Extreme Value Theorems of Uncertain Process with Application to Insurance Risk Model Baoding Liu Uncertainty Theory Laboratory Department of Mathematical Sciences Tsinghua University, Beijing 100084, China
[email protected]
http://orsc.edu.cn/liu
Abstract Uncertain process is a sequence of uncertain variables indexed by time. This paper presents a series of extreme value theorem of uncertain independent increment process and provides uncertainty distribution of first hitting time. This paper also proposes an insurance risk model with uncertain claims. Finally, a concept of ruin index is defined and a ruin index formula is given. Keywords: uncertainty theory, uncertain process, finance, insurance, risk
1
Introduction
When historical data are not available to estimate a probability distribution, we have to invite some domain experts to evaluate their belief degree that each event will occur. Since human beings usually overweight unlikely events, the belief degree may have much larger variance than the real frequency. Perhaps some people think that the belief degree is subjective probability. However, Liu [9] showed that it is inappropriate because probability theory may lead to counterintuitive results in this case. In order to deal with this phenomena, uncertainty theory was founded by Liu [3] in 2007 and refined by Liu [8] in 2010. Nowadays uncertainty theory has become a branch of mathematics for modeling human uncertainty. As an extension of uncertainty theory, the concept of uncertain process was given by Liu [4] as a sequence of uncertain variables indexed by time. Two years later, Liu [8] proved an elementary renewal theorem for determining the average renewal number, and a renewal reward theorem for determining the long-run reward rate. In addition, Yao and Li [15] introduced the concept of alternating renewal process and proved an alternating renewal theorem for determining the availability rate. Based on uncertain process, the concept of uncertain integral was proposed by Liu [4] in 2008 in order to integrate an uncertain process with respect to the canonical process. One year later, Liu [5] recast his work via the fundamental theorem of uncertain calculus and thus produced the techniques of chain rule, change of variables, and integration by parts. Since then, the theory of uncertain calculus was well developed. After the concept of uncertain differential equation was proposed by Liu [4], an existence and uniqueness theorem of solution of uncertain differential equation was proved by Chen and Liu [1] under linear growth condition and Lipschitz continuous condition. In order to solve uncertain differential equations, Chen and Liu [1] obtained an analytic solution to linear uncertain differential equations. In addition, Liu [13] presented a spectrum of analytic methods to solve some special classes of nonlinear uncertain differential equations. Furthermore, a numerical method was designed by Yao and Chen [16] for solving general uncertain differential equations. Uncertain differential equations were first introduced into finance by Liu [5] in which an uncertain stock model was proposed and European option price formulas were documented. Besides, Chen [2] derived American option price formulas for this type of uncertain stock model. In addition, Peng and Yao [14] presented a different uncertain stock model and obtained the corresponding option price formulas. Yu [17] proposed an uncertain stock model with jumps. Since then, uncertain stock models were well developed. Uncertain differential equations were also employed to model uncertain currency markets by Liu and Chen [12] in which the exchange rate was assumed to follow a geometric canonical process and an uncertain currency model was proposed, and were applied to optimal control by Zhu [18] in which 1
Zhu’s equation of optimality is proved to be a necessary condition for extremum of uncertain optimal control model. This paper will prove a series of extreme value theorem of uncertain independent increment process and provide the uncertainty distribution of first hitting time. Based on the theory of uncertain renewal reward process, this paper will also present an uncertain insurance model by assuming the claim is a renewal reward process, and prove a ruin index theorem that provides a formula for calculating ruin index.
2
Preliminaries
Uncertainty theory (Liu [3][8]) starts with the concept of uncertain measure that is a set function satisfying normality, duality, subadditivity and product axioms (Liu [5]). On the basis of uncertain measure, an uncertain variable is defined as a measurable function ξ from an uncertainty space to the set of real numbers. In order to describe the uncertain variable ξ in practice, the concept of uncertainty distribution is defined by Φ(x) = M {ξ ≤ x} . (1) The expected value of an uncertain variable ξ is defined as its average value in the sense of uncertain measure, i.e., Z +∞ Z 0 E[ξ] = M{ξ ≥ r}dr − M{ξ ≤ r}dr (2) −∞
0
provided that at least one of the two integrals is finite. From the concepts of uncertain measure, uncertain variable, uncertainty distribution and expected value operator, we may define variance, moments, distance, entropy, etc. Let ξ1 , ξ2 , · · · , ξn be independent uncertain variables with uncertainty distributions Φ1 , Φ2 , · · · , Φn , respectively. If the function f (x1 , x2 , · · · , xn ) is strictly increasing with respect to x1 , x2 , · · ·, xm and strictly decreasing with respect to xm+1 , xm+2 , · · · , xn , then ξ = f (ξ1 , ξ2 , · · · , ξn ) is an uncertain variable with inverse uncertainty distribution −1 −1 −1 Ψ−1 (α) = f (Φ−1 1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α)).
(3)
This is the operational law of uncertain variables. For example, let ξ1 and ξ2 be independent uncertain variables with regular uncertainty distributions Φ1 and Φ2 , respectively. Since x1 + x2 is a strictly increasing function with respect to (x1 , x2 ), the sum ξ = ξ1 + ξ2 is an uncertain variable with inverse uncertainty distribution −1 Ψ−1 (α) = Φ−1 (4) 1 (α) + Φ2 (α). In addition, since x1 − x2 is strictly increasing with respect to x1 and strictly decreasing with respect to x2 , the inverse uncertainty distribution of the difference ξ1 − ξ2 is −1 Ψ−1 (α) = Φ−1 1 (α) − Φ2 (1 − α).
(5)
Furthermore, Liu and Ha [11] proved that the uncertain variable ξ = f (ξ1 , ξ2 , · · · , ξn ) has an expected value Z 1 −1 −1 −1 E[ξ] = f (Φ−1 (6) 1 (α), · · · , Φm (α), Φm+1 (1 − α), · · · , Φn (1 − α))dα. 0
An uncertain process is essentially a sequence of uncertain variables indexed by time. Let T be an index set and let (Γ, L, M) be an uncertainty space. An uncertain process is defined by Liu [4] as a measurable function from T × (Γ, L, M) to the set of real numbers, i.e., for each t ∈ T and any Borel set B of real numbers, the set {Xt ∈ B} = {γ ∈ Γ Xt (γ) ∈ B} (7) is an event. For each fixed γ ∗ , the function Xt (γ ∗ ) is called a sample path of the uncertain process Xt . An uncertain process Xt is said to be sample-continuous if almost all sample paths are continuous with respect to t. An uncertain process Xt is said to have independent increments if Xt0 , Xt1 − Xt0 , Xt2 − Xt1 , · · · , Xtk − Xtk−1 2
(8)
are independent uncertain variables where t0 is the initial time and t1 , t2 , · · ·, tk are any times with t0 < t1 < · · · < tk . That is, an independent increment process means that its increments are independent uncertain variables whenever the time intervals do not overlap. Let ξ1 , ξ2 , · · · be iid positive uncertain variables. Define S0 = 0 and Sn = ξ1 + ξ2 + · · · + ξn for n ≥ 1. Then the uncertain process Nt = max n Sn ≤ t (9) n≥0
is called a renewal process (Liu [4]). When ξ1 , ξ2 , · · · have a common uncertainty distribution Φ, Liu [8] proved that Nt has an uncertainty distribution t , ∀x ≥ 0 (10) Υt (x) = 1 − Φ bxc + 1 where bxc represents the maximal integer less than or equal to x. Let (ξ1 , η1 ), (ξ2 , η2 ), · · · be a sequence of pairs of uncertain variables. We shall interpret ηi as the rewards (or costs) associated with the i-th interarrival times ξi for i = 1, 2, · · ·, respectively. Assume ξ1 , η1 , ξ2 , η2 , · · · are also independent uncertain variables. Then Rt =
Nt X
ηi
(11)
i=1
is called a renewal reward process (Liu [8]), where Nt is the renewal process with uncertain interarrival times ξ1 , ξ2 , · · · Assume those interarrival times and rewards have uncertainty distributions Φ and Ψ, respectively. Then Rt has an uncertainty distribution x t . (12) ∧Ψ Υt (x) = max 1 − Φ k≥0 k+1 k Here we set x/k = +∞ and Ψ(x/k) = 1 when k = 0. For exploring the recent developments of uncertainty theory and uncertain process, the interested readers may consult Liu [10].
3
Extreme Value Theorems
This section will present a series of extreme value theorem of sample-continuous uncertain independent increment processes. Note that discrete-time process will be considered sample-continuous. Let us first prove two lemmas for the extreme values of a finite sequence of uncertain variables. Lemma 1 Let ξ1 , ξ2 , · · · , ξn be independent uncertain variables. Assume that Si = ξ 1 + ξ 2 + · · · + ξ i
(13)
have uncertainty distributions Ψi for i = 1, 2, · · · , n, respectively. Then the maximum S = S1 ∨ S2 ∨ · · · ∨ Sn
(14)
Υ(x) = Ψ1 (x) ∧ Ψ2 (x) ∧ · · · ∧ Ψn (x).
(15)
has an uncertainty distribution
Proof: Assume that the uncertainty distributions of the uncertain variables ξ1 , ξ2 , · · · , ξn are Φ1 , Φ2 , · · · , Φn , respectively. Define f (x1 , x2 , · · · , xn ) = x1 ∨ (x1 + x2 ) ∨ · · · ∨ (x1 + x2 + · · · + xn ). Then f is a strictly increasing function and S = f (ξ1 , ξ2 , · · · , ξn ). 3
It follows from the operational law that S has an uncertainty distribution Υ(x) =
Φ1 (x1 ) ∧ Φ2 (x2 ) ∧ · · · ∧ Φn (xn )
sup f (x1 ,x2 ,···,xn )=x
= min
sup
1≤i≤n x1 +x2 +···+xi =x
Φ1 (x1 ) ∧ Φ2 (x2 ) ∧ · · · ∧ Φi (xi )
= min Ψi (x). 1≤i≤n
Thus (15) is verified. Lemma 2 Let ξ1 , ξ2 , · · · , ξn be independent uncertain variables. Assume that Si = ξ 1 + ξ 2 + · · · + ξ i
(16)
have uncertainty distributions Ψi for i = 1, 2, · · · , n, respectively. Then the minimum S = S1 ∧ S2 ∧ · · · ∧ Sn
(17)
Υ(x) = Ψ1 (x) ∨ Ψ2 (x) ∨ · · · ∨ Ψn (x).
(18)
has an uncertainty distribution
Proof: Assume that the uncertainty distributions of the uncertain variables ξ1 , ξ2 , · · · , ξn are Φ1 , Φ2 , · · · , Φn , respectively. Define f (x1 , x2 , · · · , xn ) = x1 ∧ (x1 + x2 ) ∧ · · · ∧ (x1 + x2 + · · · + xn ). Then f is a strictly increasing function and S = f (ξ1 , ξ2 , · · · , ξn ). It follows from the operational law that S has an uncertainty distribution Υ(x) =
Φ1 (x1 ) ∧ Φ2 (x2 ) ∧ · · · ∧ Φn (xn )
sup f (x1 ,x2 ,···,xn )=x
= max
sup
1≤i≤n x1 +x2 +···+xi =x
Φ1 (x1 ) ∧ Φ2 (x2 ) ∧ · · · ∧ Φi (xi )
= max Ψi (x). 1≤i≤n
Thus (18) is verified. Theorem 1 (Extreme Value Theorem) Let Xt be a sample-continuous independent increment process with an uncertainty distribution Φt (x) at each time t. Then the supremum sup Xt
(19)
0≤t≤s
has an uncertainty distribution Ψ(x) = inf Φt (x). 0≤t≤s
Proof: Let 0 = t1 < t2 < · · · < tn = s be a partition of the closed interval [0, s]. It is clear that Xti = Xt1 + (Xt2 − Xt1 ) + · · · + (Xti − Xti−1 ) for i = 1, 2, · · · , n. Since the increments Xt1 , Xt2 − Xt1 , · · · , Xtn − Xtn−1 4
(20)
are independent uncertain variables, it follows from Theorem 1 that the maximum max Xti
1≤i≤n
has an uncertainty distribution min Φti (x).
1≤i≤n
Since Xt is sample-continuous, we have max Xti → sup Xt
1≤i≤n
0≤t≤s
and min Φti (x) → inf Φt (x)
1≤i≤n
0≤t≤s
as n → ∞. Thus the supremum sup0≤t≤s Xt has the uncertainty distribution Ψ. The theorem is proved. Theorem 2 (Extreme Value Theorem) Let Xt be a sample-continuous independent increment process with an uncertainty distribution Φt (x) at each time t. Then the infimum inf Xt
(21)
Ψ(x) = sup Φt (x).
(22)
0≤t≤s
has an uncertainty distribution 0≤t≤s
Proof: Let 0 = t1 < t2 < · · · < tn = s be a partition of the closed interval [0, s]. It is clear that Xti = Xt1 + (Xt2 − Xt1 ) + · · · + (Xti − Xti−1 ) for i = 1, 2, · · · , n. Since the increments Xt1 , Xt2 − Xt1 , · · · , Xtn − Xtn−1 are independent uncertain variables, it follows from Theorem 2 that the minimum min Xti
1≤i≤n
has an uncertainty distribution max Φti (x).
1≤i≤n
Since Xt is sample-continuous, we have min Xti → inf Xt
1≤i≤n
0≤t≤s
and max Φti (x) → sup Φt (x)
1≤i≤n
0≤t≤s
as n → ∞. Thus the infimum inf 0≤t≤s Xt has the uncertainty distribution Ψ. The theorem is proved.
4
First Hitting Time
This section will give the uncertainty distribution of first hitting time for sample-continuous independent increment process. 5
Theorem 3 Let Xt be a sample-continuous independent increment process with a continuous uncertainty distribution Φt (x) at each time t. Then the first hitting time τz that Xt reaches the level z has an uncertainty distribution, inf Φt (z), if z > X0 1 − 0≤t≤s Υ(s) = (23) sup Φt (z), if z < X0 . 0≤t≤s
Proof: Note that Xt is a sample-continuous independent increment process. When z > X0 , it follows from the extreme value theorem that Υ(s) = M{τz ≤ s} = M sup Xt ≥ z = 1 − inf Φt (z). 0≤t≤s
0≤t≤s
When z < X0 , it follows from the extreme value theorem that Υ(s) = M{τz ≤ s} = M inf Xt ≤ z = sup Φt (z). 0≤t≤s
0≤t≤s
The theorem is verified.
5
Strictly Increasing Function of Uncertain Process
Theorem 4 Let Xt be a sample-continuous independent increment process with an uncertainty distribution Φt (x) at each time t. If f is a strictly increasing function, then the supremum sup f (Xt )
(24)
0≤t≤s
has an uncertainty distribution Ψ(x) = inf Φt (f −1 (x)). 0≤t≤s
(25)
Proof: Since f is a strictly increasing function, f (Xt ) ≤ x if and only if Xt ≤ f −1 (x). It follows from the extreme value theorem that Ψ(x) = M sup f (Xt ) ≤ x 0≤t≤s
=M
sup Xt ≤ f −1 (x)
0≤t≤s
= inf Φt (f −1 (x)). 0≤t≤s
The theorem is proved. Example 1: Let Xt be a sample-continuous independent increment process with an uncertainty distribution Φt (x) at each time t. Then the supremum sup exp(Xt )
(26)
0≤t≤s
has an uncertainty distribution Ψ(x) = inf Φt (ln x) 0≤t≤s
(27)
because f (x) = exp(x) and f −1 (x) = ln x. Example 2: Let Xt be a sample-continuous and positive independent increment process with an uncertainty distribution Φt (x) at each time t. Then the supremum sup ln Xt 0≤t≤s
6
(28)
has an uncertainty distribution Ψ(x) = inf Φt (exp(x)) 0≤t≤s
(29)
because f (x) = ln x and f −1 (x) = exp(x). Example 3: Let Xt be a sample-continuous and nonnegative independent increment process with an uncertainty distribution Φt (x) at each time t. Then the supremum sup Xt2
(30)
√ Ψ(x) = inf Φt ( x)
(31)
0≤t≤s
has an uncertainty distribution
0≤t≤s
because f (x) = x2 and f −1 (x) =
√
x.
Theorem 5 Let Xt be a sample-continuous independent increment process with an uncertainty distribution Φt (x) at each time t. If f is a strictly increasing function, then the infimum inf f (Xt )
(32)
Ψ(x) = sup Φt (f −1 (x)).
(33)
0≤t≤s
has an uncertainty distribution 0≤t≤s
Proof: Since f is a strictly increasing function, f (Xt ) ≤ x if and only if Xt ≤ f −1 (x). It follows from the extreme value theorem that Ψ(x) = M inf f (Xt ) ≤ x 0≤t≤s
=M
inf Xt ≤ f
−1
0≤t≤s
(x)
= sup Φt (f −1 (x)). 0≤t≤s
The theorem is proved. Example 4: Let Xt be a sample-continuous independent increment process with an uncertainty distribution Φt (x) at each time t. Then the infimum inf exp(Xt )
(34)
Ψ(x) = sup Φt (ln x).
(35)
0≤t≤s
has an uncertainty distribution 0≤t≤s
Example 5: Let Xt be a sample-continuous and positive independent increment process with an uncertainty distribution Φt (x) at each time t. Then the infimum inf ln Xt
(36)
Ψ(x) = sup Φt (exp(x)).
(37)
0≤t≤s
has an uncertainty distribution 0≤t≤s
Example 6: Let Xt be a sample-continuous and nonnegative independent increment process with an uncertainty distribution Φt (x) at each time t. Then the infimum inf Xt2
0≤t≤s
7
(38)
has an uncertainty distribution
√ Ψ(x) = sup Φt ( x).
(39)
0≤t≤s
Theorem 6 Let Xt be a sample-continuous independent increment process with a continuous uncertainty distribution Φt (x) at each time t. If f is a strictly increasing function and z is a given level, then the first hitting time τz that f (Xt ) reaches the level z has an uncertainty distribution, inf Φt (f −1 (z)), if z > f (X0 ) 1 − 0≤t≤s Υ(s) = (40) sup Φt (f −1 (z)), if z < f (X0 ). 0≤t≤s
Proof: Note that Xt is a sample-continuous independent increment process and f is a strictly increasing function. When z > f (X0 ), it follows from the extreme value theorem that Υ(s) = M{τz ≤ s} = M sup f (Xt ) ≥ z = 1 − inf Φt (f −1 (z)). 0≤t≤s
0≤t≤s
When z < f (X0 ), it follows from the extreme value theorem that Υ(s) = M{τz ≤ s} = M inf f (Xt ) ≤ z = sup Φt (f −1 (z)). 0≤t≤s
0≤t≤s
The theorem is verified.
6
Strictly Decreasing Function of Uncertain Process
Theorem 7 Let Xt be a sample-continuous independent increment process with a continuous uncertainty distribution Φt (x) at each time t. If f is a strictly decreasing function, then the supremum sup f (Xt )
(41)
0≤t≤s
has an uncertainty distribution Ψ(x) = 1 − sup Φt (f −1 (x)).
(42)
0≤t≤s
Proof: Since f is a strictly decreasing function, f (Xt ) ≤ x if and only if Xt ≥ f −1 (x). It follows from the extreme value theorem that Ψ(x) = M sup f (Xt ) ≤ x = M inf Xt ≥ f −1 (x) 0≤t≤s
0≤t≤s
=1−M
−1 inf Xt < f (x) = 1 − sup Φt (f −1 (x)).
0≤t≤s
0≤t≤s
The theorem is proved. Example 7: Let Xt be a sample-continuous independent increment process with a continuous uncertainty distribution Φt (x) at each time t. Then the supremum sup exp(−Xt )
(43)
0≤t≤s
has an uncertainty distribution Ψ(x) = 1 − sup Φt (− ln x) 0≤t≤s
because f (x) = exp(−x) and f −1 (x) = − ln x. 8
(44)
Example 8: Let Xt be a sample-continuous and positive independent increment process with a continuous uncertainty distribution Φt (x) at each time t. Then the supremum sup 0≤t≤s
1 Xt
(45)
has an uncertainty distribution 1 Ψ(x) = 1 − sup Φt x 0≤t≤s
(46)
because f (x) = 1/x and f −1 (x) = 1/x. Theorem 8 Let Xt be a sample-continuous independent increment process with a continuous uncertainty distribution Φt (x) at each time t. If f is a strictly decreasing function, then the infimum inf f (Xt )
(47)
Ψ(x) = 1 − inf Φt (f −1 (x)).
(48)
0≤t≤s
has an uncertainty distribution 0≤t≤s
Proof: Since f is a strictly decreasing function, f (Xt ) ≤ x if and only if Xt ≥ f −1 (x). It follows from the extreme value theorem that Ψ(x) = M inf f (Xt ) ≤ x = M sup Xt ≥ f −1 (x) 0≤t≤s
=1−M
0≤t≤s
−1 sup Xt < f (x) = 1 − inf Φt (f −1 (x)). 0≤t≤s
0≤t≤s
The theorem is proved. Example 9: Let Xt be a sample-continuous independent increment process with a continuous uncertainty distribution Φt (x) at each time t. Then the infimum inf exp(−Xt )
(49)
Ψ(x) = 1 − inf Φt (− ln x).
(50)
0≤t≤s
has an uncertainty distribution 0≤t≤s
Example 10: Let Xt be a sample-continuous and positive independent increment process with a continuous uncertainty distribution Φt (x) at each time t. Then the infimum inf
0≤t≤s
1 Xt
(51)
has an uncertainty distribution Ψ(x) = 1 − inf Φt 0≤t≤s
1 . x
(52)
Theorem 9 Let Xt be a sample-continuous independent increment process with a continuous uncertainty distribution Φt (x) at each time t. If f is a strictly decreasing function and z is a given level, then the first hitting time τz that f (Xt ) reaches the level z has an uncertainty distribution, sup Φt (f −1 (z)), if z > f (X0 ) 0≤t≤s Υ(s) = (53) 1 − inf Φt (f −1 (z)), if z < f (X0 ). 0≤t≤s
9
Proof: Note that Xt is an independent increment process and f is a strictly decreasing function. When z > f (X0 ), it follows from the extreme value theorem that Υ(s) = M{τz ≤ s} = M sup f (Xt ) ≥ z = sup Φt (f −1 (z)). 0≤t≤s
0≤t≤s
When z < f (X0 ), it follows from the extreme value theorem that Υ(s) = M{τz ≤ s} = M inf f (Xt ) ≤ z = 1 − inf Φt (f −1 (z)). 0≤t≤s
0≤t≤s
The theorem is verified.
7
Uncertain Insurance Model
Assumed that a is the initial capital of an insurance company, b is the premium rate, bt is the total income up to time t, and the uncertain claim process is a renewal reward process Rt =
Nt X
ηi
(54)
i=1
with iid uncertain interarrival times ξ1 , ξ2 , · · · and iid uncertain claim amounts η1 , η2 , · · · Then the capital of the insurance company at time t is Zt = a + bt − Rt (55) and Zt is called an insurance risk process. Z. t
... .......... ... .... ...... ...... .. ..... ... ..... ... ... ...... .... ........... .... ...... ... ........ ... ..... . . . . ... . ... . . .. ... ...... ... .. ..... . ... . . ... . . .. ... .... . . ... . . .. ... .... .... ... . . . . . . ...... . . . ... . . . . . . . .. ... ...... ... .... ... ........ . . ... .......... .... . . ... ..... .. ... ......... .... ... . . ... . . ............ . . ... . . ....... .......... .. ... . . ... . . . . ... . ..... .. .. .. .. ... ... ...... ... .. .. .. ... .......... ... ... .. .. .. ... .......... ... .. .. .. ... .. ... .. .. .. ... .. ... .. .. .. ... .. ... .. ... .. .. .. ... ... .. .. .. .. ... ... .. .. .. .. ... ... .. .. .. . . . ............................................................................................................................................................................................................................................................................................... ... .... .... . ..... ...... ... 1 2 3 4 . ... . ... . ... ... ... . .... ...... ... .. ... .......
a
0
S
S
S
S
t
Figure 1: An Insurance Risk Process Ruin index is the uncertain measure that the capital of the insurance company becomes negative. Definition 1 Let Zt be an insurance risk process. Then the ruin index is defined as the uncertain measure that Zt is less than 0 at some time t, i.e., Ruin = M inf Zt < 0 . (56) t≥0
It is clear that the ruin index is a special case of the risk index in the sense of Liu [7]. Theorem 10 (Ruin Index Theorem) Let Zt = a + bt − Rt be an insurance risk process where a and b are positive numbers, and Rt is a renewal reward process with iid uncertain interarrival times ξ1 , ξ2 , · · · 10
and iid uncertain claim amounts η1 , η2 , · · · If ξ1 and η1 have continuous uncertainty distributions Φ and Ψ, respectively, then the ruin index is x x−a Ruin = max sup Φ . (57) ∧ 1−Ψ k≥1 x≥0 kb k Proof: For each positive integer k, it is clear that the arrival time of the kth claim is Sk = ξ 1 + ξ 2 + · · · + ξ k whose uncertainty distribution is Φ(s/k). Define an uncertain process indexed by k as follows, Yk = a + bSk − (η1 + η2 + · · · + ηk ). It is easy to verify that Yk is an independent increment process with respect to k. In addition, Yk is just the capital at the arrival time Sk and has an uncertainty distribution x z+x−a Fk (z) = sup Φ . ∧ 1−Ψ kb k x≥0 Since a ruin occurs only at the arrival times, we have Ruin = M inf Zt < 0 = M min Yk < 0 . t≥0
k≥1
It follows from the extreme value theorem that Ruin = max Fk (0) = max sup Φ k≥1
k≥1 x≥0
x−a kb
x ∧ 1−Ψ . k
The theorem is proved. Definition 2 Let Zt be an insurance risk process. Then the ruin time is determined by τ = inf t ≥ 0 Zt < 0 .
(58)
If Zt ≥ 0 for all t ≥ 0, then we define τ = +∞. Note that the ruin time is just the first hitting time that the total capital Zt becomes negative. Since inf t≥0 Zt < 0 if and only if τ < +∞, the relation between ruin index and ruin time is Ruin = M inf Zt < 0 = M{τ < +∞}. t≥0
Acknowledgments This work was supported by National Natural Science Foundation of China Grants No.70833003 and No.91024032.
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