HSC Research Report. Asymptotic behavior of the finite time ruin probability of a gamma Lévy process HSC/07/01. Zbigniew Michna* Aleksander Weron**

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1 HSC/7/ HSC Research Report Asymptotic ehavior of the finite time rin proaility of a gamma Lévy process Zigniew Michna* Aleksander Weron** * Department of Mathematics, Wrocław University of Economics, Poland ** Hgo Steinhas Center, Wrocław University of Technology, Poland Hgo Steinhas Center Wrocław University of Technology Wy. Wyspiańskiego 27, 5-37 Wrocław, Poland

2 o. l V ) A CT A CA H YP IS O P L O CA N I B o N 5 ASYMPTOTIC BEHAVIOR OF THE FINITE TIME RUIN PROBABILITY OF A GAMMA LÉVY PROCESS Zigniew Michna Department of Mathematics, Wrocław University of Economics Komandorska 8/2, Wrocław, Poland Aleksander Weron Hgo Steinhas Center, Institte of Mathematics and Compter Science Wrocław University of Technology Wyrzeże Wyspiańskiego 27, 5-37 Wrocław, Poland Received Octoer 25, 26) In this paper we consider a jmp-diffsion type approximation of the classical risk process y a gamma Lévy process. We derive here the asymptotic ehavior lower and pper onds) of the finite time rin proaility for any gamma Lévy process. PACS nmers: 5.4. a, 2.5.Ey, 5.2. y, a. Introdction In examining the natre of the risk associated with a portfolio of siness in econophysics [], it is often of interest to assess how the portfolio may e expected to perform over an extended period of time. One approach concerns the se of rin theory [2, 3]. Rin theory is concerned with the excess of the income with respect to a portfolio of siness) over the otgo, or claims paid. This qantity, referred to as insrer s srpls, varies in time. Specifically, rin is said to occr if the insrer s srpls reaches a specified lower ond, e.g. mins of the initial capital. One measre of risk is the proaility of sch an event, clearly reflecting the volatility inherent in the siness. In addition, it can serve as a sefl tool in long range planning for the se of insrer s fnds. The recent increasing interplay etween actarial and financial mathematics has led to srge of risk theoretic modelling, [4]. Presented at the XIX Marian Smolchowski Symposim on Statistical Physics, Kraków, Poland, May 4 7, )

3 882 Z. Michna, A. Weron Unfortnately, the rin proailities in infinite and finite time can only e calclated for a few special cases of the claim amont distrition. Ths, finding a reliale approximation, especially in the ltimate case when the straightforward Monte Carlo approach can not e tilized, is really important from a practical point of view, see [5]. Here we se another jmpdiffsion type) approximation of the classical risk process y a gamma Lévy process, [6] and [7] which permits to find asymptotic ehavior of the finite time rin proaility. We give the exact forms of the constants C,C 2 and the fnction g where C lim inf IPsp t T Zt) ct) > )/g) lim sp IPsp t T Zt) ct) > )/g) C 2 for any T > and c >. The finite time rin proaility is an important qantity in risk theory. Compting asymptotic, onds and exact forms of rin proaility is the key task of risk theory see e.g. [8] and [9]). Rin proaility of Brownian motion, stale processes, compond Poisson processes and Lévy processes is one of the most important prolems of flctations theory in proaility. Let {Zt) : t [,]} e a gamma Lévy process that is a stochastic process starting from with stationary, independent increments and Z) having gamma distrition with shape parameter a > and scale parameter >. Precisely, random variale Z) has the following density distrition fnction { if y, fy) a Γa) ya exp y ) if y >. ) Shortly, we say that Z is a gamma Lévy process with shape parameter a and scale parameter. The aim of the paper is to find an asymptotic ehavior of the following proaility IPsp Zt) ct) > ), 2) for any c > and. In or considerations a certain series representation will e crcial. For example, a reslt from [] showing that a gamma random variale can e otained as a shot noise variale, see also [], gives the following representation Zt) k exp Γ ) k V k I{U k t}, 3) a where t and {Γ k } k is a seqence of arrival epochs in a Poisson process with nit arrival rate, {V k } k is a seqence of iid standard with parameter eqal ) exponential random variales and {U k } k is a seqence of iid random variales niformly distrited on [,]. These seqences are independent.

4 Asymptotic Behavior of the Finite Time Rin Proaility of Or comptation will rely on conditioning on Γ. It is easy to conclde that a gamma Lévy process Z nder condition Γ x, where x > Γ has exponential distrition with parameter eqal ) can e expressed as follows Z x t) A x t) + Y x t), 4) where A x t) e x/a V I{U t}, Y x t) d e x/a k exp Γ k/a)v k I{U k t} is a gamma Lévy process with shape parameter a, scale parameter e x/a and the processes A x and Y x are independent. We write that g) h) for if lim [g)/h)]. We will need the following property of the incomplete gamma fnction s p e s ds p e + O )), 5) for where p IR which implies that sp e s ds p e. 2. Main reslt We derive here the asymptotic properties lower and pper onds) of the finite time rin proaility of a gamma Lévy process. Proposition 2. Let Z e a gamma Lévy process with shape parameter a and scale parameter. Then for any c > where and C lim inf lim sp IPsp Zt) ct) > ) g) IPsp Zt) ct) > ) g) 6) C 2, 7) g) a exp /), 8) C exp c/) a Γa), 9) C 2 exp c/) c a 2 Γa). ) Let s note that IPsp Zt) ct) > ) IPsp Z x t) ct) > )e x dx. )

5 884 Z. Michna, A. Weron First we derive the pper ond. Since the process Y x has non-decreasing trajectories we get the following pper ond IPsp Z x t) ct) > ) IPsp A x t) ct + Y x t)) > ) IPsp A x t) ct) + Y x ) > ). 2) The density distrition of the random variale Y x ) is the following { if y, f x y) e x a Γa) ya exp y ex/a) if y >. Ths sing independence A x and Y x 2) can e compted as follows IPsp A x t) ct)+y x ) > ) + 3) IPsp A x t) ct) > y)f x y)dy IPsp A x t) ct) > y)f x y)dy f x y)dy. 4) First let s consider the integrand fnction in 4) for y < IPsp A x t) ct) > y) IPe x/a V cu > y) IPe x/a V cs > y)ds IPV > + cs y e x/a )ds exp + cs y ) e x/a ds exp y ) e x/a c e x/a exp exp c ) ex/a s ds y ) [ e x/a exp c ex/a)].

6 Asymptotic Behavior of the Finite Time Rin Proaility of Now we are in a position to calclate 4) IPsp A x t) ct) > y)f x y)dy c e x/a exp y ) e x/a [ exp c ex/a)] e x a Γa) ya exp y ex/a) dy e x c a Γa) e x/a exp [ ex/a) exp c ex/a)] e x a ac a Γa) e x/a exp ex/a)[ exp c ex/a)]. y a dy Ths y ) we shold integrate 4) with exponential density and sing the last calclations we get IPsp A x t) ct) > y)f x y)dy e x dx a ac a Γa) sstitting s /)e x/a we contine a+ c a Γa) / e x/a exp ex/a)[ exp c ex/a)] dx, s 2 e s ds / s 2 e s+c/) ds, sstitting w s + c/) in the second integral we get a+ c a Γa) sing 5) we otain / exp c/) c a 2 Γa) s 2 e s ds + c ) +c)/ w 2 e w dw, a exp /) + O/)). 5)

7 886 Z. Michna, A. Weron Now let s integrate 4) f x y)dy e x dx dy a Γa) f x y)e x dx dy y a exp y ex/a) dx, sstitting s y/) e x/a we proceed a a Γa) a a Γa) a Γa) Γa) / / / dy y a y/ ds s e s s e s ds s y a dy s e s a s a a )ds a s a e s ds a Γa) / s e s ds a Γa) a e / + O/)) a Γa) a e / + O/)) a Γa) a exp /) O/), 6) where in the second last eqality we se 5). Comining 5) and 6) we get 7). Now we consider the following lower ond for the finite time rin proaility IPsp Zt) ct) > ) IPZ) c > ) IPZ) > + c) a Γa) +c y a e y/ dy

8 Asymptotic Behavior of the Finite Time Rin Proaility of Γa) Γa) +c)/ s a e s ds ) + c a e +c)/ exp c/) a Γa) a exp /), where in the third eqality we sstitte s y/ and in the second last we se 5) which gives 6). 3. Conclsions Oserve that if we admit c, then C a Γa) and C 2 and Proposition 2. gives the exact asymptotic proaility ecase IPsp Zt) > ) IPZ) > ) a Γa) Γa) / y a e y/ dy s a e s ds a Γa) a exp /), a Γa) where in the second last eqality we sstitte s y/ and in the last one we se 5). If we consider gamma Lévy process Z on the non-negative half-line that is {Zt) : t [, )} it is easy to conclde the following reslt. Let Z e a gamma Lévy process with shape parameter a and scale parameter. Then for any T > and c > C lim inf lim sp IPsp t T Zt) ct) > ) g) IPsp t T Zt) ct) > ) g) 7) C 2, 8) where g) at exp /), 9)

9 888 Z. Michna, A. Weron and Indeed, since C exp ct/) at ΓaT) 2) C 2 exp ct/) ct at 2 ΓaT). 2) IPsp Zt) ct) > ) IPsp ZTt) ctt) > ) t T and the process Z t) ZTt) is a gamma Lévy process with shape parameter at and scale parameter we need to pt a : at and c : ct in Proposition 2.. Let s notice finally that C and C 2 as c. The plot of the fnctions C C c) and C 2 C 2 c) for a 2, and T is given in Fig.. C, C c Fig.. Plot of the constants C and C 2 as a fnction of c for a 2, and T C thin line, C 2 thick line ).

10 Asymptotic Behavior of the Finite Time Rin Proaility of REFERENCES [] R. Cont, P. Tankov, Financial Modelling with Jmp Processes, Chapman & Hall, CRC Press, London 24. [2] J. Grandell, Aspects of Risk Theory, Springer-Verlag, Berlin 99. [3] J.M.P. Alin, M. Snden, On the Asymptotic Behavior of Lévy Processes. Part I: Sexponential and Exponential Processes, Preprint 26. [4] P. Cizek, W. Härdle, R. Weron, Statistical Tools for Finance and Insrance, Springer-Verlag, Berlin, Heidelerg 25. [5] K. Brnecki, P. Miśta, A. Weron, Acta. Phys. Pol. B 36, ). [6] F. Dfresne, H.U. Gerer, E.S.W. Shi, Astin Blletin 2, 77 99). [7] H. Yang, L. Zhang, Adv. Appl. Proa. 33, 28 2). [8] S. Asmssen, Rin Proailities, World Scientific, Singapore 2. [9] T. Rolski, H. Schmidli, V. Schmidt, J. Tegels, Stochastic Processes for Insrance and Finance, John Wiley and Sons, New York 999. [] L. Bondesson, Adv. Appl. Proa. 4, ). [] J. Rosiński in Lévy processes Theory and Applications, Eds O.E. Barndorff- Nielsen, T. Mikosch, S. Resnick, Birkhaser, Boston 2, p. 4.

11 HSC Research Report Series 27 For a complete list please visit Asymptotic ehavior of the finite time rin proaility of a gamma Lévy process y Zigniew Michna and Aleksander Weron