Modern Stochastics: Theory and Applications

The notion of the renewal risk model traces back to Sparre Andersen [2], and it has played a key role in classical Risk Theory as a natural generalization of the classical Cramér–Lundberg risk model. In its standard framework both claims and interarrival times form two sequences of i.i.d. random variables, which are also assumed to be mutually independent. However, these independence assumptions can be too restrictive as far as actuarial applications are considered, and generalizations to dependence scenarios must be developed.

In practice, insurance portfolios tend to be inhomogeneous and a mixture of smaller homogeneous ones that are identified by the realizations of a random variable (or vector) Θ. Such portfolios are usually modelled via (compound) mixed counting processes, with the typical example being that of the (compound) mixed Poisson one (MPP for short). In the case of (compound) MPPs, the sequence of the interarrival times W is P-conditionally independent and P-conditionally exponentially distributed, something that leads to an interesting dependence structure between the interarrival times and reveals a connection between (compound) MPPs and exchangeability.

Mixed renewal processes (MRPs for short) serve as a proper generalization of MPPs, which has not been much considered in the literature, but they are interesting from both theoretical and applied point of view. Their study is mathematically challenging, as they are not, in general, Markov processes (see [14], Theorem 3, and [19], Proposition 3.2) and they have a close connection to exchangeable stochastic processes (see [14], Corollary on p. 20, and [17], Theorem 4.2). As far as practical models are considered, MRPs seem to have interesting applications in both life insurance (see, e.g., [8]) and nonlife insurance mathematics (see, e.g., [29] and [30]). It is worth noticing that Segerdahl [29] was the first who considered MRPs, in an actuarial context, as an alternative to the classical Pólya–Lundberg process (i.e. a mixed Poisson-gamma one).

The change of measures technique has been successfully applied to various theories such as queues and fluid flows (Asmussen [3, 4], Palmowski and Rolski [21, 22]), ruin theory (Dassios and Embrechts [9], Asmussen [3], Asmussen and Albrecher [5], Schmidli [25–27]), simulation (Boogaert and De Waegenaere [6], Ridder [23]) and pricing of insurance risks (premium calculation principles) (Delbaen and Haezendonck [10], Lyberopoulos and Macheras [16], Macheras and Tzaninis [20, 31]). The process of interest is usually Markovian and, under a suitably chosen new probability measure, it is again a Markov process with some “nicer” desired properties.

In [31], the same problem was investigated for the class of compound MRPs. In the same paper, given an aggregate claims process S being a MRP under P, a full characterization of all probability measures Q, which are progressively equivalent to P and preserve the structure of S, but with some better desired properties, was provided, see [31] Theorem 4.5 and Corollary 4.8 as well as Proposition 4.15. In the present paper by utilizing the aforementioned change of measures technique an explicit formula and bounds for the probability of ruin in a mixed renewal risk model are obtained.

Part of Proposition 4.15 from [31] is Proposition 3.1, formulated for the purposes of the present paper and being the starting point for applications to the ruin problem. Proposition 3.1, as well as Proposition 3.2, extend the corresponding results for the renewal risk model (see, e.g., [27], Lemmas 8.4 and 8.6, respectively) to the compound MRPs.

A first consequence of Proposition 3.1 is Proposition 4.1, where an upper bound for the ruin probability within finite time is obtained. Another implication of Proposition 3.1 is Proposition 4.2, where it is proven, that if the net profit condition is fulfilled under an original measure P, and S is a compound MRP under P then under the new measure resulting from Proposition 3.1 the process S will be of the same type, except that the net profit condition will no longer be fulfilled and ruin will always occur within finite time. Thus, the ruin problem becomes easier to handle, since by Proposition 4.2 under the new measure the probability of ruin is equal to 1, something that gives the opportunity to express the ruin probability under P as a quantity under the new measure, see Theorem 4.1, and to find upper and lower bounds for it, see Corollary 4.2.

Throughout this paper, unless stated otherwise, $(\varOmega ,\varSigma ,P)$ is an arbitrary but fixed probability space. Given a topology $\mathfrak{T}$ on Ω write $\mathfrak{B}(\varOmega )$ for its Borel σ-algebra on Ω, i.e. the σ-algebra generated by $\mathfrak{T}$. The measure theoretic terminology is standard and generally follows [7]. For the definitions of real-valued random variables and random variables, cf., e.g., [7], p. 308. Notation ${P_{X}}:={P_{X}}(\theta ):=\mathbf{K}(\theta )$ denotes that X is distributed according to the law $\mathbf{K}(\theta )$, where $\theta \in D\subseteq {\mathbb{R}^{d}}$ ($d\in \mathbb{N}$) is the parameter of the distribution. Denote again by $\mathbf{K}(\theta )$ the distribution function induced by the probability distribution $\mathbf{K}(\theta )$. Notation $\mathbf{Ga}(b,a)$, where $a,b\in (0,\infty )$, stands for the law of gamma distribution (cf., e.g., [28], p. 180). In particular, $\mathbf{Ga}(b,1)=\mathbf{Exp}(b)$ stands for the law of exponential distribution. For two real-valued random variables X and Y write $X=Y$ P-a.s. if $\{X\ne Y\}$ is a P-null set. If $A\subseteq \varOmega $, then ${A^{c}}:=\varOmega \setminus A$, while ${\chi _{A}}$ denotes the indicator (or characteristic) function of the set A. For a map $f:D\to E$ and for a nonempty set $A\subseteq D$ denote by $f\upharpoonright A$ the restriction of f to A. Write ${\mathbb{E}_{P}}[X\mid \mathcal{F}]$ for a version of a conditional expectation (under P) of a P-integrable random variable X given a σ-subalgebra $\mathcal{F}$ of Σ. For $X:={\chi _{E}}$ with $E\in \varSigma $ set $P(E\mid \mathcal{F}):={\mathbb{E}_{P}}[{\chi _{E}}\mid \mathcal{F}]$. For the unexplained terminology of Probability and Risk Theory, see [28].

Given two measurable spaces $(\varOmega ,\varSigma )$ and $(\varUpsilon ,H)$, a function k from $\varOmega \times H$ into $[0,1]$ is a Σ-H-Markov kernel if it has the following properties: In particular, given a real-valued random variable X on Ω and a d-dimensional random vector Θ on Ω, a conditional distribution of X over Θ is a $\sigma (\varTheta )$-$\mathfrak{B}$-Markov kernel denoted by ${P_{X\mid \varTheta }}:={P_{X\mid \sigma (\varTheta )}}$ and satisfying for each $B\in \mathfrak{B}$ the condition

Clearly, for every ${\mathfrak{B}_{d}}$-$\mathfrak{B}$-Markov kernel k, the map $K(\varTheta )$ from $\varOmega \times \mathfrak{B}$ into $[0,1]$ defined by means of is a $\sigma (\varTheta )$-$\mathfrak{B}$-Markov kernel. Then for $\theta =\varTheta (\omega )$ with $\omega \in \varOmega $ the probability measures $k(\theta ,\bullet )$ are distributions on $\mathfrak{B}$ and so one may write $\mathbf{K}(\theta )(\bullet )$ instead of $k(\theta ,\bullet )$. Consequently, in this case $K(\varTheta )$ will be denoted by $\mathbf{K}(\varTheta )$.

For any real-valued random variables $X,Y$ on Ω the conditional distributions ${P_{X|\varTheta }}$ and ${P_{Y|\varTheta }}$ are $P\upharpoonright \sigma (\varTheta )$-equivalent (in symbols, ${P_{X|\varTheta }}={P_{Y|\varTheta }}$ $P\upharpoonright \sigma (\varTheta )$-a.s.), if there exists a P-null set $M\in \sigma (\varTheta )$ such that for any $\omega \notin M$ and $B\in \mathfrak{B}$ the equality ${P_{X|\varTheta }}(\omega ,B)={P_{Y|\varTheta }}(\omega ,B)$ holds true.

For the definition of a P-conditionally (stochastically) independent process over $\sigma (\varTheta )$ as well as of a P-conditionally identically distributed process over $\sigma (\varTheta )$, cf., e.g., [31], p. 4. Recall that a process is P-conditionally (stochastically) independent or identically distributed given Θ, if it is conditionally independent or identically distributed over the σ-algebra $\sigma (\varTheta )$.

Henceforth, unless stated otherwise, Θ is a d-dimensional random vector on Ω with values in $D\subseteq {\mathbb{R}^{d}}$ ($d\in \mathbb{N}$). Furthermore, simply write “conditionally” in the place of “conditionally given Θ” whenever conditioning refers to Θ.

A family $N:={\{{N_{t}}\}_{t\in {\mathbb{R}_{+}}}}$ of random variables from $(\varOmega ,\varSigma )$ into $(\overline{\mathbb{R}},\mathfrak{B}(\overline{\mathbb{R}}))$ is called a counting (or claim number) process, if there exists a P-null set ${\varOmega _{N}}\in \varSigma $ such that the process N restricted on $\varOmega \setminus {\varOmega _{N}}$ takes values in ${\mathbb{N}_{0}}\cup \{\infty \}$, has right-continuous paths, presents jumps of size (at most) one, vanishes at $t=0$ and increases to infinity. Without loss of generality one may and do assume, that ${\varOmega _{N}}=\varnothing $. Denote by $T:={\{{T_{n}}\}_{n\in {\mathbb{N}_{0}}}}$ and $W:={\{{W_{n}}\}_{n\in \mathbb{N}}}$ the (claim) arrival process and (claim) interarrival process, respectively (cf., e.g., [28], Section 1.1, p. 6, for the definitions) associated with N. Note also that every arrival process induces a counting process, and vice versa (cf., e.g., [28], Theorem 2.1.1).

Furthermore, let $X:={\{{X_{n}}\}_{n\in \mathbb{N}}}$ be a sequence of positive real-valued random variables on Ω, and for any $t\ge 0$ define Accordingly, the sequence X is said to be the claim size process, and the family $S:={\{{S_{t}}\}_{t\in {\mathbb{R}_{+}}}}$ of real-valued random variables on Ω is said to be the aggregate claims process induced by the pair $(N,X)$. Recall that a pair $(N,X)$ is called a risk process, if N is a counting process, X is P-i.i.d. and the processes N and X are P-independent (see [28], Chapter 6, Section 6.1).

Recall that a counting process N is a P-mixed renewal process with mixing parameter Θ and interarrival time conditional distribution $\mathbf{K}(\varTheta )$ (written P-MRP$(\mathbf{K}(\varTheta ))$ for short), if the induced interarrival process W is P-conditionally independent and (see also [17], Definition 3.1, or [19], Definition 3.2(b)). In particular, if the distribution ${P_{\varTheta }}$ of Θ is degenerate at some point ${\theta _{0}}\in D$, then the counting process N becomes a P-renewal process with interarrival time distribution $\mathbf{K}({\theta _{0}})$ (written P-RP$(\mathbf{K}({\theta _{0}}))$ for short).

Accordingly, an aggregate claims process S induced by a P-risk process $(N,X)$ such that N is a P-MRP$(\mathbf{K}(\varTheta ))$ is called a compound mixed renewal process with parameters $\mathbf{K}(\varTheta )$ and ${P_{{X_{1}}}}$ (P-CMRP$(\mathbf{K}(\varTheta ),{P_{{X_{1}}}})$ for short). In particular, if ${P_{\varTheta }}$ is degenerate at ${\theta _{0}}\in D$, then S is called a compound renewal process with parameters $\mathbf{K}({\theta _{0}})$ and ${P_{{X_{1}}}}$ (P-CRP$(\mathbf{K}({\theta _{0}}),{P_{{X_{1}}}})$ for short).

Throughout what follows denote again by $\mathbf{K}(\varTheta )$ and $\mathbf{K}(\theta )$ the conditional distribution function and the distribution function induced by the conditional probability distribution $\mathbf{K}(\varTheta )$ and the probability distribution $\mathbf{K}(\theta )$, respectively.

The following conditions for the quadruplet $(P,W,X,\varTheta )$ (or, if no confusion arises, for the probability measure P) will be useful throughout the paper:

Since conditioning is involved in the definition of (compound) mixed renewal processes, it is natural to expect that regular conditional probabilities (or disintegrations) will play a key. To this purpose recall the following definition (cf., e.g., [31], Definition 3.2).

The family ${\{{P_{\theta }}\}_{\theta \in D}}$ is consistent with Θ if, for each $B\in \mathfrak{B}(D)$, the equality ${P_{\theta }}({\varTheta ^{-1}}(B))=1$ holds for ${P_{\varTheta }}$-almost every $\theta \in B$.

A rcp ${\{{P_{\theta }}\}_{\theta \in D}}$ of P over ${P_{\varTheta }}$ consistent with Θ is essentially unique, if for any other rcp ${\{{\widetilde{P}_{\theta }}\}_{\theta \in D}}$ of P over ${P_{\varTheta }}$ consistent with Θ there exists a ${P_{\varTheta }}$-null set $N\in \mathfrak{B}(D)$ such that for any $\theta \notin N$ the equality ${P_{\theta }}={\widetilde{P}_{\theta }}$ holds true.

Regular conditional probabilities seem to have a bad reputation when it comes to applications, and that is probably due to the facts that their own existence is not always guaranteed (see [18], Examples 4 and 5) and their construction usually involves manipulations with Radon–Nikodým derivatives. Nevertheless, as the spaces used in applied Probability Theory are mainly Polish ones, such rcps always exist (see [11], Theorem 6), and in fact they can be explicitly constructed for the class of (compound) mixed renewal processes (see [31], Proposition 4.1).

From now on the family ${\{{P_{\theta }}\}_{\theta \in D}}$ is a rcp of P over ${P_{\varTheta }}$ consistent with Θ.

Let S be the aggregate claims process induced by the counting process N and the claim size process X. Fix on arbitrary $u\in \varUpsilon $ and $t\in {\mathbb{R}_{+}}$, and define the function ${r_{t}^{u}}:\varOmega \times D\to \mathbb{R}$ by means of ${r_{t}^{u}}(\omega ,\theta ):=u+c(\theta )\cdot t-{S_{t}}(\omega )$ for any $(\omega ,\theta )\in \varOmega \times D$, where c is a positive $\mathfrak{B}(D)$-measurable function. For arbitrary but fixed $\theta \in D$, the process ${r^{u}}(\theta ):={\{{r_{t}^{u}}(\theta )\}_{t\in {\mathbb{R}_{+}}}}$ defined by ${r_{t}^{u}}(\theta ):={r_{t}^{u}}(\omega ,\theta )$ for any $\omega \in \varOmega $, is called the reserve process induced by the initial reserve u, the premium intensity or premium rate $c(\theta )$ and the aggregate claims process S (see [28], Section 7.1, pp. 155–156, for the definition).

Define the real-valued function ${R_{t}^{u}}(\varTheta )$ on Ω by means of ${R_{t}^{u}}(\varTheta ):={r_{t}^{u}}\circ (i{d_{\varOmega }}\times \varTheta )$. The process ${R^{u}}(\varTheta ):={\{{R_{t}^{u}}(\varTheta )\}_{t\in {\mathbb{R}_{+}}}}$ is called the reserve process induced by the initial reserve u, the stochastic premium intensity or stochastic premium rate $c(\varTheta )$ and the aggregate claims process S.

$\mathcal{F}:={\{{\mathcal{F}_{t}}\}_{t\in {\mathbb{R}_{+}}}}$, where ${\mathcal{F}_{t}}:=\sigma ({\mathcal{F}_{t}^{S}}\cup \sigma (\varTheta ))$, denotes the canonical filtration generated by S and Θ; ${\mathcal{F}_{\infty }^{S}}:=\sigma ({\textstyle\bigcup _{t\in {\mathbb{R}_{+}}}}{\mathcal{F}_{t}^{S}})$ and ${\mathcal{F}_{\infty }}:=\sigma ({\mathcal{F}_{\infty }^{S}}\cup \sigma (\varTheta ))$. For the definition of a $(P,\mathcal{Z})$-martingale, where $\mathcal{Z}={\{{\mathcal{Z}_{t}}\}_{t\in {\mathbb{R}_{+}}}}$ is a filtration for $(\varOmega ,\varSigma )$, cf., e.g., [28], p. 25. A $(P,\mathcal{Z})$-martingale ${\{{Z_{t}}\}_{t\in {\mathbb{R}_{+}}}}$ is P-a.s. positive, if ${Z_{t}}$ is P-a.s. positive for each $t\ge 0$. For $\mathcal{Z}=\mathcal{F}$ write “martingale” instead of “$(P,\mathcal{Z})$-martingale”, for simplicity.

Henceforth, $\varUpsilon :=(0,\infty )$, $\varOmega :={\varUpsilon ^{\mathbb{N}}}\times {\varUpsilon ^{\mathbb{N}}}\times D$, $\varSigma :=\mathfrak{B}(\varOmega )$, $\mathbf{K}(\varTheta )$ and ${\{{P_{\theta }}\}_{\theta \in D}}$ are as in [31], Proposition 4.1, $P\in {\mathcal{M}_{S,\mathbf{K}(\varTheta )}}$, and assume that ${\mathbb{E}_{P}}[{W_{1}}|\varTheta ]\in \varUpsilon \hspace{2.5pt}P\upharpoonright \sigma (\varTheta )$-a.s.

The following proposition is a part of Proposition 4.15 from [31]. Since it is the basic tool for the proofs of the results, it is restated exactly in the form needed for the purposes of the present paper.

In the following example Proposition 3.1 is applied for an initial probability measure $P\in {\mathcal{M}_{S,\mathbf{Exp}(\varTheta )}}$, where Θ is a positive real valued random variable, i.e. when S is a compound mixed Poisson process (cf., e.g., [16], p. 4, for its definition).

The following proposition extends Lemma 8.6 of [27].

In this section the change of measures technique for compound mixed renewal processes appearing in Proposition 3.1 is applied to the ruin problem. In the first result a bound for the finite time ruin probability is proven. In order to present it denote by $\tau :={\tau _{u}}$ the ruin time of the reserve process ${R^{u}}(\varTheta )$ ($u\in {\mathbb{R}_{+}}$) (cf., e.g., [27], p. 84, for the definition) and by $\psi (u,t):=P(\{{\inf _{v\le t}}{R_{v}^{u}}(\varTheta )<0\})=P(\{\tau \le t\})$ the finite time ruin probability (cf., e.g., [5], p. 115) for the reserve process ${R^{u}}(\varTheta )$ with respect to P.

Recall that for any arbitrary but fixed $\theta \in D$ the function ${\psi _{\theta }}:\varUpsilon \to [0,1]$ defined by ${\psi _{\theta }}(u):={P_{\theta }}(\{{\inf _{t\in {\mathbb{R}_{+}}}}{r_{t}^{u}}(\theta )<0\})$ is called the probability of ruin for the reserve process ${r^{u}}(\theta )$ with respect to ${P_{\theta }}$ (see [28], Section 7.1, p. 158, for the definition). The function $\psi :\varUpsilon \to [0,1]$ defined by $\psi (u):=P(\{{\inf _{t\in {\mathbb{R}_{+}}}}{R_{t}^{u}}(\varTheta )<0\})$ is called the probability of ruin for the reserve process ${R^{u}}(\varTheta )$ with respect to P.

When considering infinite time ruin probabilities in a mixed renewal risk model one has to assume that the conditional net profit condition holds, in order to avoid a P-a.s. ruin (see [30], Lemma 5.3). Note that since S is a P-CMRP$(\mathbf{K}(\varTheta ),{P_{{X_{1}}}})$ and conditions (a1) and (a2) are valid, one may apply [31], Proposition 3.2, along with [15], Lemma 3.5, in order to show that condition ($NP{C_{\varTheta }}$) is equivalent to where ${L_{\ast \ast }}\in \mathfrak{B}(D)$ is the ${P_{\varTheta }}$-null set appearing in Proposition 3.1. Fix an arbitrary $\theta \notin {L_{\ast \ast }}$. If condition ($NP{C_{\theta }}$) holds true for any $\theta \notin {L_{\ast \ast }}$ and $r\in {\mathbb{R}_{+}}$ is as in Proposition 3.1, it follows by, e.g., [27], p. 133, that there exists an adjustment coefficient $R(\theta )\in \varUpsilon $ with respect to ${P_{\theta }}$.

Throughout what follows assume that condition ($NP{C_{\varTheta }}$) holds true and that $R(\theta )\in \varUpsilon $ is an adjustment coefficient with respect to ${P_{\theta }}$ for any $\theta \notin {L_{\ast \ast }}$.

The next result is an immediate consequence of Proposition 4.1 and extends the celebrated Lundberg inequality to the case of CMRPs.

The proof follows immediately by Proposition 4.1 for $\underline{y}=0$, $\overline{y}\to \infty $ and $r=R(\theta )$.

It should be clear by Proposition 4.1 and Corollary 4.1 that one cannot, in general, expect to obtain exponential bounds in the case of CMRPs, as the adjustment coefficient depends on the random vector Θ. However, applying Proposition 3.1 one can obtain an explicit formula for the probability of ruin in infinite time, see Theorem 4.1, which can lead to exponential bounds, see Corollary 4.2. In order to formulate Theorem 4.1, one needs to establish the validity of the following proposition, which is a consequence of Proposition 3.1, and allows one to construct a probability measure ${Q^{{R^{\ast }}}}$, being singular to the original probability measure P and such that ruin occurs ${Q^{{R^{\ast }}}}$-a.s.

In the following example, the assumptions ${R^{\ast }}<\infty $ and ${\mathbb{E}_{P}}\big[{e^{{R^{\ast }}{X_{1}}}}\big]<\infty $ of Proposition 4.2 hold.

In the example that follows, an explicit formula for the probability of ruin in the interesting case of constant premiums is obtained by applying Theorem 4.1.

The following result shows that Theorem 4.1, along with Proposition 3.1, yields upper and lower bounds of the probability of ruin under P.

It is worth noting that in the Cramér–Lundberg risk model one can construct exponential martingales, and using the stopping theorem one is able to prove upper bounds for the ruin probabilities. However, this technique does not give the opportunity to prove a lower bound. A method to find also lower bounds for the ruin probabilities is the “change of measure technique” for a compound mixed renewal process S developed above.

In the next two examples, an explicit computation for the ruin probabilities of the reserve process with respect to the probability measures P and ${P_{\theta }}$ is undertaken by applying the change of measures technique of Proposition 3.1 and Theorem 4.1.