带扩散风险模型破产概率的拉普拉斯变换方法_英文_.pdf

A LAPLACE TRANSFORMAT I ON M ETHOD FOR EVALUAT I ON OF THE RU I N PROBAB I L ITY FOR R ISK MODEL W ITH D IFFU SI ON L iu Taisheng College of M athem atics N ankaiU niversity T ianjin 300071 A bstract The present paper dealsw ith the classical risk model which is perturbed by diffusion and which is actually a SPLP ie spectrally L vy process W ith some well2known results w ith respect to SPLP the Laplace tran2 sformation of7 u the ruin function is given nevertheless when the clai m distributions are exponential and m ixture exponential the precise expression of ruin function is also given Key words risk model SPLP Laplace transformation Poisson process ruin function partial fraction principle CLC number F224 9 0 Some basic results Def in ition 1 L et 8 1 be a probability space Then a t 2adapted process Xt t 0 X0 0 a s is called aL vy process if 1 Xt t 0has stationary increments 2 Xt t 0has independent increments 3 Xt t 0is stochastical continuous Lemma 1 A ny L vy process has a version which is right continuous and has left li m it is also a L vy process Def in ition 2 A L vy process is spectrally positive denoted by SPLP if it has only positive jumps or if the L vy measure is concentrated on R Lemma 2 Suppose Yt t 0is a SPLP on 8 1 and Yt is not a subordinator 0 is the exponent ofYt i eEx Yt e t see reference 1 0 and is the largest non2negative root of 0 M supt 0Yt qx Pr M x Then if we have 0 e xq xdx 1 1 1 Remark 1 The proof of Lemma 2 is very sophisticated it can be found in reference 1 1 TheM odel Let U t u ct N t k 1Z k W t whereuis the initial surplus c 0 is the safety loading c 0 is the rate at which the prem ium s are received N t k 1 Zkrepresents the aggregate clai m s between ti mes 0 andt N t is a Poission processw ith pa2 rameter is a positive constant W t is a standard Brownian motion It is easy to see thatYt N t k 1 Zk ct W t has stationary independent increments and only positive 第33卷 第4期南 开 大 学 学 报 自然科学 Vol 33 4 2000年12月A cta Scientiarum N aturalium U niversitatis N ankaiensisDec 2000 Received 1999206207 jump so it is a SPLP Ee Yt Ee N t k 1 Zk ct W t e t where c 2 2 2 1 0 e xp x d x c 2 0 xe xp x d x 2 0 x 2e xp x d x In which 0 0 c 0 EZk so 0 is a convex function and 0 on 0 so 0 has an unique non2negative root and 0 In addition we supposeEZk and V ar Zk 0 when 0 7 n i 1 1 i n k 17i k 1 i 0 has only negative roots so the above inverseLaplace transformation is correct N evertheless we can prove that it hasndifferent real roots References 1 Doney R A H itting probability for spectrally positive L vy process J L ondonM ath Soc 1991 44 2 566 576 2 Dufrense F Gerber H U The surpluses i mmediately before and at ruin Insurance M athem atics and Econom ics 1989 7 193 199 3 Feller A n I