The concept of measuring distances between probability measures is a fundamental one with applications across several areas of mathematics. In ruin theory, the main objective is to model the surplus of an insurance business using a stochastic process and evaluate its ruin probability. This characteristic is a typical measure for the solvency of a portfolio. An extension of this, which accounts for the severity of ruin, is the distribution of the deficit at ruin (given that ruin occurs). In general, explicit expressions for the ruin probability and/or deficit at ruin are known only in some cases, for instance, when the claim sizes have exponential or phase-type distribution (see [1], pg 14-15). Therefore, several theoretical approaches have been proposed to approximate, bound, estimate and numerically compute the ruin probability.

In insurance mathematics, stochastic models are used to idealize input and output elements to approximate real insurance activities. The problem of stability stated as seeking an appropriate measure of closeness between the ideal and real input element in order to estimate the corresponding deviation in the output. Several theoretical approaches have been developed to analyze optimal choices of metrics between input elements. The problem of stability of the aggregate claim amount over a finite time horizon is formally analyzed in [3]. The estimation of the ruin probability in univariate risk models, using the strong stability method has been investigated in [18] and [6]. Then, the application of this approach has been extended in various directions. For instance, the stability of the ruin probability in a Markov modulated risk model with Lévy process with investments was studied in [24]. A two-dimensional classical risk model was considered in [4] using the strong stability method (see also, [27, 2] and [15]). Continuity properties of the surplus process in multidimensional renewal risk models were studied in [12]. Furthermore, a functional approach that was used to obtain approximations and bounds for some ruin-related quantities was developed in [20] and [19]. In the discrete-time setup, approximation techniques involving some integral operator were proposed in [7] and a general formula yielding approximations based on negative binomial mixtures were presented in [26].

Recently, the Banach contraction principle and fixed point results for contractive operators have been applied in the theory of risk models. An approximation of the ruin probability under a Markov modulated classical risk model based on the Banach contraction principle was presented in [8]. It was shown in [16] that the ultimate ruin probability can be expressed as the fixed point of a contraction mapping in terms of q-scale functions. Similar contractive approaches have been examined in several recent works (see for instance [9] and [25]). Gordienko and Vázquez-Ortega in [11] proposed continuity inequalities for ruin probability using properties of contractive mappings. Extending this approach, we derive new continuity bounds for the ruin probability and the deficit at ruin in terms of various choices of probability metrics and we also investigate properties of contractive operators and fixed point results in the context of a continuous time surplus process perturbed by diffusion. To the best of our knowledge, no existing work has employed probability metrics in the setting of a surplus process perturbed by diffusion.

The goal in this paper is to propose appropriate probability metrics that allow us to obtain suitable continuity estimates for ruin-related quantities, such as probability of ruin with and without diffusion and the deficit at ruin, in the classical of risk theory. The paper is organized as follows: in the next section, we introduce the main quantities of interest in the classical risk model of risk theory, and we present the concept of the continuity problem for the ruin probability. In Section 3, we derive a continuity inequality for ruin probabilities and provide an upper bound on the supremum distance between two deficits at ruin. In Section 4, we derive upper bounds for a ruin-related quantity in the classical risk process perturbed by diffusion and obtain an approximation-using the Banach fixed point theorem (BFPT) to this quantity. Proofs are given in Section 5, while Section 6 contains numerical examples that illustrate the validity of the results. The final section summarizes the paper.

The paper concerns the compound Poisson model with risk process (1) U ( t ) = u + c t − ∑ n = 1 N ( t ) X n , t ≥ 0 , where u ≥ 0 is the initial surplus, c is the premium rate and N ( t ) denotes the number of claims up to time t. The X n ’s represent the claim sizes and they are assumed to be independent identically distributed positive random variables (r.v.’s) with a common distribution function F ( x ) = P r ( X ≤ x ), tail F ‾ ( x ) = 1 − F ( x ) = P r ( X > x ) and mean E X < + ∞. Also, N ( t ) is a Poisson process with rate λ, independent of X n . We further assume c = λ ( 1 + θ ) E X, where θ > 0 is the relative security loading. Let (2) ψ ( u ) : = P r ( inf t ≥ 0 U ( t ) < 0 | U ( 0 ) = u ) , u ≥ 0 , be the ruin probability in infinite time. In the classical risk model (1), it is well-known that ψ ( u ) satisfies (3) ψ ( u ) = P r ( L > u ) = ∑ n = 1 ∞ θ 1 + θ 1 1 + θ n F e ∗ n ‾ ( u ) , u ≥ 0 , where F e ∗ n ‾ ( u ) = P r ( X e ( 1 ) + X e ( 2 ) + ⋯ X e ( n ) > u ) is the tail of the nth-fold convolution of F e ( u ) = ∫ 0 u F ‾ ( z ) d z / ∫ 0 ∞ F ‾ ( z ) d z with itself. The variable L here is the maximal aggregate loss in the surplus process. The distribution F e is known as the equilibrium distribution associated with F.

In general, the continuity problem is based on the following implication. Let U ( t , α ) be a risk process governed by a parameter α = ( λ , c , F ) with the ruin probability ψ α . If A denotes the space of possible values of the parameter α then one can view the ruin probability as a mapping ψ : A → Ψ, where Ψ is the functional space of all possible functions ψ α . Assume that A and Ψ are metric spaces with metrics δ and ν, respectively. In such terms, the problem of interest is reduced to investigation of ( δ , ν )-continuity of this mapping: that is if the implication (4) δ ( α , α ˜ ) → 0 , then ν ( ψ α , ψ α ˜ ) → 0 , for α , α ˜ ∈ A, holds at a fixed point α.

The metrics δ and ν should be computationally convenient and reflect the core of the problem. In the following, we consider various choices for these metrics. Other such metrics that may be suitable to obtain bounds of this form for ruin probabilities and ruin-related quantities can be found in [17]. If we can find an inequality (5) ν ( ψ α , ψ α ˜ ) ≤ w ( δ ( α , α ˜ ) ) , where w ( s ) ≥ 0 , w ( s ) → s → 0 0 and w ( 0 ) = 0, then the inequality (5) is called a continuity inequality (estimate) and provides the possibility of bounding ν ( ψ α , ψ α ˜ ) in terms of the distance δ ( α , α ˜ ) (see e.g. [18]).

In applications, the vector parameter α = ( λ , c , F ) that governs the risk model is usually unknown. Therefore, the intensity of claim arrivals λ and the distribution of claim sizes F are approximated by some parameter λ ˜ and distribution F ˜, respectively, for which the ruin probability ψ ˜ can be found. This allows local estimations of the ruin probability alterations to be obtained with respect to disturbances of the parameter λ and distribution F. Throughout the paper, we make the assumption that μ : = ∫ 0 ∞ x d F ( x ) < ∞ and μ ˜ : = ∫ 0 ∞ x d F ˜ ( x ) < ∞ hold, as well as the net profit conditions: (6) ϕ = λ μ c < 1 , and ϕ ˜ = λ ˜ μ ˜ c < 1 .

In the classical risk model the number of claims N ( t ) in (1) follows a homogeneous Poisson process with intensity λ > 0. An extension of this is to add a diffusion term to account for additional uncertainties in the aggregate claims or the premium income. Gerber (1970) introduced the classical model perturbed by diffusion, with surplus process (12) U ( t ) = u + c t − S ( t ) + σ B ( t ) , t ≥ 0 , where the dispersion parameter σ > 0 and { B ( t ) : t ≥ 0 } is a standard Wiener process that is independent of the compound Poisson process { S ( t ) : t ≥ 0 } and of the individual claim sizes.

Dufresne and Gerber in [5] studied three kinds of probabilities based on (12): ψ d ( u ) = P r ( T < ∞ , U ( T ) = 0 | U ( 0 ) = u ) is the probability for ruin that is caused by oscillation, ψ s ( u ) = P r ( T < ∞ , U ( T ) < 0 | U ( 0 = u ) is the probability that ruin is caused by a claim and ψ t ( u ) = P r ( T < ∞ | U ( 0 ) = u ), the probability of ruin. We have that ψ t ( u ) = ψ d ( u ) + ψ s ( u ), with ψ d ( 0 ) = 1 and ψ s ( 0 ) = 1.

It was shown in [5] showed that ψ t ( u ) = P r ( L ∗ > u ) is the tail probability of the maximal aggregate loss L ∗ = m a x { u − U ( t ) : t ≥ 0 }. The r.v. L ∗ can be decomposed as L ∗ = L o , 0 + L c , 1 + ⋯ + L c , N + L o , N = ∑ n = 1 ∞ L o , n − 1 + L c , N + L o , N , with L ∗ = L o , 0 if N = 0, where L o , N and L c , N are the amounts that result in the ( n + 1 )-th and n-th record highs of the aggregate loss process { u − U ( t ) } due to oscillation and a claim, respectively, and N it the number of record highs of the process { u − U ( t ) } caused by a claim. In addition, the r.v’s L o , 0 , L o , 1 , L o , 2 ... are identically distributed (as L o ) with common distribution function H 1 ( u ) = 1 − e − ( c / D ) u , where D = σ 2 / 2 and L c , 1 , L c , 2 , L c , 3 ... are identically distributed (as L c ) with common distribution H 2 ( u ) = F e ( u ). Also, N , L o , 0 , L c , 1 , L o , 1 , L c , 2 , L o , 2 ... are independent. For further details and a probabilistic viewpoint of ψ t (see [5] and [29]).

Furthermore, Tsai in [28] showed that (13) K ‾ ( u ) = P r ( L K > u ) = 1 1 + θ ψ d ( u ) + ψ s ( u ) = ∑ n = 1 ∞ θ 1 + θ θ 1 + θ n A ∗ n ‾ ( u ) , u ≥ 0 , with K ‾ ( 0 ) = 1 / ( 1 + θ ), where A ‾ ( x ) = 1 − H 1 ∗ H 2 ( x ) = 1 − ∫ 0 x H 1 ‾ ( x − t ) d H 2 ( t ) is the distribution function of L o + L c with density a ( x ). Since L ∗ = L K + L o , we have that P r ( L ∗ > u ), the ruin probability for surplus process (12), is a compound geometric convolution, given by (14) ψ t ( u ) = K ∗ H 1 ‾ ( u ) = ∑ n = 0 ∞ θ 1 + θ θ 1 + θ n A ∗ n ∗ H 1 ‾ ( u ) , u ≥ 0 . Similarly, the expression for K ‾ ( u ) in (13) can be considered as K ‾ ( u ) = P r ( L K > u ) where L K = L o , 0 + L c , 1 + ⋯ + L o , N − 1 = ∑ n = 1 N L o , n − 1 + L c , n , with L K = 0 if N = 0. When the diffusion term is removed (i.e., σ = 0), then all L o s disappear, implying model (12) reduce to the non-perturbed classical risk model and both ψ t ( u ) and K ( u ) reduce to (3).

In this section, we provide proofs of all results stated in Sections 3 and 4. Proof of Theorem 3.4.

The following integral equation for ruin probabilities is commonly known (see e.g. [22]), (19) ψ ( u ) = λ c ∫ u ∞ F ‾ ( t ) d t + ∫ 0 u ψ ( u − t ) F ‾ ( t ) d t . It is easy to check that for the operators (20) T x ( u ) = λ c ∫ u ∞ F ‾ ( t ) d t + ∫ 0 u x ( u − t ) F ‾ ( t ) d t , u ≥ 0 , and (21) T ˜ x ( u ) = λ ˜ c ∫ u ∞ F ˜ ‾ ( t ) d t + ∫ 0 u x ( u − t ) F ˜ ‾ ( t ) d t , u ≥ 0 , we have T D γ ⊂ D γ and T ˜ D γ ⊂ D γ .

By (20) and (21) for every x , y ∈ D γ , ν γ ( T x , T y ) = λ c ∫ 0 ∞ ( 1 + u ) γ ∫ 0 u x ( u − t ) F ‾ ( t ) d t − ∫ 0 u y ( u − t ) F ‾ ( t ) d t d u ≤ λ c ∫ 0 ∞ ∫ 0 u ( 1 + u ) γ F ‾ ( t ) x ( u − t ) − y ( u − t ) d t d u = λ c ∫ 0 ∞ ∫ t ∞ ( 1 + u ) γ F ‾ ( t ) x ( u − t ) − y ( u − t ) d u d t , where the last equality was obtained by changing the order of integration. Hence, setting u = t + z and using the inequality ( 1 + z + t ) γ ≤ ( 1 + z ) γ ( 1 + t ) γ , for all z ≥ 0, t ≥ 0 and γ ≥ 0, it follows that ν γ ( T x , T y ) ≤ λ c ∫ 0 ∞ ∫ t ∞ ( 1 + u ) γ F ‾ ( t ) x ( u − t ) − y ( u − t ) d u d t = λ c ∫ 0 ∞ ∫ 0 ∞ ( 1 + z + t ) γ F ‾ ( t ) x ( z ) − y ( z ) d z d t ≤ λ c ∫ 0 ∞ ( 1 + t ) γ F ‾ ( t ) ∫ 0 ∞ ( 1 + z ) γ x ( z ) − y ( z ) d z d t = ν γ ( x , y ) λ c ∫ 0 ∞ ( 1 + t ) γ F ‾ ( t ) d t = ν γ ( x , y ) λ c M γ X . Therefore, the operators (20) and (21) are contractive on D γ with modules λ μ / c and λ ˜ μ ˜ / c, respectively, since net profit conditions hold (see equation (6)) Now, integrating by parts and noting that γ ≥ 0, we get M γ X = ∫ 0 ∞ ( 1 + t ) γ F ‾ ( t ) d t = F ‾ ( t ) ( 1 + t ) γ + 1 γ + 1 0 ∞ + ∫ 0 ∞ f ( t ) ( 1 + t ) γ + 1 γ + 1 d t = E ( X + 1 ) γ + 1 − 1 γ + 1 .

Also, it is well known that E ( | X | p ) < ∞ ⇔ E ( | X − a | p ) < ∞, ∀ a ∈ R and 0 < p < ∞. Therefore, M γ X exists if the moments E X ( γ + 1 ) of the claim-size distribution are finite for γ ≥ 0.

According to (19), ψ and ψ ˜ are the unique fixed points of T and T ˜ that is ψ = T ψ and ψ ˜ = T ˜ ψ ˜. Now, ν γ ( ψ , ψ ˜ ) = ν γ ( T ψ , T ˜ ψ ˜ ) ≤ ν γ ( T ψ , T ψ ˜ ) + ν γ ( T ψ ˜ , T ˜ ψ ˜ ) ≤ λ c M γ X ν γ ( ψ , ψ ˜ ) + ν γ ( T ψ ˜ , T ˜ ψ ˜ ) , or, (22) ν γ ( ψ , ψ ˜ ) ≤ c c − λ M γ X ν γ ( T ψ ˜ , T ˜ ψ ˜ ) , where M γ X = ∫ 0 ∞ ( 1 + t ) γ F ‾ ( t ) d t = 1 γ + 1 E ( X + 1 ) γ + 1 − 1 . In view of (20) and (21), for each ψ ∈ D γ , we have ν γ ( T ψ , T ˜ ψ ) ≤ ∫ 0 ∞ ( 1 + u ) γ λ c ∫ u ∞ F ‾ ( t ) d t + ∫ 0 u ψ ( u − t ) F ‾ ( t ) d t − λ c ∫ u ∞ F ˜ ‾ ( t ) d t + ∫ 0 u ψ ( u − t ) F ˜ ‾ ( t ) d t d u + ∫ 0 ∞ ( 1 + u ) γ λ − λ ˜ c ∫ u ∞ F ˜ ‾ ( t ) d t + ∫ 0 u ψ ( u − t ) F ˜ ‾ ( t ) d t d u = I 1 + I 2 . For the first term I 1 on the last inequality, we have I 1 ≤ λ c ∫ 0 ∞ ( 1 + u ) γ ∫ u ∞ F ‾ ( t ) − F ˜ ‾ ( t ) d t + ∫ 0 u ψ ( u − t ) F ‾ ( t ) − F ˜ ‾ ( t ) d t d u = λ c ∫ 0 ∞ ∫ 0 t ( 1 + u ) γ F ‾ ( t ) − F ˜ ‾ ( t ) d u d t + ∫ 0 ∞ ∫ t ∞ ( 1 + u ) γ ψ ( u − t ) F ‾ ( t ) − F ˜ ‾ ( t ) d u d t , where for the first term inside the square bracket it follows that (23) ∫ 0 ∞ F ‾ ( t ) − F ˜ ‾ ( t ) ∫ 0 t ( 1 + u ) γ d u d t = ∫ 0 ∞ ( 1 + t ) γ + 1 − 1 γ + 1 F ‾ ( t ) − F ˜ ‾ ( t ) d t ≤ 1 γ + 1 ν γ + 1 ( F , F ˜ ) , and (24) ∫ 0 ∞ ∫ t ∞ ( 1 + u ) γ ψ ( u − t ) F ‾ ( t ) − F ˜ ‾ ( t ) d u d t = ∫ 0 ∞ F ‾ ( t ) − F ˜ ‾ ( t ) × ∫ 0 ∞ ( 1 + z + t ) γ ψ ( z ) d z d t ≤ ∫ 0 ∞ ( 1 + t ) γ F ‾ ( t ) − F ˜ ‾ ( t ) d t × ∫ 0 ∞ ( 1 + z ) γ ψ ( z ) d z ≤ ν γ ( F , F ˜ ) M γ L . Therefore, by (23) and (24) it follows that (25) I 1 ≤ 1 γ + 1 ν γ + 1 ( F , F ˜ ) + ν γ ( F , F ˜ ) M γ L . Similarly, for the term I 2 , we have I 2 = | λ − λ ˜ | c ∫ 0 ∞ F ˜ ‾ ( t ) ∫ 0 t ( 1 + u ) γ d u d t + ∫ 0 ∞ F ˜ ‾ ( t ) ∫ t ∞ ( 1 + u ) γ ψ ( u − t ) d u d t = | λ − λ ˜ | c ∫ 0 ∞ F ˜ ‾ ( t ) ( 1 + t ) γ + 1 − 1 γ + 1 d t + ∫ 0 ∞ F ˜ ‾ ( t ) ∫ 0 ∞ ( 1 + z + t ) γ ψ ( z ) d z d t ≤ | λ − λ ˜ | c ∫ 0 ∞ F ˜ ‾ ( t ) ( 1 + t ) γ + 1 γ + 1 d t + M γ L ∫ 0 ∞ F ˜ ‾ ( t ) ( 1 + t ) γ d t ≤ | λ − λ ˜ | c ∫ 0 ∞ F ˜ ‾ ( t ) ( 1 + t ) γ + 1 d t + M γ L ∫ 0 ∞ F ˜ ‾ ( t ) ( 1 + t ) γ + 1 d t Hence, (26) I 2 ≤ | λ − λ ˜ | c M γ + 1 X ˜ 1 + M γ L . By (22), (25) and (26) the result follows.  □