Thi in tun implie that any quantitie of inteet that may be deduced fom the Gebe-Shiu function alo depend on α only though ome initial value In paticula, the pobability of eventual uin and the deficit at uin, which ae mentioned above, ae elated to α though thee initial condition Conequently, it become impotant to pecify both the fom of the Gebe-Shiu function and the elated initial condition A long a the expected dicounted penalty function i concened, we popoe two olution tategie In egad to the initial condition, Section 3 in Cai and Dickon (22) povide a patial anwe Namely, when the foce of inteet i zeo, m() i pecified and may eve a an initial condition Since thee i no complete olution to the peviouly mentioned intego-diffeential equation atified by the Gebe-Shiu function in the actuaial liteatue, ou fit olution tategy i though it Laplace tanfom Thi lead to a imple linea odinay diffeential equation that might be olved explicitly fo paticula cae of the penalty function a long a an initial condition i povided In paticula, thi might be m() when Ou econd olution tategy i elated to the cae when claim ae exponentially ditibuted We deduce an explicit fom of the Gebe-Shiu function without impoing additional etiction on the model paamete The latte i a notable impovement ove elevant eult in the cuent actuaial liteatue Finally, ou finding make u believe that although we povide an explicit expeion of the Gebe-Shiu function when claim have exponential ditibution, it i moe pactical to deduce the value of m(u) fo a pecific initial capital u numeically though the Laplace tanfom m in equation (43) REFERENCES Cai, J (27) On the Time Value of Abolute Ruin with Debit Inteet, Advance in Applied Pobability, 39: Cai, J and DCM Dickon (22) On the Expected Dicounted Penalty Function at Ruin of a Suplu Poce with Inteet, Inuance: Mathematic and Economic, 3:

19 196 SENDOVA AND ZANG Cai, J, R Feng, and GE Willmot (29a) Analyi of the Compound Poion Suplu Model with Liquid Reeve, Inteet and Dividend, ASTIN Bulletin, 39(1): Cai, J, R Feng, and GE Willmot (29b) The Compound Poion Suplu Model with Inteet and Liquid Reeve: Analyi of the Gebe-Shiu Dicounted Penalty Function, Methodology and Computing in Applied Pobability, 11: Gebe, HU and ESW Shiu (1998) On the Time Value of Ruin, Noth Ameican Actuaial Jounal, 2: Lin, XS, GE Willmot, and S Dekic (23) The Claical Rik Model with a Contant Dividend Baie: Analyi of the Gebe-Shiu dicounted penalty function, Inuance: Mathematic and Economic, 33: Mitic, I-R, KP Sendova, and CC-L Tai (21) On a Multi-Thehold Compound Poion Poce Petubed by Diffuion, Statitic and Pobability Lette, 8: Petovki, IG (1966) Odinay Diffeential Equation, New Yok: Dove Publication Polyanin, AD and VF Zaitev (23) Handbook of Exact Solution fo Odinay Diffeential Equation (2nd), Boca-Raton: Chapman and Hall/CRC Spiegel, MR (1965) Theoy and Poblem of Laplace Tanfom, New Yok: Schaum Publihing Yang, H, Z Zhang, and C Lan (28) On the Time Value of Abolute Ruin fo a Multi-Laye Compound Poion Model unde Inteet Foce, Statitic and Pobability Lette, 78:

20 Repoduced with pemiion of the copyight owne Futhe epoduction pohibited without pemiion