32 G; F ; (1) {X, X(i), i = 1, 2,..., X, (2) {M(t), t α Poisson, t ; (3) {Y, Y (i), i = 1, 2,..., Y, (4) {N(t), t β Poisson, t ; (5) {W (t), t, σ ; (6

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1 212 2 Chinese Journal of Applied Probability and Statistics Vol.28 No.1 Feb. 212 Poisson ( 1,, 211; 1 2,3 2 2,, 2197) ( 3,, 2197) Poisson,,.,. : :,,,,. O , ( 1 6]). 4] Cai Poisson,, 6] Fang Luo Poisson,., Poisson. M(t) N(t) S t = u + X(i) Y (i) + σw (t) = u + U t H(t) = i=1 i=1 = u + H(t) Z(t), t, S = u, (1.1) M(t) i=1 U t = M(t) i=1 N(t) X(i) i=1 X(i) + σw (t), Z(t) = Y (i) + σw (t), N(t) i=1 Y (i), t, (167132, 18711, )(BK286) (11711) (21JCXM-2-8) ,

2 32 G; F ; (1) {X, X(i), i = 1, 2,..., X, (2) {M(t), t α Poisson, t ; (3) {Y, Y (i), i = 1, 2,..., Y, (4) {N(t), t β Poisson, t ; (5) {W (t), t, σ ; (6) {X(i), i = 1, 2,..., {Y (i), i = 1, 2,..., {M(t), t, {N(t), t {W (t), t. τ (1.1), τ = inf{t : S t < inf φ = ; Φ(u) (1.1) u, Φ(u) = P{τ < S = u = P{S t < t S = u; Ψ(u) (1.1) u, Ψ(u) = 1 Φ(u). Poisson, X t = e δt( u + t ) e δs dus, t, X = u, δ, (1.2) δ, δ. T (1.2), T = inf{t : X(t) < ; ϕ(u) (1.2) u, ϕ(u) = P{T < X = u = P{X t < t X = u, ψ(u) (1.2) u, ψ(u) = 1 ϕ(u).,,,,. T s T s = inf{x t <, X h >, < h < t; T d T d = inf{t : X t =, X h >, < h < t, T = min{t s, T d, ϕ s ϕ d ϕ s (u) = P{T s < X = u, ϕ d (u) = P{T d < X = u. ψ s (u) = 1 ϕ s (u), ψ d (u) = 1 ϕ d (u). ϕ(u) = ϕ d (u) + ϕ s (u), u. (1.3)

3 Poisson 33, ϕ d () = ϕ() = 1, ϕ s () =. (1.4) ψ(u) = 1 ϕ(u) ψ(u) = ψ s (u) + ψ d (u) 1, ψ() =. (1.5), P{T s < T < = ϕ s (u)/ϕ(u); P{T d < T < = ϕ d (u)/ϕ(u)., ϕ(u, t) ϕ(u, t) = P{T < t X = u, ψ(u, t) = 1 ϕ(u, t). ϕ s (u, t) = P{T s < t X = u, ψ s (u, t) = 1 ϕ s (u, t); ϕ d (u, t) = P{T d < t X = u, ψ d (u, t) = 1 ϕ d (u, t), ϕ(u, t) = ϕ d (u, t) + ϕ s (u, t), u. (1.6) 2 ; 3 ϕ s (u) ϕ d (u), ϕ(u) ; 4, ϕ s (u), ϕ d (u) ϕ(u); 5.,, 2. EU t = EM(t)EX(t) EN(t)EY (t) + σew (t) = (αe(x) βe(y ))t > {U t, t. α (, 1/2), α Hölder, T >, ω Ω, K = K(T, ω), s < t T, W (t)(ω) W (s)(ω) K t s α, P(Ω ) = 1. 7]. 2.3 lim S t =, t a.s.. Anscombe ( 8] 2.5.3). 2.4 lim Ψ(u) = 1, a.s.. (2.1) u

4 lim S t =, t, t, U t >, T >, t > T, U t >. T 2.2 inf U t, inf U t >, a.s.. t<t t u, S t, (2.1). : 3.1 a.s.. 3., ψ s (u), u, ψ s (u) (α+β)ψ s (u) uδψ s(u) 1 2 σ2 ψ s (u) = α ψ s (+ ) = 1, ψ s () = 1, α 1 2 σ2 ψ s () = α h(t) = ue δt + σ ψ s (u+x)dg(x)+β t ψ s (x)dg(x). ψ s (u y)df (y). (3.1) (3.2) e δs dw s u. (3.3) (, t], (1.2) X t. M(t) N(t) Poisson, (, t] o(t). (1) M(t) N(t), (1 αt)(1 βt) + o(t). (2) M(t) N(t), αt(1 βt) + o(t). (3) M(t)N(t), (1 αt)βt + o(t). (4) M(t) ( N(t)) M(t) N(t), (3), y > u + h(t), ψ s (u + h(t) y) =, { ψ s (u) = (1 αt)(1 βt)e{ψ s u + h(t)] + αt(1 βt)e { +h(t) + βt(1 αt)e ψ s u + h(t) y]df (y) + o(t). ψ s u + h(t) + x]dg(x)

5 Poisson 35 { (α + β)te{ψ s u + h(t)] = E{ψ s u + h(t)] ψ s (u) + αte ψ s u + h(t) + x]dg(x) Y (t) = h(t) + u, (3.3) dψ s u + h(t)] = dψ s Y (t)] = ψ s u + h(t)] = ψ s Y (t)] { +h(t) + βte ψ s u + h(t) y]df (y) + o(t). (3.4) dy (t) = uδe δt dt + σe δt dw t, Y () = u. = ψ s (u) + + E{ψ s u + h(t)] = ψ s (u) + { uδe δt ψ sy (t)] σ2 e 2δt ψ s Y (t)] dt + σe δt ψ sy (t)]dw t. t t t { uδe δx ψ sy (x)] σ2 e 2δx ψ s Y (x)] dx σe δx ψ sy (x)]dw x. (3.4) t, t, (3.5) (α + β)ψ s (u) = uδψ s(u) σ2 ψ s (u) + α (3.1). ( uδe δx E{ψ sy (x)] + 1 ) 2 σ2 e 2δx E{ψ s Y (x)] dx. (3.5) ψ s (u + x)dg(x) + β ψ s (u y)df (y), ϕ s (u) ϕ(u) Φ(u), Φ(+ ) =, ψ s (+ ) = 1, (3.5) ψ s () = 1; (3.1), u. 3.2 ψ d (u), u, ψ d (u) = α (α + β)ψ d (u) uδψ d (u) 1 2 σ2 ψ d (u) ψ d (u + x)dg(x) + β ψ d (+ ) = 1, ψ s (u y)df (y) + βf (u), (3.6) ψ d () =, 1 2 σ2 ψ d () = α ψ d (x)dg(x). (3.7)

6 36 y) = 1, (3), y > u + h(t), ψ s (u + h(t) (3.6). ψ d (u) = (1 αt)(1 βt)e{ψ d u + h(t)] { + αt(1 βt)e ψ d u + h(t) + x]dg(x) { +h(t) + βt(1 αt)e + βt(1 αt)e{f u + h(t)] + o(t). ψ d u + h(t) y]df (y) ϕ d (u) ϕ(u) Φ(u), Φ(+ ) =, ψ d (+ ) = 1, (1.4) ψ d () = ; (3.6), u. : , u >, ψ(u) (α + β)ψ(u) uδψ (u) 1 2 σ2 ψ (u) = α (1.5), ψ(+ ) = 1, ψ() =, 1 2 σ2 ψ () = α ψ(u + x)dg(x) + β ψ(x)dg(x). ψ(u y)df (y), (3.8) ψ d (u) + ψ s (u) = ψ(u) + 1, ψ d (u) + ψ s(u) = ψ (u), ψ d (u) + ψ s (u) = ψ (u)., (3.1) (3.6), (3.8), (3.2) (3.7), (3.9) (3.9) Gλ 1, F λ 2, 3.1, u >, ψ s (u) (δλ 2 + βλ 1 αλ 2 δλ 1 uδλ 1 λ 2 )ψ s(u) + 2δ (α + β) 1 ] 2 σ2 λ 1 λ 2 uδ(λ 1 λ 2 ) ψ s (u) + uδ 1 ] 2 σ2 (λ 1 λ 2 ) ψ s (u) σ2 ψ s (u) =. (4.1)

7 Poisson 37 ψ s (+ ) = 1, ψ s () = 1, α 1 2 σ2 ψ s () = α F G, (3.1) = α (α + β)ψ s (u) uδψ s(u) 1 2 σ2 ψ s (u) ψ s (u + x)λ 1 e λ 1x dx + β (4.2), x 1 = u + x, y 1 = u y, = α ψ s (x)λ 1 e λ 1x dx. (α + β)ψ s (u) uδψ s(u) 1 2 σ2 ψ s (u) (4.3) u u ψ s (x 1 )λ 1 e λ 1(x 1 u) dx 1 + β ψ s (u y)λ 2 e λ 2y dy. (4.2) ψ s (y 1 )λ 2 e λ 2(u y 1 ) dy 1. (4.3) (α + β)ψ s(u) δψ s(u) uδψ s (u) 1 2 σ2 ψ s (u) + αλ 1 ψ s (u) βλ 2 ψ s (u) = λ 1 α (4.4) u ψ s (u + x)λ 1 e λ 1x dx λ 2 β ψ s (u y)λ 2 e λ 2y dy. (4.4) (α + β)ψ s (u) 2δψ s (u) uδψ s (u) 1 2 σ2 ψ s (u) + αλ 1 ψ s(u) = λ 2 1α βλ 2 ψ s(u) + λ 2 1αψ s (u) + λ 2 2βψ s (u) (4.2) (4.4) (4.5), ψ s (u + x)λ 1 e λ 1x dx λ 2 2β (δλ 2 + βλ 1 αλ 2 δλ 1 uδλ 1 λ 2 )ψ s(u) + 2δ (α + β) 1 ] 2 σ2 λ 1 λ 2 uδ(λ 1 λ 2 ) ψ s (u) + uδ 1 ] 2 σ2 (λ 1 λ 2 ) ψ s (u) σ2 ψ s (u) =. ψ s (u y)λ 2 e λ 2y dy. (4.5) (4.1). (3.2), ψ s (u).

8 u >, ψ d (u) Gλ 1, F λ 2, (δλ 2 + βλ 1 αλ 2 δλ 1 uδλ 1 λ 2 )ψ d (u) + 2δ (α + β) 1 ] 2 σ2 λ 1 λ 2 uδ(λ 1 λ 2 ) ψ d (u) + uδ 1 ] 2 σ2 (λ 1 λ 2 ) ψ d (u) σ2 ψ d (u) =, 4.1. (3.7), ψ d (u). 4.3 ψ d (+ ) = 1, ψ d () =, 1 2 σ2 ψ d () = α ψ d (x)λ 1 e λ1x dx , u >, ψ(u) (δλ 2 + βλ 1 αλ 2 δλ 1 uδλ 1 λ 2 )ψ (u) + 2δ (α + β) 1 ] 2 σ2 λ 1 λ 2 uδ(λ 1 λ 2 ) ψ (u) + uδ 1 ] 2 σ2 (λ 1 λ 2 ) ψ (u) σ2 ψ (u) =, ψ(+ ) = 1, ψ() =, 1 2 σ2 ψ () = α 3.1. ψ(x)λ 1 e λ 1x dx ψ s (u, t) u, t. u, ψ s (u, t) = α (α + β)ψ s (u, t) ψ s(u, t) uδ ψ s(u, t) ψ s (u + x, t)dg(x) + β 1 2 σ2 2 ψ s (u, t) 2 ψ s (u y, t)df (y). (5.1)

9 Poisson 39 ψ s (+, t) = 1, ψ s (u, + ) = ψ s (u). (, ], (1.2) X t. M(t) N(t) Poisson, (, ] (1) M(t) N(t), (1 α )(1 β ) + o( ). (2) M(t) N(t), α (1 β ) + o( ). (3) M(t)N(t), (1 α )β + o( ). (4) M(t) ( N(t)) (, ], M(t) N(t), o( ). (3), y > u + h( ), ψ s (u + h( ) y, t ) =, ψ s (u, t) = (1 α )(1 β )E{ψ s u + h( ), t ] { + α (1 β )E ψ s u + h( ) + x, t ]dg(x) { +h( ) + β (1 α )E (α + β) E{ψ s u + h( ), t ] { = E{ψ s u + h( ), t ] ψ s (u, t) + α E ψ s u + h( ) y, t ]df (y) + o( ). ψ s u + h( ) + x, t ]dg(x) { +h( ) + β E ψ s u + h( ) y, t ]df (y) + o( ). (5.2) Y (t) = h(t) + u, (3.3) dy ( ) = uδe δ d + σe δ dw, Y () = u. = dψ s u + h( ), t ] = dψ s Y ( ), t ] { uδe δ ψ sy ( ), t ] + ψ sy ( ), t ] σ2 e 2δ 2 ψ s Y ( ), t ] 2 d + σe δ ψ sy ( ), t ] dw.

10 4 = ψ s u + h( ), t ] = ψ s Y ( ), t ] { uδe δx ψ sy (x), t x] + ψ sy (x), t x] + ψ s (u, t) + σe δx ψ sy (x), t x] dw x σ2 e 2δx 2 ψ s Y (x), t ] 2 dx E{ψ s u + h( ), t ] ( { = ψ s (u, t) + uδe δx ψs Y (x), t x] { ψs Y (x), t x] E + E + 1 { 2 σ2 e 2δx 2 ψ s Y (x), t x] ) E 2 dx. (5.2),, (α + β)ψ s (u, t) = ψ s(u, t) (5.1). 1 + α t, (5.1) (3.1). + uδ ψ s(u, t) ψ s (u + x, t)dg(x) + β ψ s (u, t) =, t= σ2 2 ψ s (u, t) 2 ψ s (u y, t)df (y). 5.2 ψ d (u, t) u, t. u, ψ d (u, t) = α (α + β)ψ d (u, t) ψ d(u, t) uδ ψ d(u, t) ψ d (u + x, t)dg(x) + β 1 2 σ2 2 ψ d (u, t) 2 βf (u) ψ d (u y, t)df (y). (5.3) ψ d (+, t) = 1, ψ d (u, + ) = ψ d (u).

11 Poisson , (3), y > u+h( ), ψ d (u + h( ) y, t ) = 1, ψ d (u, t) = (1 α )(1 β )E{ψ d u + h( ), t ] { + α (1 β )E ψ d u + h( ) + x, t ]dg(x) (5.3). 2 { +h( ) + β (1 α )E + β (1 α )E{F (u + h( )) + o( ). t, (5.3) (3.6). : 5.1 ψ d (u, t) =, t= ψ d u + h( ) y, t ]df (y) , u >, ψ(u, t) = α (α + β)ψ(u, t) (1.6), ψ(u, t) ψ(u + x, t)dg(x) + β ψ d (u, t) + ψ s (u, t) = ψ(u, t) + 1, ψ d (u, t) + ψ s(u, t) = (5.1) (5.3) (5.4). ψ(u, t) uδ ψ(+, t) = 1, ψ(u, + ) = ψ(u). ψ(u, t), 1 2 σ2 2 ψ(u, t) 2 ψ(u y, t)df (y). (5.4) ψ d (u, t) + ψ s(u, t) ψ(u, t) =, 2 ψ d (u, t) ψ s (u, t) 2 = 2 ψ(u, t) 2. 3 t, (5.4) (3.9). ψ(u, t) =, t=

12 42 1] Asmussen, S., Ruin Probabilities, Singapore, World Scientific, 2. 2] Cai, J. and Dickson, D.C.M., On the expected discounted penalty function at ruin of a surplus process with interest, Insurance: Mathematics and Economics, 3(22), ] Palsen, J. and Gjessing, H.K., Ruin theory with stochastic economic environment, Advances in Applied Probability, 29(1997), ] Cai, J. and Yang, H.L., Ruin in the perturbed compound poisson risk process under interest force, Advances in Applied Probability, 37(3)(25), ] Sundt, B. and Teugels, J.L., Ruin estimates under interest force, Insurance: Mathematics and Economics, 16(1995), ],, Poisson,, 22(2)(26), ] Applebaum, D., Lévy Processes and Stochastic Calculus, England, Cambridge University Press, 24. 8] Paul, E., Claudia, K. and Thomas, M., Modelling Extremal Events for Insurance and Finance, New York: Springer-Verlag, The Survival Probability for the Perturbed Double Compound Poisson Risk Process under Constant Interest Force Wei Guanghua 1 Gao Qibing 2,3 Wang Xiaoqian 2 ( 1 Department of Basic Courses, Jinling Institute of Technology, Nanjing, 211 ) ( 2 School of Mathematics and Computer Science, Nanjing Normal University, Nanjing, 2197 ) ( 3 Department of mathematics, Southeast University, Nanjing, 2197 ) In this paper, we consider the perturbed double compound Poisson risk process under constant interest force. Exponential type upper bounds are obtained for the ultimate ruin probability of this risk model by the way of martingale. For infinite time and finite time survival probabilities, we obtain the respective integro-differential equations. When the premiums are exponentially distributed, some differential equations are derived for infinite time survival probability. Keywords: Double compound Poisson risk process, Brown motion, jump-diffusion process, survival probability, integro-differential equations. AMS Subject Classification: 6G5.

: 29 : n ( ),,. T, T +,. y ij i =, 2,, n, j =, 2,, T, y ij y ij = β + jβ 2 + α i + ɛ ij i =, 2,, n, j =, 2,, T, (.) β, β 2,. jβ 2,. β, β 2, α i i, ɛ i

: 29 : n ( ),,. T, T +,. y ij i =, 2,, n, j =, 2,, T, y ij y ij = β + jβ 2 + α i + ɛ ij i =, 2,, n, j =, 2,, T, (.) β, β 2,. jβ 2,. β, β 2, α i i, ɛ i 2009 6 Chinese Journal of Applied Probability and Statistics Vol.25 No.3 Jun. 2009 (,, 20024;,, 54004).,,., P,. :,,. : O22... (Credibility Theory) 20 20, 80. ( []).,.,,,.,,,,.,. Buhlmann Buhlmann-Straub