On the Total Duration of Negative Surplus of a Risk Process with Two-step Premium Function

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Aailable at http://pame/pages/398asp ISSN: 93-9466 Vol, Isse (December 7), pp 7 (Preiosly, Vol, No ) Applications an Applie Mathematics (AAM): An International Jornal Abstract On the Total Dration of

1 Aailable at ISSN: Vol, Isse (December 7), pp 7 (Preiosly, Vol, No ) Applications an Applie Mathematics (AAM): An International Jornal Abstract On the Total Dration of Negatie Srpls of a Risk Process with Two-step Premim Fnction Palina Joranoa Faclty of Mathematics an Informatics, Shmen Uniersity 5 Uniersitetska St Shmen 97, Blgaria palina_kj@abbg Receie Jly 4, 7; accepte September 6, 7 We consier a risk resere process whose premim rate reces from c to c when the resere comes aboe some critical ale In the moel of Cramer-Lnberg with initial capital, we obtain the probability that rin oes not occr before the first p-crossing of leel When <, following H Gerber an E Shi (997), we erie the probability that starting with initial capital rin occrs an the seerity of rin is not bigger than Frther we express the probability of rin in the two step premim fnction moel - (,), by the last two probabilities Or assmptions imply that the srpls process will go to infinity almost srely This entails that the process will stay below zero only temporarily We erie the istribtion of the total ration of negatie srpls an obtain its Laplace transform an mean ale As a conseqence of these reslts, ner certain conitions in the Moel of Cramer-Lnberg we obtain the expecte ale of the seerity of rin In the en of the paper we gie examples with exponential claim sizes Keywors: Srpls process; Probability of rin; Total ration of negatie srpls AMS Sbject Classification No: 9B3; 6G99 Introction In this paper Y We enote by Y, Y, the claim amonts an sppose that they are positie, inepenent an ientically istribte (ii) ranom ariables (rs) with istribtion fnction (f) F which is absoltely continos, an with finite mean EY µ < We assme that the times between claim arrials X, X, are inepenent an exponentially istribte rs with parameter λ > an the seqences {Y i : i N} an {X i : i N} are inepenent 7

2 AAM: Intern J, Vol, No (December 7) [Preiosly, Vol, No ] 8 In the classical risk theory sally the time of rin, the probability of rin, an the eficit at rin are examine A goo treatment in this area is Granel (99) In the Cramer-Lnberg moel the premims are receie at a constant rate c > per nit time In this case the ration of negatie srpls hae been inestigate in Reis (993) an in Dickson an Reis (996) The joint istribtion of the time of rin, the srpls immeiately before rin, an the eficit at rin is examine by Gerber an Shi (997) The istribtion of the time of rin τ () for the cases when the initial capital, in terms of the f of the aggregate claim amont is expresse by the following Seal s formlae (see eg Rolski (999), Theorem 56): ct N ( t) P(τ () > t) ct P( Yi y) y; () i an if > then N ( t) N ( t) Yi i P(τ () > t) P( Yi + ct) - c P(τ () > t - ) P y y i c Here we enote by N (t) the nmber of claim arrials p to time t E Kolkoska, et al (5) proe the existence of local time of renewal risk process with continos claim istribtion They obtain the Laplace fnctional of the occpation measre of the risk process We consier a particlar case of a collectie risk moel with risk epenent premims The risk process is efine by t N ( t) R (t) + c (R (s)) s - Y i, t, i where c, y c( y) c, y > an is a fixe, nonnegatie real nmber T(,) τ() t Fig Case <

3 9 Palina Joranoa Sch a moel is calle two-step premim fnction moel (see eg Asmssen (996)) For examples of sample paths see Figre an Figre τ() T(, ) Fig Case Withot loss of generality we assme that min(c, c ) c Or first step will be to reqire the basic net profit conition EY min(c, c ) > EX This leas to (, ) < for all <, where (, ) is the probability of rin in the twostep premim fnction moel with initial capital Or assmptions imply also that P( R(t) for t ) The latter means that the ration of a single perio of negatie srpls an the nmber of own-crosses of leel zero by the srpls process are almost srely finite Preliminary Reslts In this section we consier the Cramer-Lnberg moel with initial capital an premim EY income rate c > EX We enote by y F I (y) µ ( - F(x)) x the integrate tail istribtion of F, by () the probability of rin an by () the srial probability The following three formlae can be fon in any basic textbook on Risk theory λ () -, () cµ () () + ( - y) y (( - ()) F I (y)), () () () + () ( - ()) k *k F I () (3) k Let q (, ) be the probability that rin oes not occr before the first p-crossing of leel

4 AAM: Intern J, Vol, No (December 7) [Preiosly, Vol, No ] In this section we assme that < Becase of () - q (, ) + () q (, ) (see eg Asmssen (996)) we hae q (, ) q (, ) (), (4) ( ) ( ) q (, ) (5) ( ) q (, ) In the following lemma we erie an integral eqation an explicit formla for q (, ) Lemma : q (, ) q (, ) + Proof: + q (, ) + k k ( ( q ( y, ) y (( - ()) F I (y)), (6) ()) ()) k k F F * k I * k I ( ) (7) ( ) We iie () to (), then we se (5) an obtain (6) Now we iie (3) to (), then we se (4) an obtain q (, ) + k ( ()) k F * k I If we gather (8) an (5) we come to (7) (8) ( ) We enote by G (, ) the probability that starting with initial capital rin occrs an the seerity of rin is not bigger than Then, becase F is absoltely continos G (, ), for all Let s remin that G (, ) () F I () (9) See eg N Bowers et al (987) or the key formla in Gerber an Shi (997) They also obtain that starting with initial capital zero, this is the probability that rin occrs an the srpls immeiately before rin is less than It is well known that it coincies with the n efectie f of the first ascening laer height of the ranom walk S n (Y k - cx k ) The sm in the enominator of (7) is jst the renewal fnction k* ( G (, )) As is k known it eqals the expecte ale of the nmber of ascening laer heights of the corresponing terminating renewal process before the first p-crossing of leel It is also the mean nmber of minimms of the risk resere process with initial capital before its first own-crossing of leel - k

5 Palina Joranoa Haing in min the paper of Gerber an Shi (997) we obtain G (, ) Lemma : For Proof: G (, ) { () - ( + ) () + ( + - z) z G (, z) - ()G (,)} For by () an (9) the lemma is obiosly tre We will consier the case when > By R (, t) we enote the risk resere p to time t an by τ () the time of rin By (46) in Gerber an Shi (997) for the fnction Γ(, x) which is efine as the soltion of the eqation Γ(, x) Γ( - z) z G (, z) + I { < x} (x), where, x A I A (x) elsewhere we hae P R (, τ ( ) ), R (, τ ( )) (x, y; τ () < ) P (x, y; τ R, τ () ), R (, τ ()) () < )Γ(, x) ( By (3) in Gerber an Shi (997) (x, y; τ () < ) µ () P Y (x + y) P R (, τ () ), R (, τ ( )) This means that P R (, τ ( ) ), R (, τ ( )) (x, y; τ () < ) µ () P Y (x + y) Γ(, x) () By Dickson s formlae (See Dickson (99)) an (5) in Gerber an Shi (997) we hae Γ(, x) () ( ), x > ( x) ( ), < x We sbstitte this expression in the eqation () So we obtain for x > P (x, y; τ R (, τ ( ) () < ) µ ()( ( )) P ), R (, τ ( )) Y () (x + y) () an for < x P (x, y; τ R (, τ ( ) () < ) µ ()( ( x) ( )) P ), R (, τ ( )) Y () (x + y) () Now we are reay to obtain G (, ) It is tre, that G (, ) P(τ () <, -R (,τ ()) ) P R (, τ ( )) (y; τ () < ) y

6 AAM: Intern J, Vol, No (December 7) [Preiosly, Vol, No ] Integrating the joint ensity fnction with respect to x we obtain the ensity of the seerity of rin, so G (, ) P R (, τ ( ) ), R (, τ ( )) (x, y; τ () < ) x y By changing the orer of integration an applying () an () we can say that µ () G (, ) () - µ () () ( ( - x) - ()) F (x) x + ( ( - x) - ()) F (x + ) x - ( - x) F (x) x ( + - x) F (x) x + () { () () ( - ()) F (x) x ( - ()) F (x+ ) x ( ) FI ( ) ( ( ))( FI ( )) + µ µ F (x) x+( - ()) + F (x) x ( - x) x F I (x) + () ( - F I ()) - () () + - ( + - x) x ( () F I (x)) + () () ( - F I ()) - () ( - F I ( + ))} Becase of (9) an () G (, ) () So we complete the proof { () - ( + ) + ( + - z) z G (, z) - ()G (,)} 3 The Total Dration of Negatie Srpls In the classical Cramer-Lnberg moel with initial capital, Reis (993) notes that, gien that rin occrs, the ration of the first single perio of a negatie srpls (we enote it by η, ()) coincies in istribtion with the time of rin It is also ientically istribte with the other perios of negatie srplses, conitionally that they are positie This istribtion col be fon by Seal s formla () For t ct ( ) P(τ () t τ () < ) { ( ) } () N t P Y i y y ct In the moel of Cramer-Lnberg we efine T (, ) to be the time of the first p-crossing of leel an as before, G (, y) to be the probability that starting with initial capital rin occrs an the seerity of rin is not bigger than y By Dickson an Reis (996), for arbitrary initial capital > ct P(η, () t η, () > ) ( (, ) ) (, ), ( ) P T y t yg y t i

7 3 Palina Joranoa In the two-step premim fnction moel with initial capital the single sojorn time of the risk resere process in (-, ] col be consiere like the single sojorn time in (-, ] of the risk resere process in the moel of Cramer-Lnberg with premim income rate c For all k,,, gien that the ration of k-th negatie srpls is positie it has the same istribtion whether we consier the moel of Cramer-Lnberg with premim income rate c or the two step premim fnction moel Frther instea of lower inex we will write when the iscsse qantity is eqal to the certain qantity in the moel of Cramer-Lnberg with premim income rate c Analogosly we se lower inex Let η k (), k,, be the ration of k-th negatie srpls in the two-step premim fnctions moel with initial capital Gien that η (), η (), are positie they are inepenent an η (), η 3 (), are ientically istribte P(η () x η () > ) P (η () x η () > ) (3) for x R When > P(η () x η () > ) P (η () x η () > ) (3) for x R When k, 3, an P(η k () x η k () > ) P(η () x η () > ), x R (33) We assme that η The total ration of negatie srpls η(, ) in the two-step premim fnctions moel can be presente as ranom sm N (,,) η(, ) η i (), i where N(,, ) is the nmber of occasions on which the srpls process falls below zero Let s note that N(,, ) an η (), η (), are not inepenent To come to the istribtion of η(, ) we hae to etermine the istribtion of N(,, ) It is not ifficlt to obtain that P (N(,, ) ) - (, ), (34) an for k,, P (N(,, ) k) (, ) k- (, )( - (, )) (35) In the following theorem we express (, ) in ifferent cases for an Theorem : For the two-step premim fnction moel with net-profit conition () a) If, then (,) () ( ) () ; (36) ( ) () () ( ) + () ()

8 AAM: Intern J, Vol, No (December 7) [Preiosly, Vol, No ] 4 b) If <, then (,) - () ( ) () ; (37) ( ) () () ( ) + () () c) If >, then (,) ( - ) - () () ( y) G (, y) (38) ( ) () () ( ) + () () y Here () is the probability of rin in the Cramer-Lnberg moel with initial capital an premim income rate per nit time c, G (, ) is eqal to G (, ) an () - () () is the srial probability in the moel of Cramer-Lnberg with initial capital an premim income rate c Proof: We enote by R (x, t) the risk resere p to time t in the moel of Cramer-Lnberg with initial capital x an premim income rate c > We efine τ (x) to be the time of rin, T (x, ) - the time of the first p-crossing of leel an q (x, ) to be the probability that rin oes not occr before the first p-crossing of leel Analogosly we enote by R (x, t) the risk resere p to time t in the moel of Cramer- Lnberg with initial capital x an premim income rate c > τ (x) is the time of rin in this moel, an (x) is the probability of rin Let T (x, ) be the time of the first pcrossing of leel In the two step premim fnction moel with initial capital an critical leel x we efine θ (, x, t) to be the nmber of own-crossings of leel p to time t We consier the following three grops of eents: A(, ) starting with initial capital there is no rin before the first p-crossing of leel, B(, ) τ() <, θ (,, τ()) an the time of the own-crossing of the leel coincies with the time of rin an C(, ) τ() <, θ (,, τ()) an the time of the own-crossing of the leel oes not coincie with the time of rin a) At this point we sppose that the risk resere process starts with initial capital By the Theorem of Total Probability we hae (, ) i P (τ () <, θ (,, τ ()) k) (39) To fin P (B(, )) we consier only the risk-resere process R (, t) It is not ifficlt to realize, that P(B(, )) P(τ () <, -R (,τ ()) > ) () - P(τ () <, -R (,τ ()) ) By (9)

9 5 Palina Joranoa P(B(, )) () - ()F I () ()( - F I ()) () F () (3) By the Theorem of Total Probability an (9) we obtain P(C(, )) P(T ( y, ) >τ ( - y)) y P(-R (,τ ()) y,τ () < ) () ( q ( y, )) y F I (y) In iew of (5), a formla eqialent to the aboe one is ( y) P(C(, )) ()F I () - () y F I (y) ( ) I Now we se () an obtain P(C(, )) ()F I () - () ( ) () ( ) () By istingishing whether or not rin occrs at the first time when the srpls falls below the initial capital, the law of total probability yiels P(τ () <, θ (,,τ ()) ) P(B(, )) + P(C(, )) ( ) () () () ( ) () - () ( ) () ( ) () Let s fin P(τ () <, θ (,,τ ()) ) Then P(A(, )) P(T ( y, ) τ ( - y)) y P(-R (,τ ()) y,τ () < ) Analogosly to P(C(, )) we obtain ( ) () P(A(, )) () q ( y, )) y F I (y) () ( ) () By istingishing whether or not rin occrs at the secon time when the srpls falls below the initial capital an applying the law of total probability, we obtain P(τ () <, θ (,,τ ()) ) P(A(, ))(P(B(, )) + P(C(, ))) Analogosly P(τ () <, θ (,,τ ()) k) P k - (A(, ))(P(B(, )) + P(C(, ))) Finally applying (39) we can calclate (, ) P k - (A(, ))(P(B(, )) + P(C(, ))) k P(B(, )) + P(C(, )) () + P(A(, )) - P(A(, )) - P(A(, )) () ( ) () - ( ) () () ( ) + () ()

10 AAM: Intern J, Vol, No (December 7) [Preiosly, Vol, No ] 6 ( ) () ( ) () () () ( ) + () () b) Analogosly to Asmssen (996) we obtain (, ) - q (, ) + (, ) q (, ) Now, we replace (, ) with the expression in a) an q (, ) by (5) an come to (37) c) As aboe, by istingishing whether or not rin occrs at the first time, when the srpls falls below the critical leel, or there is no rin before the first p-crossing of leel an applying the law of total probability, we obtain (, ) P(B(, )) + P(C(, )) + P(A(, )) (, ) (3) We fon (, ) in a) To fin P(B(, )) we consier an axiliary risk-resere process R with initial capital - an premim income rate c P(B(, )) P(τ ( - ) <, -R ( -, τ ( - )) > ) ( - ) - P(τ ( - ) <, -R ( -, τ ( - )) ) ( - ) - G ( -, ) By Total probability, (5) an () P(C(, )) ( q ( y, )) y G (, y) ( y) G (, ) - y G (, y), ( ) an ( y) P(A(, )) q ( y, ) y G (, y) y G (, y) ( ) Sbstitting these expressions in (3) an sing (5), we obtain (, ) ( - ) P (A(, )) (, ) ( - ) - () () ( y) yg (, y) ( ) () () ( ) + () () Note: As a conseqence we obtaine the obios reslt that for, it is tre that (, ) () It is interesting to note, that if then (, ) (, ) (, ) (, ) ( ) ( ) () ( ) ( )( )

11 7 Palina Joranoa 3 If we compare the two - step premim fnction moel with initial capital an critical leel an the Moel of Cramer -Lnberg with initial capital - an premim income rate c it is interesting to mention that the following relation hols ( - ) P(A(, )) + P(B(, )) + P(C(, )) Now we are reay to obtain or main reslt Theorem : For the two-step premim fnction moel with net-profit conition (), initial capital an critical leel i) for x, P(η (, ) x) an for x, P(η (, ) < x) ( - (, )) + (, )( - (, )) K *K * (k - ) (x)( (, )) k -, (3) i where K (x) P(η () x η () > ) an K (x) P(η () x η () > ) are etermine by (3), (3) an (33) ii) for x > y f ( x) (, ) e yg (, y) Ee -xη(, ) ( ) ( - (, )) ( + ), Y f ( x) µ ( Ee ) (, ) f ( x) where the fnction f(s) satisfies the eqation s c f (s) + λ (E e -Y f (s) - ), s < iii) If DY <, then Proof: Eη (, ) (, ) c () ( ) x µ E( Y ) (, ) (, ) + xg x (, ) i) By total probability (34), (35) an P(η(, ) < x) P( we hae (3) N (,,) i η () < x), i ii) By (36) in Dickson an Reis (996) for all > an for all x > E ( η () > ) ( ) e x ( ) e η y f ( x) G (, y), where the fnction f(s) satisfies the eqation s c f (s) + λ (E e -Y f (s) - ), for s < y By Reis (993) we hae for k,3, E ( xη η k () > ) E ( () e η () > ) x k () e η where f(s) is the same fnction as aboe Y f ( x) µ ( Ee ), f ( x)

12 AAM: Intern J, Vol, No (December 7) [Preiosly, Vol, No ] 8 These formlae an the law of the total expectation yiel N (,,) ηi ( ) Ee -xη(, ) E i e - (, ) + (, )( - (, ))E( x ( ) e η η () > ) k xη ( ) (, ) Ee ( ( ) ) xη () Ee yf( x) y ( ) Yf ( x) k - xη (, )( E( () e η () > )) k η > ( - (, )) + (, ) ( η () > ) (, ) e G ( y, ) ( - (, )) + µ ( Ee ) (, ) f( x) iii) By Reis (993), we hae for all > E (η () η () > ) c () ( ) x x G (, x) If the ariance of Y is finite, by Reis (993) we hae for k,3, µ E( Y ) E (η k () η k () > ) () By these formlae an the law of the total expectation we obtain Eη(, ) E N (,,) i η () i c i ( i k E (η k () η k () > )) (, )( - (, )) i - (, ) ( i ) µ E( Y ) i (, )( - (, )) ( x xg (, x) ) (, ) c () ( ) + c () i (, )( - (, )) xg x ( x, ) ( (, )) c() ( ) µ EY ( ) µ EY ( ) + c()( (, )) c()( (, )) (, ) µ E( Y ) µ E( Y + x xg (, x) c () ( ) ( (, )) (, ) µ E( Y ) (, ) + x xg (, x) c () ( ) (, ) )

13 9 Palina Joranoa 4 A Nmerical Example Let P(Y < x) e x, x >, x µ As it is known, in this case the integrate tail istribtion of F is also exponential with the cµ same parameter µ, - an λ () - µ + e + Accoring to the aboe lemmas q (, ) an + e + e q (, ) µ + µ + µ + + e Becase of () an by (9) + µ e G (, ) + When we sbstitte these expressions in Lemma we obtain µ + e µ G (, ) ( e ) + So we obtain the well known reslt that in this case, G (, ) () F I () (4) Let s remin, that (4) is not correct for any claim size istribtion fnction F In general case, as is shown in Bowers et al (987) or in Gerber an Shi (997), (4) is correct only for When we apply Theorem for the exponential claim sizes, we obtain a) If then ( ) ( )( ) (, ), ( )( ) c µ where c -, µ - an λ λ

14 AAM: Intern J, Vol, No (December 7) [Preiosly, Vol, No ] () µ + e + ; b) If > (, ) ( ) () ( ) ( )( () ()) + () () µ ( ) + e µ ( ) + e ( ) ( )( ) (, ) It is interesting to note that when the claim sizes are exponentially istribte an > (, ) ( ) (, ) ( ) Let λ an µ 4 By net profit conition () min(c, c ) shol be greater than 5 Table an Table show ales of (, ) an Eη(, ) for ifferent cases of c, c, an Table Vales of (, ) an Eη(, ) for c 6 an c 3; 35 or 4 Table Vales of (, ) an Eη(, ) for c 3 an c 3; 4 or 5 c c (, ) Eη (, ) c c (, ) Eη (, ) Let s remin, that when c c or moel coincies with the Moel of Cramer-Lnberg In this case, has no effect neither on the probability of rin nor on the expecte ale of the total ration of negatie srpls

15 Palina Joranoa When we compare rows 4, 7, 6 an 7 from Table crresponingly with rows 4, 7, an 3 from Table we can see that in the two-step premim fnction moel, when the critical leel is large enogh, the calclate ales o not epen on c The reason for this is that in this case the probability of the the eent the risk resere process will rich this critical leel before the last own-crossing of zero leel becames as smaller as ecreases Acknowlegement: The paper is partially spporte by NSFI-Blgaria, Grant No VU-MI-5/5 I wish to thank to anonymos referees for their remarks REFERENCES Asmssen, S (996) Rin Probabilities, Worl Scientific, Singapore Bowers, NI, HU Gerber, CJ Hickman, DA Jones an CJ Nesbitt, (987) Actarial Mathematics, Society of Actaries, Itaca, IL Dickson, DCM (99) On the Distribtion of the Srpls Prior to Rin, Insrance: Mathematics an Economics,, pp9 7 Dickson DCM an A D Egiio os Reis, (996) On the Distribtion of the Dration of Negatie Srpls, Scan Actarial Jornal,, pp48 64 Egiio os Reis, A D (993) How Long is the Srpls Below Zero?, Insrance: Mathematics an Economics,, pp3 38 Gerber, HU an E S W Shi, (997) The Joint Distribtion of the Time of Rin, the Srpls Immeiately Before Rin, an the Deficit at Rin, Insrance: Mathematics an Economics,, pp 9 37 Granel, J (99) Aspects of Risk Theory, Springer, Berlin time of classical risk process, Insrance: Mathematics an Economics, 37, pp Kolkoska, E, JO Lopes-Mimbela an JV Morales, (5) Occpation measre an local Rolski, T, H Schmili, V Schmit an J Tegels, (999) Stochastic Processes for Insrance an Finance, John Wiley, Chichester

n s n Z 0 on complex-valued functions on the circle. Then sgn n + 1 ) n + 1 2) s

n s n Z 0 on complex-valued functions on the circle. Then sgn n + 1 ) n + 1 2) s . What is the eta invariant? The eta invariant was introce in the famos paper of Atiyah, Patoi, an Singer see [], in orer to proce an inex theorem for manifols with bonary. The eta invariant of a linear