0 引言

重大疾病或重大交通事故的发生常常伴随着巨大的财产损失.基于此，保险公司设计了重疾险和第三方保险，承诺在首次出险后一段时间内再次出险仍将进行二次赔付.风险模型中，首次赔付和二次赔付分别称为主索赔和延迟索赔，通常认为其分布是重尾的^[1-2].重尾分布下带常数利息力的风险模型不仅刻画了造成保险公司破产的大额索赔，而且还反映了盈余资本在常数利息下的累积^[3-4].针对此模型，文献[5]研究了索赔额属于L∩D族且负相依下的有限时破产概率.文献[6]在索赔额属于C族且上尾渐近独立情形下得到了破产概率渐近表达式.文献[7]给出了索赔到达间隔由独立变为WLOD(宽下象限相依)情形下的有限时和终极破产概率.文献[8-10]则将保费推广为随机过程，给出了相应的有限时破产概率.文献[11]又将延迟索赔增加到模型中，但保费仍为常数，在索赔到达为Poisson过程条件下，得到了新的破产概率等价式.文献[12]将文献[11]中索赔额由独立变为负相依，索赔到达推广为更新过程，但索赔额分布缩小到L∩D族.本文继续考虑带有延迟索赔的风险模型，将索赔额扩展到广义负相依，又将保费总收入推广为非负非降的随机过程，在索赔到达为更新过程的情形下，得到了有限时破产概率的渐近等价式.文中所建模型包含了保费收入为线性过程且索赔额负相依这一特殊情形，同时也更符合实际.

若无特别说明，本文支撑均定义在[0, ∞)上，且文中极限关系均为对x→∞而言.对于任意两个正函数a(·)和b(·)，约定a(x)~b(x)表示lim (a(x)/b(x))=1，a(x)=o(b(x))表示lim (a(x)/b(x))=0；任意两个正二元函数a(·, ·)和b(·, ·)，约定a(x, t)~b(x, t)表示对任意非空集合Δ，关于t一致有$\mathop {\lim }\limits_{x \to \infty } \mathop {\sup }\limits_{t \in \mathit{\Delta }} \left( {a\left( {x, t} \right)/b\left( {x, t} \right)} \right) = 1 $，同样地，a(x, t) < b(x, t)或b(x, t)>a(x, t)，表示$ \mathop {\lim }\limits_{x \to \infty } \mathop {\sup }\limits_{t \in \mathit{\Delta }} \left( {a\left( {x, t} \right)/b\left( {x, t} \right)} \right) \le 1$成立^[13].

1 模型和引理

模型中，用x≥0表示保险公司的初始资本，δ>0表示常数利息力.假设{θ_i, i≥1}独立同分布，表示主索赔到达间隔序列，则$ {S_k} = \mathop \sum \limits_{i = 1}^k {\theta _i}$为第k次主索赔到达时刻，构成更新计数过程N(t)=sup{k≥1:S_k≤t}, t≥0.风险盈余过程可表示为

$ {U_\delta }\left( t \right) = x{{\rm{e}}^{\delta t}} + \int_0^t {{{\rm{e}}^{\delta \left( {t - s} \right)}}} C{\rm{d}}s - \sum\limits_{i = 1}^{N\left( t \right)} {{X_i}{{\rm{e}}^{\delta \left( {t - {S_i}} \right)}}} \\- \sum\limits_{i = 1}^{N\left( t \right)} {{Y_i}{{\rm{e}}^{\delta \left( {t - {S_i} - {W_i}} \right)}}} {I_{\left\{ {{S_i} + {W_i} \le t} \right\}}}, t \ge 0, $

(1)

其中：{C(t), t≥0}表示到时刻t为止的保费总收入，满足C(0)=0，C(t) < ∞；X_i和Y_i分别表示第i次主索赔和延迟索赔；W_i表示相应的延迟时间间隔；I_A表示集合A的示性函数.

通常定义有限时破产概率为

$ \psi \left( {x, T} \right) = P\left( {\mathop {{\rm{inf}}}\limits_{0 \le t \le T} {U_\delta }\left( t \right) < 0|{U_\delta }\left( 0 \right) = x} \right), T \ge 0. $

(2)

定义1^[5]  设随机变量X，分布为F，若对∀l，有$\mathop {\lim }\limits_{x \to \infty } \left( {\bar F\left( {x + l} \right)/\bar F\left( x \right)} \right) = 1 $，称分布F属于L族，记作F∈L.若对∀n≥2，有$ \mathop {\lim }\limits_{x \to \infty } ({{\bar F}^{n \cdot }}\left( x \right)/\bar F\left( x \right)) = n$，称分布F属于S族，记作F∈S. F^n·为F的n重卷积.若对∀0 < y < 1，有$\mathop {\lim }\limits_{x \to \infty } \sup \left( {\bar F\left( {xy} \right)/\bar F\left( x \right)} \right) < \infty $，称分布F属于D族，记作F∈D.对上述各重尾分布子族，有包含关系L∩D⊂S⊂L成立.

定义2^[10]  随机变量序列{X_i, i≥1}，对每个n=1, 2, …和所有x₁, x₂，…，x_n，若存在M>0，有$P(\mathop \cap \limits_{i = 1}^n \{ {X_i} \le {x_i}\} ) \le M\mathop \prod \limits_{i = 1}^n P({X_i} \le {x_i}) $, 称其为下广义负相依(LEND)；若存在M>0，有$ P(\mathop \cap \limits_{i = 1}^n \{ {X_i} > {x_i}\} ) \le M\mathop \prod \limits_{i = 1}^n P({X_i} > {x_i})$，称其为上广义负相依(UEND)；若两式同时满足，称{X_i, i≥1}是广义负相依(END)的；特别地，当M=1时，随机变量序列{X_i, i≥1}称为负相依(ND)的.

广义负相依不仅包含了负相依，而且还包含了某些正相依序列，具有更广泛的研究价值^[14].

定义3^[13]  对任意分布F，称J_F⁺=inf {-log F_*(y)/log y, y>1}为F的上Matuszewska指数.其中F_*(y)=lim inf (F(xy)/F(x)).

假定模型(1)满足:(A₁)索赔到达过程的均值函数有限，即E[N(t)]=λ_t. (A₂)主索赔额序列{X_i, i≥1}是END，共同分布为F.延迟索赔序列{Y_i, i≥1}是END，共同分布为G. {W_i, i≥1}独立同分布，共同分布为K. (A₃)随机变量序列{X_i, i≥1}，{Y_i, i≥1}，{W_i, i≥1}及{N(t), t≥0}两两相互独立.并记$ {H_i} = {X_i} + {Y_i}{{\rm{e}}^{ - \delta {W_i}}}{I_{\{ {S_i} + {W_i} \le t\} }}, \tilde C\left( t \right) = \smallint _0^t{{\rm{e}}^{ - \delta s}}C{\rm{d}}s$.

引理1^[13]  若F∈D, 则对任意β>J_F⁺，存在正数C和d，有1) (F(y)/F(x))≤C(x/y)^β，x≥y≥d.2) x^-β=o(F(x)).

引理2^[15]  若{X_i, i≥1}是END，{g_k, k≥1}均是非增或均是非降的，则{g_k(x_k), k≥1}仍是END.

引理3^[16]  设非负随机变量X₁和X₂，分布分别为F₁, F₂且均属于L∩D，若对于(i, j)分别为(1, 2)或(2, 1)，关系式$ \mathop {\lim }\limits_{{x_i} \wedge {x_{j \to \infty }}} P({X_i} > {x_i}|{X_j} > {x_j}) = 0$成立，则F₁*F₂∈L∩D，且$ \overline {{F_1}*{F_2}} \sim{{\bar F}_1} + {{\bar F}_2}$.

引理4  在假设(A₂), (A₃)下，若F, G∈L∩D，则{H_i, i≥1}是END，且对∀i≥1有H_i∈L∩D.

证明  因为{X_i, i≥1}和{Y_i, i≥1}均为END，故由文献[17]的引理5可知，{X_i+Y_iI_{Si+Wi≤t}, i≥1}是END.又对∀i≥1, P(0 < e^-δW_i < 1)=1，由引理2可得{Y_ie^-δW_i, i≥1}是END.从而可知{H_i, i≥1}是END.由引理3及文献[18]的定理2.2、定理3.3可得H_i∈L∩D, ∀i≥1.

引理5  设任意非负随机变量Z与{X_i, i≥1}，{Y_i, i≥1}，{W_i, i≥1}，{N(t), t≥0}两两相互独立，在引理4条件下，对任意的T及正整数m₀，有

$ \sum\limits_{k = 1}^{{m_0}} P \left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {S_i}}}} > x + Z, N\left( T \right) = k} \right) \sim \sum\limits_{k = 1}^{{m_0}} {\sum\limits_{i = 1}^k P \left( {{H_i}{{\rm{e}}^{ - \delta {S_i}}} > x, N\left( T \right) = k} \right)} . $

(3)

证明  对k=1, 2, …, m₀，由已知独立性条件有

$ P\left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {S_i}}} > x} + Z, N\left( T \right) = k} \right) =\\ \int\limits_{0 \le {s_1} \le \cdots \le {s_k} \le T, {s_{k + 1}} > T} {\int_0^\infty {P\left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x + z} } \right)} \cdot }\\ P\left( {Z \in {\rm{d}}z} \right)P\left( {{S_1} \in {\rm{d}}{s_1}, \cdots , {S_{k + 1}} \in {\rm{d}}{s_{k + 1}}} \right). $

(4)

取1≤i≠j≤k，由引理4知∃M>0，有P(H_ie^-δs_i>x+z, H_je^-δs_j>x+z)≤MP(H_ie^-δs_i>x+z)P(H_je^-δs_j>x+z)=o(P(H₁e^-δs₁>x)).则

$ P\left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x + z} } \right) \ge \sum\limits_{i = 1}^k {P\left( {{H_i}{{\rm{e}}^{ - \sigma {s_i}}} > x + z} \right)} - \\ M\sum\limits_{1 \le i < j \le k} {P\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x + z} \right)} P\left( {{H_j}{{\rm{e}}^{ - \delta {s_j}}} > x + z} \right) \sim \\ \sum\limits_{i = 1}^k {P\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x + 1} \right)} \sim \sum\limits_{i = 1}^k {P\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x} \right)} . $

(5)

(5) 式中最后一步等价式可由L族定义得到.又对∀L>0，有

$ P\left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {s_i}}}} > x + z} \right) \le P\left( {\bigcup\limits_{i = 1}^k {\left\{ {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x + z - L} \right\}} } \right) +\\ P\left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x + z} , } \right.\\ \left. {\bigcap\limits_{i = 1}^k {\left\{ {{H_i}{{\rm{e}}^{ - \delta {s_i}}} \le x + z - L} \right\}} } \right) = {\mathit{\Delta }_1}{ + }{\mathit{\Delta }_2}. $

(6)

对于Δ₁，再次利用L族性质有

$ {\mathit{\Delta }_1} \le \sum\limits_{i = 1}^k {P\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x + z - L} \right)} \sim \sum\limits_{i = 1}^k {P\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x} \right).} $

(7)

对于Δ₂，由D族定义，存在x₀>0及常数c₀，当x>x₀时

$ \begin{array}{l} {\mathit{\Delta }_2} = P\left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {s_i}}}} > x + z, \left( {x + z} \right)/k < \mathop {\max }\limits_{1 \le i \le k} {H_i}{{\rm{e}}^{ - \delta {s_i}}} \le x + z - L} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\le \sum\limits_{i = 1}^k {\sum\limits_{j = 1, j \ne i}^k {P\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > L/\left( {k - 1} \right)} \right.;} } \\ \left. {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{H_j}{{\rm{e}}^{ - \delta {s_j}}} > \left( {x + z} \right)/k} \right) \le \sum\limits_{i = 1}^k {\sum\limits_{j = 1, j \ne i}^k {MP\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > } \right.} } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {L/\left( {k - 1} \right)} \right){c_0}P\left( {{H_j}{{\rm{e}}^{ - \delta {s_j}}} > x + z} \right) = o\left( {\sum\limits_{i = 1}^k {P\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x} \right)} } \right). \end{array} $

(8)

结合式(5)~(8)可得$ P(\mathop \sum \limits_{i = 1}^k {H_i}{{\rm{e}}^{ - \delta {s_i}}} > x + z)\sim\mathop \sum \limits_{i = 1}^k P({H_i}{{\rm{e}}^{ - \delta {s_i}}} > x)$.再由(4)知(3)成立.

2 主要结果

定理1  对于模型(1)，在假设条件(A₁)~(A₃)下，若F, G∈L∩D，则有限时破产概率满足

$ \psi \left( {x, T} \right) \sim \int_0^T {\bar F\left( {x{{\rm{e}}^{\delta s}}} \right){\rm{d}}{\lambda _s} + } \int_0^T {\int_0^{T - s} {\bar G\left( {x{{\rm{e}}^{\delta \left( {s + t} \right)}}} \right)} {\rm{d}}K\left( t \right){\rm{d}}{\lambda _s}} . $

(9)

证明  由式(1)知索赔折现过程满足$x - \mathop \sum \limits_{i = 1}^{N\left( T \right)} {H_i}{{\rm{e}}^{ - \delta {S_i}}} \le {U_\delta }\left( t \right){{\rm{e}}^{ - \delta t}} \le x + \tilde C\left( T \right) - \mathop \sum \limits_{i = 1}^{N\left( t \right)} {H_i}{{\rm{e}}^{ - \delta {S_i}}} $.再由(2)式得破产概率满足$\psi \left( {x, T} \right) \le P(\mathop \sum \limits_{i = 1}^{N\left( T \right)} {H_i}{{\rm{e}}^{ - \delta {S_i}}} > x) $，且$\psi \left( {x, T} \right) \ge P(\mathop \sum \limits_{i = 1}^{N\left( t \right)} {H_i}{{\rm{e}}^{ - \delta {S_i}}} > x + \tilde C\left( T \right), \exists t \in \left[ {0, T} \right]) = $$P(\mathop \sum \limits_{i = 1}^{N\left( T \right)} {H_i}{{\rm{e}}^{ - \delta {S_i}}} > x + \tilde C\left( T \right)) $.一方面，任取正整数m₀，有

$ \psi \left( {x, T} \right) \le \left( {\sum\limits_{k = 1}^{{m_0}} + \sum\limits_{k = {m_0} + 1}^\infty {} } \right)P\left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {s_i}}}} > x, N\left( T \right) = k} \right) = {I_1} + {I_2}. $

(10)

对I₁，由引理5知，在Z≡0时，有

$ {I_1} \sim \sum\limits_{k = 1}^{{m_0}} {\sum\limits_{i = 1}^k P } \left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x, N\left( T \right) = k} \right)\\ \le \sum\limits_{i = 1}^\infty P \left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x, {S_k} \le T} \right) = \int_0^T {P\left( {{H_1}{{\rm{e}}^{ - \delta s}} > x} \right){\rm{d}}{\lambda _s}.} $

(11)

对I₂，由引理1，存在常数C>0, D>0及p>J_F⁺，有

$ \begin{array}{l} {I_2} \le \sum\limits_{k = {m_0} + 1}^\infty P \left( {\sum\limits_{i = 1}^k {{H_i} > x} } \right)P\left( {N\left( T \right) = k} \right) \le \sum\limits_{{m_0} < k \le x/D} \\{\sum\limits_{i = 1}^k P \left( {{H_i} > \left( {x/k} \right)} \right)P\left( {N\left( T \right) = k} \right)} + \sum\limits_{k > x/D} {P\left( {N\left( T \right) = k} \right)} \le \\ \sum\limits_{{m_0} < k \le x/D} {k \cdot C \cdot {k^p} \cdot P} \left( {{H_1} > x} \right)P\left( {N\left( T \right) = k} \right) + {\left( {x/D} \right)^{ - p - 1}}E\left( {{{\left( {N\left( T \right)} \right)}^{P + 1}}} \right)\\{I_{\left\{ {N\left( T \right) > x/D} \right\}}} \le CP\left( {{H_1} > x} \right)E\left( {{{\left( {N\left( T \right)} \right)}^{p + 1}}} \right){I_{\left\{ {N\left( T \right) > {m_0}} \right\}}}. \end{array} $

又由文献[14]知，对∀t∈[0, T]，当m₀→∞时，有(E((N(T))^p+1)I_{N(T)>m0})/λ_t→0.可得

$ \mathop {\lim }\limits_{{m_0} \to \infty } \mathop {\lim }\limits_{x \to \infty } \sup \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} {I_2}/\int_0^T P \left( {{H_1}{{\rm{e}}^{ - \delta s}} > x} \right){\rm{d}}{\lambda _s} = 0. $

(12)

从而由式(10)~(12)知$ \psi \left( {x, T} \right) \prec \smallint _0^TP({H_1}{{\rm{e}}^{ - \delta s}} > x){\rm{d}}{\lambda _s}$.另一方面，对上述m₀,

$ \begin{array}{l} \psi \left( {x, T} \right) \ge \sum\limits_{k = 1}^{{m_0}} P \left( {\sum\limits_{i = 1}^k {{H_i}{{\rm{e}}^{ - \delta {S_i}}}} > x + \tilde C\left( T \right), N\left( T \right) = k} \right) \sim \\ \sum\limits_{k = 1}^{{m_0}} {\sum\limits_{i = 1}^k P } \left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x, N\left( T \right) = k} \right) = \\ \left( {\sum\limits_{k = 1}^\infty {\sum\limits_{i = 1}^k - \sum\limits_{k = {m_0} + 1}^\infty {\sum\limits_{i = 1}^k {} } } } \right)P\left( {{H_i}{{\rm{e}}^{ - \delta {s_i}}} > x, N\left( T \right) = k} \right) = \int_0^T {P\left( {{H_1}{{\rm{e}}^{ - \delta s}} > x} \right){\rm{d}}{\lambda _s} - {J_2}.} \end{array} $

(13)

类似I₂证明方法，有J₂≤P(H₁>x)EN(T)I_{N(T)>m0}.从而

$ \mathop {\lim }\limits_{{m_0} \to \infty } \mathop {\lim }\limits_{x \to \infty } \mathop {\sup }\limits_{t \in \left[ {0, T} \right]} \left( {{J_2}/\int_0^T P \left( {{H_1}{{\rm{e}}^{ - \delta s}} > x} \right){\rm{d}}{\lambda _s}} \right) = 0. $

(14)

结合式(13)和(14)知$ \psi \left( {x, T} \right) > \smallint _0^TP({H_1}{{\rm{e}}^{ - \delta s}} > x){\rm{d}}{\lambda _s}$.因此

$ \psi \left( {x, T} \right) \sim \int_0^T P \left( {{H_1}{{\rm{e}}^{ - \delta s}} > x} \right){\rm{d}}{\lambda _s} \sim \int_0^T\\ {\bar F\left( {s{{\rm{e}}^{\delta s}}} \right){\rm{d}}{\lambda _s} + \int_0^T {\int_0^{T - s} {\bar G\left( {s{{\rm{e}}^{\delta \left( {s - t} \right)}}} \right){\rm{d}}K\left( t \right){\rm{d}}{\lambda _s}, } } } $

上式第2个等价关系可由引理3得到.从而定理得证.

3 结论

与以往模型相比，本文考虑了延迟索赔的影响，并在保费随机，索赔额相依情形下得到了破产概率渐近等价式，该结果对保险公司在新的承保模式下进行风险管控具有一定的指导意义.