Martingale Approach to Derive Lundberg-Type Inequalities

: In this paper, we ﬁnd the upper bound for the tail probability P (cid:0) sup n (cid:62) 0 ∑ ni = 1 ξ i > x (cid:1) with random summands ξ 1 , ξ 2 , . . . having light-tailed distributions. We ﬁnd conditions under which the tail probability of supremum of sums can be estimated by quantity (cid:36) 1 exp {− (cid:36) 2 x } with some positive constants (cid:36) 1 and (cid:36) 2 . For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model.

In the case of exponential function ϕ, Theorem 1 implies the following upper estimation for the tail probability of r.v. M ∞ . Corollary 1. Let {ξ 1 , ξ 2 , . . .} be a sequence of i.i.d. r.v.s. If Eξ 1 < 0 and E e hξ 1 < ∞ for some positive h then there exist positive constants 1 and 2 such that for all nonnegative x.
If a sequence {ξ 1 , ξ 2 , . . .} consists of independent but possibly differently distributed r.v.s, then the similar estimate to that in (1) also holds. The following assertion is proved in [3] (see Lemma 1). Theorem 2. Let {ξ 1 , ξ 2 , . . .} be a sequence of independent r.v.s such that Then the estimate (1) holds for all positive x and some positive constants 1 and 2 .
In Theorem 3 of [4] the following more general assertion was proved using classical ideas of Chernoff [5] and Hoeffding [6].  for some a > 0, b ∈ N, c > 0, ε 0, h > 0, d 1 1 and d 2 1. If It should be noted that conditions of Theorem 3 are weaker than the conditions of Theorem 2. In addition, the assertion of Theorem 3 provides an algorithm to calculate two positive constants controlling the exponential upper bound. For this reason, conditions of the last theorem have more explicit form.
In this paper we extend the above results by deriving the more precise upper bounds for probability P M ∞ > x under less restrictive requirements. In addition, from these upper bounds we derive the so called Lundberg-type exponential estimates for ruin probabilities of the nonhomogeneous renewal risk models. Results on upper bounds for P M ∞ > x are presented in Section 2, and the versions of the Lundberg-type inequalities are given in Section 3. Section 4 deals with proofs of the main results, and finally, Section 5 addresses to several applications of the results obtained.
It should be noted that the problem under consideration and the method used are related with the problem of upper bound for probability P(S n > x), where S n is a sum of independent or dependent random variables. It is natural that for probability P(S n > x) more sharp upper bounds can be obtained comparing with the upper bound for P(max 1 k n S k > x). The pioneer exponential-type inequalities for probability P(S n > x) were derived by Bernstein [7,8] and later were improved and generalized by many authors, see [6,[9][10][11][12][13][14][15], for instance. The boundedness of summands in S n is a key requirement in these papers to get sharp exponential-type upper bounds. Upper bound for probability P(max 1 k n S k > x) can be derived from the upper estimates of P(S n > x, C) with a suitable condition C. Such a way is described in detail by Fan et al. [12] and in references therein. Unfortunately, the derived upper exponential-type estimates for P(S n > x, C) "work" under quite restrictive requirements for summands of sum S n . The main object of our research is the ruin probability of the renewal risk model. In order to obtain a good and general upper bound of this probability, we use the estimate of probability P(max 1 k n S k > x) presented in Lemma 1. In this lemma the requirements for summands of S n are minimal.

Upper Bounds for Tail of Maximum of Sums
The first theorem of this section gives the upper estimate for probability P M ∞ > x under less requirements than in Theorems 2 and 3 by supposing that random variables ξ 1 , ξ 2 , . . . satisfy the net profit condition, have a finite exponential moment and a negligible left tail on average.
then the estimate (1) holds for all positive x and some positive constants 1 and 2 .
The second theorem provides an algorithm to obtain numerical expressions of constants 1 and 2 in the estimate (1). The assertion of theorem below is similar to that in Theorem 3. However, we derive more precise expressions of constants using the sharp initial inequality of Lemma 1 below. Theorem 5. Let {ξ 1 , ξ 2 , . . .} be a sequence of independent r.v.s such that: Let, in addition: with some δ ∈ (0, 1/2), then for all positive x The last theorem shows what upper bound can be derived for tail of maximum of sums in the case when the cumulant generating functions (see [16], for instance) can be successfully estimated for all r.v.s {ξ 1 , ξ 2 , . . .}. Theorem 6. Let {ξ 1 , ξ 2 , . . .} be a sequence of independent r.v.s. If for all h ∈ (0, h * ) and n ∈ N with some h * > 0 and function ϕ not depending on n, then for all positive x and h ∈ (0, h * ).

Remark 1.
The last estimation (6) implies the possibility to get more sharp estimate than the standard exponential for P ( for that x.

Exponential Estimates for Ruin Probabilities
In this section, we present three corollaries from Theorems 4-6 on the Lundberg-type inequalities for the ultimate ruin probability of an inhomogeneous renewal risk model. We say that the insurer's surplus R(t) varies according to an inhomogeneous renewal risk model (IRRM) if equation holds for all t 0 with the initial insurer's surplus x 0, a constant premium rate p > 0, a sequence of independent, non negative and possibly differently distributed claim amounts {Z 1 , Z 2 , . . .} and with the renewal counting process The ultimate ruin probability (or simply ruin probability) are the main characteristics of the renewal risk model.
The equality (8) shows that results of Theorems 4-6 can be directly applied to derive exponential estimates for ψ(x) of IRRM which are traditionally called Lundberg-type inequalities. In this section, we present three versions of the Lundberg-type inequality for IRRM.
for some D * 1, and with some positive h * and some function Λ not depending on n, then for all positive x and h ∈ (0, h * ).

Proofs
In this section, we prove all main results presented in Sections 2 and 3. Statements of Section 2 can be derived from the following lemma.
holds for all x 0 and h 0.
The assertion of this lemma can be proved using different ways. Here, we present two different proofs of the lemma. The first proof is based on the martingale property of special transform of sum of random variables and on the maximal inequality for submartingales, see, for instance, Exercise 7 on page 110 and Theorem 1 on page 492 of [43]. We found such proof in the unpublished manuscript [44]. The second more direct proof is based on the fundamental Wald's equality for not necessarily identically distributed random variables. For various versions of the Wald's equalities see [45][46][47], § 2 of Chapter VII in [43,[48][49][50] among others.

Proof. (I).
For N ∈ N and h 0 let us define If E N (h) = ∞, then obviously that for all nonnegative x, h. If E N (h) is finite, then for each n ∈ {0, 1, . . . , N}, we define the following nonnegative r.v.
M n := e hS n E e hS n .
Since M 0 = 1 and Consequently inequality (10) is satisfied again, because for arbitrary positive x and h. The estimate (9) of Lemma 1 follows now immediately due to the following relations

Proof. (II).
In this part we present another way to prove the inequality (10). It is enough to prove this estimate to obtain the new proof way because of the standard derivation of (9) from (10) we presented in the first part. The inequality (10) is evident if N = 1. Let us suppose that N 2 and for the sequence {S 1 , S 2 , . . . , S N } define stopping time τ N by the following equation N, if S n x for n ∈ {1, 2, . . . , N}.

If E N (h) is finite, then we have
because of the Wald's fundamental equality for collection of independent but not necessary identically distributed r.v.s {ξ 1 , ξ 2 , . . . , ξ N } and stopping time τ N , see [46,47]. Hence the estimate (10) follows and this ends another proof of the lemma.

Proof of Theorem 4.
According to the estimate (9) of Lemma 1 we have for all x 0, y ∈ [0, h] and N 1.
Using the inequalities By choosing u = 1/ 4 √ y we get that for y ∈ [0, h/2], where k ∈ N and c 1 = c 1 (h) is a positive constant from the estimate v 2 c 1 e hv/2 , v 0.
According to the second condition of Theorem 4 where c 5 > 0 and n N * with the sufficiently large N * . From this and from the inequality (16) we get that for all x 0, y ∈ (0, h/2] and N N * . The condition (iii) of Theorem 4 implies that if y ∈ (0, y * ] and n N for some positive y * ∈ (0, h/2] and some natural N N * . Due to estimates (17) and (18), the inequality holds x 0, y ∈ (0, y * ] and N N.
If we choose y under conditions then we get the desired estimate (1) from (19) with constants Theorem 4 is proved.
Proof of Theorem 5. Due to the Lemma 1 and condition (iv) of Theorem 5 we have for all x > 0 and y ∈ (0, h]. According to the estimate (12) and the obvious inequality v 2 e v − 1, v 0, we have that for all k ∈ N and y ∈ (0, h/2]. Consequently, due to the conditions (i), (ii) and (iii) of Theorem 5. Let now y = δh with some δ ∈ (0, 1/2] satisfying condition (3). For this y we derive from (21) that the estimate (4) holds. Theorem 5 is proved.
Proof of Theorem 6. Exponential moments E e hξ k are positive for all h > 0 and k ∈ N. Hence condition (5) implies that Now the estimate (6) of Theorem 6 follows from Lemma 1 immediately.
Proof of Corollaries 2-4. All assertions follow from Theorems 4-6 immediately by supposing that

Numerical Examples
In this section we present three particular examples of IRRM. For all these models we obtain the Lundberg-type estimates for ultimate ruin probabilities using Corollaries 3 and 4. We compare the obtained bounds with the values of ψ(x) derived by the Monte Carlo method.
The first example is borrowed from the article [4]. We show that with the help of the Corollaries 3 or 4 more accurate upper bounds for the ruin probability can be obtained. Example 1. Let us consider IRRM which is generated by inter occurrence times {θ 1 , θ 2 , . . . } uniformly distributed on interval [1,3], constant premium rate p = 2 and a sequence of the claim amounts {Z 1 , Z 2 , . . .} such that In the case under consideration, we have: Consequently, for n 1, we get In addition, if 1 n 4, then This implies that for all n 5.

The obtained estimates imply conditions of Corollary 3 with
it follows from Corollary 3 that for all x 0.
We observe that in example under consideration we can get sharper upper bound for the ruin probability because distributions of the first two claims are conducive to an increase of the initial surplus. Namely, for x 0 we have where ψ denotes the ruin probability of IRRM generated by random claims { Z 1 , Z 2 , . . .} and inter occurrence times { θ 1 , θ 2 , . . .}. For all k ∈ N r.v. θ k is uniformly distributed on interval [1,3] and The upper bound for the ultimate ruin probability ψ can be derived using Corollary 3 as well as Corollary 4. We choose the latter assertion. We should establish function Λ which bound sum of cumulants of r.v.s Z k − p θ k .
Below, in Figure 1, we compare upper bounds ψ 1 (x) and ψ 2 (x) of ruin probability with its values obtained by the Monte Carlo method. The second IRRM which we present here is generated by exponentially distributed claims and inter occurrence times. We show that we can also derive the upper exponential bounds for ruin probability using Corollaries 3 and 4 again. It is obvious that EZ k = Eθ k = 1 for k ∈ {1, 3, 5, . . .} and EZ k = 2, Eθ k = 3 for k ∈ {2, 4, 6, . . .}. Hence, for n 2, we have After some calculations, we obtain that conditions of Corollary 3 hold with the following collection of constants. In addition, It is evident that the obtained estimate has the exponential form but it is quite conservative. The reason for this is the generality of the Corollary 3. The last estimate holds for wide group of IRRMs. In fact, the estimate presented in Corollary 3 is not sensitive to the structure of the model. Fortunately, in the example under consideration, the cumulant generating functions of r.v.s {Z k − pθ k } ∞ k=1 have sufficiently simple analytic expressions. Hence we can derive more sharp estimate for the model ruin probability using Corollary 4.
Namely, if k ∈ {1, 3, 5, . . .} and h ∈ [0, 1), then If k ∈ {2, 4, 6, . . .} and h ∈ [0, 1/2), then Consequently, By supposing h = 1/8 we obtain from Corollary 4 that for all initial surplus values x 0. Below, in Figure 2, we illustrate the results obtained. In the figure, we can see the values of ruin probability ψ(x) obtained by the Monte Carlo method, its conservative estimate ψ 1 (x) and its sharp estimate ψ 2 (x). The last our example shows that for particular IRRM a sharper upper bound compared to the standard exponential estimate for ruin probability can be derived. For this we need to apply Corollary 4, because using Corollary 3 we can get only the standard exponential upper estimate, and the model should be generated by random claims {Z k } ∞ k=1 having finite exponential moments {Ee hZ k } ∞ k=1 for all positive h.
In the case under consideration, we have that log E e h(Z k −pθ k ) = e h − k − 1 for all k ∈ N and h 0. Below, in Figure 3, we illustrate the results obtained. In the figure, red line is the derived upper bound for ruin probability, and green line is the values of ψ(x) obtained by the Monte Carlo method.  Author Contributions: All authors contributed equally to this work, as well as to its preparation. They have read and approved the final manuscript. All authors have read and agreed to the published version of the manuscript.

Conflicts of Interest:
The authors declare no conflict of interest.