Modern Stochastics: Theory and Applications

Given a price process ${U_{\mathbb{T}}}:={\{{U_{t}}\}_{t\in \mathbb{T}}}$, where $\mathbb{T}:=[0,T]$, $T\gt 0$, on a probability space $(\varOmega ,\varSigma ,P)$, a basic method in Mathematical Finance is to replace the initial probability measure P by an equivalent one Q, which converts ${U_{\mathbb{T}}}$ into a martingale with respect to Q. The new probability measure, often called risk-neutral or equivalent martingale measure (written EMM for short), see Section 3 for the definition, is then used for pricing and hedging contingent claims (e.g., options, futures, etc.). Note that such a method for pricing contingent claims is originated from the field of Actuarial Science (see Delbaen and Schachermayer [9], pages 149–150 for more details). However, in contrast to the situation of the classical Black–Scholes option pricing formula, where the EMM is unique, in Actuarial Mathematics which is certainly not the case, as the insurance market is not, in general, complete (see, e.g., Sondermann [31], Section 4). Thus, if ${U_{\mathbb{T}}}$ represents the liabilities of an insurance company, then there exist infinitely many equivalent martingale measures for ${U_{\mathbb{T}}}$, so that pricing is directly linked with an attitude towards risk, see [8], pages 269–270, for more details. The latter led Delbaen and Haezendonck [8] to a positive answer to the problem of characterizing all those EMMs Q which preserve the structure of a given compound Poisson process under P, see [8], Proposition 2.2. The work of Delbaen and Haezendonck played a key role in understanding the interplay between financial and actuarial pricing of insurance (see Embrechts [10] for an overview), and has influenced the studies of many researchers (see [22], page 44, and the references therein for more details). Nevertheless, such a characterization of EMMs for ${U_{\mathbb{T}}}$ does not always provide a viable pricing system in actuarial practice, since it is not appropriate for describing inhomogeneous risk portfolios. For this reason, the work of Delbaen and Haezendonck [8] was generalized by Embrechts and Meister [11] and Lyberopoulos and Macheras [19, 20] to mixed Poisson risk models. However, since the (mixed) Poisson risk model presents some serious deficiencies as far as practical models are considered (see [22], page 44, and [33], page 226, and the references therein), it seems reasonable to investigate the existence of EMMs for the price process ${U_{\mathbb{T}}}$ in the more general mixed renewal risk model.

In [33], Corollary 4.8, a characterization of all progressively equivalent probability measures that convert a compound mixed renewal process (CMRP for short) into a compound mixed Poisson process (CMPP for short) was proven. In Section 3, relying on the above result, we provide a characterization of all progressively EMMs Q for a canonical price process that convert a CMRP under P into a CMPP under Q, see Theorem 1. This theorem generalizes corresponding results of [19], Proposition 5.1(ii), and [22], Proposition 4.2. A first consequence of Theorem 1 is Theorem 2, where we find out a wide class of canonical price processes, inducing a corresponding class of EMMs, satisfying the condition of no free lunch with vanishing risk (written (NFLVR) for short), connecting in this way our results with this basic notion of Mathematical Finance.

Another implication of Theorem 1 concerning the ruin problem is discussed in Section 4, where for a given reserve process ${R^{u}}(\varTheta )$ induced by the initial reserve u, the stochastic premium intensity $c(\varTheta )$ and the aggregate claims process S (see Definition 2), we characterize all those progressively equivalent to P ℓ-martingale measures Q that convert a CMRP under P into a CMPP under Q in such a way that ruin for ${R^{u}}(\varTheta )$ occurs Q-a.s., see Theorem 3. In Section 5, we discuss some implications of our results to the pricing of actuarial risks (premium calculation principles) in an arbitrage-free insurance market. In Section 6 we present some concrete examples demonstrating how to construct mixed premium calculation principles (see Section 5 for the definition) in an insurance market possessing the property of (NFLVR) and how to obtain explicit formulas for their corresponding ruin probabilities. Finally, in the Appendix a list of symbols is presented.

$\mathbb{N}$, $\mathbb{Q}$ and $\mathbb{R}$ stand for the natural, the rational and the real numbers, respectively, while ${\mathbb{N}_{0}}:=\mathbb{N}\cup \{0\}$ and ${\mathbb{R}_{+}}:=\{x\in \mathbb{R}:x\ge 0\}$. If $d\in \mathbb{N}$ then ${\mathbb{R}^{d}}$ denotes the Euclidean space of dimension d. For a map $f:A\to E$ and for a nonempty set $B\subseteq A$ write $f\upharpoonright B$ for the restriction of f to B and ${1_{B}}$ for the indicator function of the set B.

Throughout this paper, unless stated otherwise, $(\varOmega ,\varSigma ,P)$ is a fixed but arbitrary probability space and $\varTheta :\varOmega \to D\subseteq {\mathbb{R}^{d}}$ ($d\in \mathbb{N}$) is a d-dimensional random vector. By ${\mathcal{L}^{\ell }}(P)$ we denote the family of all Σ-measurable real-valued functions f on Ω such that $\textstyle\int |f{|^{\ell }}\hspace{0.2778em}dP\lt \infty $ ($\ell \in \{1,2\}$). For any Hausdorff topology $\mathfrak{T}$ over Ω, by $\mathfrak{B}(\varOmega )$ is denoted the Borel σ-algebra on Ω, i.e., the σ-algebra generated by $\mathfrak{T}$, while $\mathfrak{B}:=\mathfrak{B}(\mathbb{R})$ and ${\mathfrak{B}_{d}}:=\mathfrak{B}({\mathbb{R}^{d}})$, where $d\in \mathbb{N}$, stand for the Borel σ-algebras of subsets of $\mathbb{R}$ and ${\mathbb{R}^{d}}$, respectively, generated by the Euclidean topology $\mathfrak{E}$ over $\mathbb{R}$ and by the product topology ${\mathfrak{E}^{d}}$ over ${\mathbb{R}^{d}}$, respectively. For any $J\subseteq {\mathbb{R}^{d}}$ denote by $\mathfrak{B}(J)$ the σ-algebra on J generated by the topology ${\mathfrak{E}^{d}}\cap J:=\{J\cap G:G\in {\mathfrak{E}^{d}}\}$. In particular, write $\mathfrak{B}(0,\infty ):=\mathfrak{B}((0,\infty ))$, for simplicity. Our measure-theoretic terminology is standard and generally follows [4]. For the definitions of real-valued random variables and random variables we refer to [4], page 308. We apply the notation ${P_{X}}:={P_{X}}(\theta ):=\mathbf{K}(\theta )$ to mean that X is distributed according to the law $\mathbf{K}(\theta )$, where $\theta \in D\subseteq {\mathbb{R}^{d}}$ is the parameter of the distribution. Notation $\mathbf{Ga}(b,a)$, where $a,b\in (0,\infty )$, stands for the law of gamma distribution (cf., e.g., [30], page 180). In particular, $\mathbf{Ga}(b,1)=\mathbf{Exp}(b)$ stands for the law of exponential distribution. For the unexplained terminology of Probability and Risk Theory we refer to [30].

Given a random variable X, a conditional distribution of X over Θ is a $\sigma (\varTheta )$-$\mathfrak{B}$-Markov kernel (see [3], Definition 36.1 for the definition) denoted by ${P_{X\mid \varTheta }}:={P_{X\mid \sigma (\varTheta )}}$ and satisfying for each $B\in \mathfrak{B}$ the equality ${P_{X\mid \varTheta }}(\bullet ,B)=P({X^{-1}}(B)\mid \sigma (\varTheta ))(\bullet )$ $P\upharpoonright \sigma (\varTheta )$-almost surely (written a.s. for short). Clearly, for every $\mathfrak{B}({\mathbb{R}^{d}})$-$\mathfrak{B}$-Markov kernel k, the map $K(\varTheta )$ from $\varOmega \times \mathfrak{B}$ into $[0,1]$ defined by 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 we 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 we say that ${P_{X|\varTheta }}$ and ${P_{Y|\varTheta }}$ are $P\upharpoonright \sigma (\varTheta )$-equivalent and we write ${P_{X|\varTheta }}={P_{Y|\varTheta }}$ $P\upharpoonright \sigma (\varTheta )$-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.

A sequence ${\{{V_{n}}\}_{n\in \mathbb{N}}}$ of real-valued random variables on Ω is said to be: We say that the process ${\{{V_{n}}\}_{n\in \mathbb{N}}}$ is P-conditionally (stochastically) independent or identically distributed given Θ, if it is conditionally independent or identically distributed over the σ-algebra $\sigma (\varTheta )$.

For the definitions of a counting (or claim number) process $N:={\{{N_{t}}\}_{t\in {\mathbb{R}_{+}}}}$, an arrival process $T:={\{{T_{n}}\}_{n\in {\mathbb{N}_{0}}}}$ induced by N, an interarrival process $W:={\{W\}_{n\in \mathbb{N}}}$ induced by T, a claim size process $X:={\{{X_{n}}\}_{n\in \mathbb{N}}}$ and an aggregate claims process $S:={\{{S_{t}}\}_{t\in {\mathbb{R}_{+}}}}$ induced by N and X, we refer to [30]. We assume that $P(\{{X_{n}}\gt 0\})=1$ for all $n\in \mathbb{N}$. 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-mutually independent (see [30], page 127).

Recall that a counting process N is said to be 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 interarrival process W is P-conditionally independent and for all $n\in \mathbb{N}$ the condition ${P_{{W_{n}}\mid \varTheta }}=\mathbf{K}(\varTheta )$ $P\upharpoonright \sigma (\varTheta )$-a.s. is valid (see [21], Definition 3.1). In particular, if the distribution 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). If N is a P-MRP$(\mathbf{K}(\varTheta ))$ then according to [33], Corollary 3.5(ii), it has zero probability of explosion, i.e., $P(\{{\sup _{n\in {\mathbb{N}_{0}}}}{T_{n}}\lt \infty \})=0$.

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). If Θ is a real-valued random variable and $\mathbf{K}(\varTheta )=\mathbf{Exp}(\varTheta )$, we say that S is a compound mixed Poisson process with parameters Θ and ${P_{{X_{1}}}}$ and write P-CMPP$(\varTheta ,{P_{{X_{1}}}})$. In particular, if the distribution of Θ 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). In the special case ${\theta _{0}}\gt 0$ and $\mathbf{K}({\theta _{0}})=\mathbf{Exp}({\theta _{0}})$ we say that S is a compound Poisson process with parameters ${\theta _{0}}$ and ${P_{{X_{1}}}}$ and write P-CPP$({\theta _{0}},{P_{{X_{1}}}})$.

Throughout what follows we 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.

Since conditioning is involved in the definition of (compound) mixed renewal processes, it is expected that regular conditional probabilities will play a fundamental role in their analysis. To this purpose, recall that if $(Z,H,R)$ is a probability space, then a family ${\{{P_{z}}\}_{z\in Z}}$ of probability measures on Σ is called a regular conditional probability (rcp for short) of P over R if for any fixed $E\in \varSigma $ the map $Z\ni z\mapsto {P_{z}}(E)$ is H-measurable, and $\textstyle\int {P_{z}}(E)\hspace{0.1667em}R(dz)=P(E)$ for every $E\in \varSigma $. If $f:\varOmega \to Z$ is an inverse-measure-preserving function (i.e., $P({f^{-1}}(B))=R(B)$ for each $B\in H$), an rcp ${\{{P_{z}}\}_{z\in Z}}$ of P over R is called consistent with f if, for each $B\in H$, the equality ${P_{z}}({f^{-1}}(B))=1$ holds for R-almost every $z\in B$ (cf., e.g., [33], Definition 3.3). We say that an rcp ${\{{P_{z}}\}_{z\in Z}}$ of P over R consistent with f is essentially unique, if for any other rcp ${\{{\widetilde{P}_{z}}\}_{z\in Z}}$ of P over R consistent with f there exists a R-null set $M\in H$ such that for any $z\notin M$ the equality ${P_{z}}={\widetilde{P}_{z}}$ holds true.

From now on $(Z,H,R):=(D,\mathfrak{B}(D),{P_{\varTheta }})$ and the family ${\{{P_{\theta }}\}_{\theta \in D}}$ is an rcp of P over ${P_{\varTheta }}$ consistent with Θ.

Let $\mathbb{T}\subseteq {\mathbb{R}_{+}}$. For a process ${Y_{\mathbb{T}}}:={\{{Y_{t}}\}_{t\in \mathbb{T}}}$ denote by ${\mathcal{F}_{\mathbb{T}}^{Y}}:={\{{\mathcal{F}_{t}^{Y}}\}_{t\in \mathbb{T}}}$ the canonical filtration of ${Y_{\mathbb{T}}}$. For $\mathbb{T}={\mathbb{R}_{+}}$ or $\mathbb{T}=\mathbb{N}$ we simply write ${\mathcal{F}^{Y}}$ instead of ${\mathcal{F}_{{\mathbb{R}_{+}}}^{Y}}$ or ${\mathcal{F}_{\mathbb{N}}^{Y}}$, respectively. Also, we write $\mathcal{F}:={\{{\mathcal{F}_{t}}\}_{t\in {\mathbb{R}_{+}}}}$, where ${\mathcal{F}_{t}}:=\sigma ({\mathcal{F}_{t}^{S}}\cup \sigma (\varTheta ))$, for the canonical filtration of S and Θ, ${\mathcal{F}_{\infty }^{S}}:=\sigma ({\textstyle\bigcup _{t\in {\mathbb{R}_{+}}}}{\mathcal{F}_{t}^{S}})$ and ${\mathcal{F}_{\infty }}:=\sigma ({\mathcal{F}_{\infty }^{S}}\cup \sigma (\varTheta ))$.

Let Q be a probability measures on $(\varOmega ,\varSigma )$. We say that P and Q are progressively equivalent, if P and Q are equivalent (in the sense of absolute continuity) on ${\mathcal{F}_{t}}$ (in symbols $Q\upharpoonright {\mathcal{F}_{t}}\sim P\upharpoonright {\mathcal{F}_{t}}$) for any $t\in {\mathbb{R}_{+}}$, see [33], Definition 3.1. If P and Q are equivalent on Σ we write $P\sim Q$.

The following conditions will be useful for our investigations:

Next, whenever condition (a1) or (a2) holds true we shall write that the quadruplet $(P,W,X,\varTheta )$ or (if no confusion arises) the probability measure P satisfies (a1) or (a2), respectively.

Recall that, for given $\mathbb{T}\subseteq {\mathbb{R}_{+}}$ a martingale in ${\mathcal{L}^{\ell }}(P)$ adapted to the filtration ${\mathcal{Z}_{\mathbb{T}}}:={\{{\mathcal{Z}_{t}}\}_{t\in \mathbb{T}}}$, or else a ${\mathcal{Z}_{\mathbb{T}}}$-martingale in ${\mathcal{L}^{\ell }}(P)$, is a family ${Z_{\mathbb{T}}}:={\{{Z_{t}}\}_{t\in \mathbb{T}}}$ of random variables in ${\mathcal{L}^{\ell }}(P)$ such that ${Z_{t}}$ is ${\mathcal{Z}_{t}}$-measurable for each $t\in \mathbb{T}$, and whenever $u\le t$ in $\mathbb{T}$ and $E\in {\mathcal{Z}_{u}}$ then ${\textstyle\int _{E}}{Z_{u}}dP={\textstyle\int _{E}}{Z_{t}}dP$. For ${\mathcal{Z}_{{\mathbb{R}_{+}}}}=\mathcal{F}$ we simply write that Z is a martingale in ${\mathcal{L}^{\ell }}(P)$. A ${\mathcal{Z}_{\mathbb{T}}}$-martingale ${\{{Z_{t}}\}_{t\in \mathbb{T}}}$ in ${\mathcal{L}^{\ell }}(P)$ is P-a.s. positive, if ${Z_{t}}$ is P-a.s. positive for each $t\in \mathbb{T}$.

Given $\varOmega :={(0,\infty )^{\mathbb{N}}}\times {(0,\infty )^{\mathbb{N}}}\times D$ and $\varSigma :=\mathfrak{B}(\varOmega )=\mathfrak{B}{(0,\infty )_{\mathbb{N}}}\otimes \mathfrak{B}{(0,\infty )_{\mathbb{N}}}\otimes \mathfrak{B}(D)$, let μ be a probability measure on $\mathfrak{B}(D)$, and let ${P_{n}}(\theta ):=\mathbf{K}(\theta )$ and ${R_{n}}:=R$ be probability measures on $\mathfrak{B}(0,\infty )$ for any $n\in \mathbb{N}$ and fixed $\theta \in D$. Assume that for any fixed $B\in \mathfrak{B}(0,\infty )$ the function $\theta \mapsto \mathbf{K}(\theta )(B)$ is $\mathfrak{B}(D)$-measurable. It then follows by [33], Proposition 4.1, that there exist:

Throughout what follows, unless stated otherwise, $(\varOmega ,\varSigma ,P)$, N, W, X, S, Θ and ${\{{P_{\theta }}\}_{\theta \in D}}$ are as above, $\varSigma ={\mathcal{F}_{\infty }}$, $\ell \in \{1,2\}$ and ${S_{t}^{(\gamma )}}:={\textstyle\sum _{j=1}^{{N_{t}}}}\gamma ({X_{j}})$ for any $t\in {\mathbb{R}_{+}}$ and for any real-valued $\mathfrak{B}(0,\infty )$-measurable function γ satisfying condition ${\mathbb{E}_{P}}[{e^{\gamma ({X_{1}})}}]=1$.

For the validity of the equality $\varSigma ={\mathcal{F}_{\infty }}$ see [33], Remark 4.2.

The following result of [33], concerning a characterization of all progressively equivalent probability measures that convert a CMRP into a CMPP, serves as a useful basic tool for our results.

In this section we find out a wide class of canonical price processes inducing a corresponding class of progressively EMMs and satisfying the condition of no free lunch with vanishing risk (written (NFLVR) for short) (see [9], Definition 8.1.2), connecting in this way our results with this basic notion in mathematical finance.

In order to present the results of this section we recall the following notions. For a given real-valued process $Y:={\{{Y_{t}}\}_{t\in {\mathbb{R}_{+}}}}$ on $(\varOmega ,\varSigma )$ a probability measure Q on Σ is called an ℓ-martingale measure for Y, if Y is a martingale in ${\mathcal{L}^{\ell }}(Q)$. We will say that Y satisfies condition (PEMM) if there exists a 2-martingale measure Q for Y, which is progressively equivalent to P. Moreover, let $T\gt 0$, $\mathbb{T}:=[0,T]$, ${\mathcal{F}_{\mathbb{T}}}:={\{{\mathcal{F}_{t}}\}_{t\in \mathbb{T}}}$, ${Q_{T}}:=Q\upharpoonright {\mathcal{F}_{T}}$, ${P_{T}}:=P\upharpoonright {\mathcal{F}_{T}}$ and ${Y_{\mathbb{T}}}:={\{{Y_{t}}\}_{t\in \mathbb{T}}}$. We will say that the process ${Y_{\mathbb{T}}}$ satisfies condition (EMM) if there exists a 2-martingale measure ${Q_{T}}$ for ${Y_{\mathbb{T}}}$, which is equivalent to ${P_{T}}$.

In the next theorems we provide a characterization of all progressively equivalent martingale measures Q on Σ converting a CMRP under P into a CMPP under Q, in such a way that they are associated to stochastic processes satisfying condition (NFLVR).

The fact that ${\{{P_{\theta }}\}_{\theta \in D}}$ is an rcp of P over ${P_{\varTheta }}$ consistent with Θ, allows for the extension of some well-known results from the Poisson or renewal risk models, to their mixed counterpart. In fact, whenever in the ${P_{\theta }}$-Cramér–Lundberg or the ${P_{\theta }}$-Sparre Andersen risk model an explicit formula for the (infinite time) ruin probability exists, then one can just mix over the involved parameter in order to obtain explicit formulas for the corresponding mixed risk models (compare Albrecher et al. [1], Sections 3 and 5). However, such explicit formulas, which are also computationally feasible, can be obtained only in certain special cases, e.g., when the claim sizes follow a gamma distribution (see Constantinescu et al. [5]), a Coxian distribution (see Landriault and Willmot [17]), or a general phase type distribution (cf., e.g., Asmussen and Albrecher [2], Chapter IX, Theorem 4.4). If we are interested in an exact value for the ruin probability in a general mixed renewal risk model, then the only method available seems to be simulation. In the setting of a finite horizon ruin probability, it is straightforward to use the Crude Monte Carlo method to simulate it, see [2], page 462. The situation is more complicated for an infinite horizon ruin probability. The difficulty is that the indicator function of the ruin event in such a case cannot be simulated in finite time: no finite segment of S can tell whether ruin will ultimately occur or not. In order to overcome this obstacle, some methods have been developed (cf., e.g., [2], Chapter XV, Sections 2–5). Among them, the most celebrated one is the change of measures technique, which gives us the opportunity to express the ruin event as a quantity under the new measure such that ruin occurs almost surely.

The most common change of measures techniques applied to the Sparre Andersen risk model arise from the martingales constructed via the so-called Backward or Forward Markovization Techniques for the reserve process (see Dassios and Embrechts [7], Section 2.3, and Dassios [6], Section 3.5, respectively, in connection with, e.g., Schmidli [28], Sections 8.1–8.3), where the martingales (and thus the new measures) are obtained as solutions of partial differential equations (see also [33], Proposition 4.15, for a simplified construction of the martingales/measures arising from the Backward Markovization Technique). These techniques have been widely used to solve various ruin related problems (cf., e.g., Embrechts et al. [12], Ng and Yang [23], Schmidli [26, 27] and Tzaninis [32]), as they allow for the construction of a suitable probability measure ${Q^{(r)}}$, where $r\gt 0$, so that ruin occurs ${Q^{(r)}}$-a.s. However, as the main assumption for their construction is the existence of the moment generating function ${M_{{P_{{X_{1}}}}}}$ of the claim size distribution (for the definition of the moment generating function of a distribution we refer to [30], page 174), heavy-tailed distributions (cf., e.g., [25], Section 2.5, for the definition and their basic properties) are naturally excluded. When P is clear from the context we write ${M_{{X_{1}}}}$ instead of ${M_{{P_{{X_{1}}}}}}$. The previous discussion raises the question of constructing a probability measure Q being progressively equivalent to P and so the ruin occurs Q-a.s., but without necessarily needing the assumption that ${M_{{X_{1}}}}$ exists.

In this section, we characterize all progressively equivalent martingale measures Q on Σ that convert a P-CMRP into a Q-CMPP, in such a way that ruin occurs Q-a.s., see Theorem 3. Such a characterization allows us to find an explicit formula for the ruin probability under P. To this purpose, we first need to prove the following auxiliary results.

In order to justify the definition of a conditional premium density we need the next lemma, which extends a well-known result in the case of a kind of compound mixed Poisson processes (cf., e.g., [13], Proposition 9.1).

Let $P\in {\mathcal{M}_{S,\mathbf{K}(\varTheta )}^{1}}$ such that $P(\{{\mathbb{E}_{P}}[{W_{1}}\mid \varTheta ]\lt \infty \})=1$. For any $\theta \notin {W_{P}}$, where ${W_{P}}\in \mathfrak{B}(D)$ is the ${P_{\varTheta }}$-null set appearing in Remark 1(b), conditions (3) and (4) imply that is, the limit ${\lim \nolimits_{t\to \infty }}\frac{{S_{t}}}{t}$ coincides with the premium density $p({P_{\theta }})$, i.e., the monetary payout per unit time, in a ${P_{\theta }}$-Sparre Andersen model (see [22], page 54, for more details). This motivates the following definitions. Throughout what follows in this section, unless stated otherwise, $P\in {\mathcal{M}_{S,\mathbf{K}(\varTheta )}^{\ast ,\ell }}$.

The next result has been proven in a more general setting for multivariate counting processes in [16], Proposition 3.39(a). However, as it is essential for the proof of the main result of the present section (see Theorem 3), we write it exactly in the form needed for our purposes. Recall that a filtration ${\{{\mathcal{G}_{t}}\}_{t\in {\mathbb{R}_{+}}}}$ for $(\varOmega ,\varSigma )$ is called right-continuous if ${\mathcal{G}_{t+}}:={\textstyle\bigcap _{s\gt t}}{\mathcal{G}_{s}}={\mathcal{G}_{t}}$ for any $t\in {\mathbb{R}_{+}}$ (cf., e.g., [25], Subsection 10.2.1, page 404).

The proof follows by a slight modification of the arguments appearing in Jacod [16], Proposition 3.39(a).

According to Theorem 1(ii), for every pair $(\beta ,\xi )\in {\mathcal{F}_{P,\varTheta }^{\ell }}\times {\mathcal{R}_{+}^{\ast ,\ell }}(D)$ there exists a unique pair $(\rho ,Q)\in {\mathfrak{M}_{+}}(D)\times {\mathcal{M}_{S,\mathbf{Exp}(\rho (\varTheta ))}^{\ast ,\ell }}$ determined by conditions (∗) and ($RP{M_{\xi }}$) such that the process ${M^{(\beta )}}(\varTheta )$, involved in condition ($RP{M_{\xi }}$), is a P-a.s. positive martingale in ${\mathcal{L}^{1}}(P)$, implying But since the ruin time ${T_{u}}(\varTheta )$ of the reserve process ${R^{u}}(\varTheta )$ is an $\mathcal{F}$-stopping time by Lemma 3(i), we may apply the Optional Stopping Theorem (cf., e.g., [28], Proposition B.2) in order to get However, the latter formula is quite complicated mainly due to the (possible) dependence between $1/{M_{{T_{u}}(\varTheta )}^{(\beta )}}(\varTheta )$ and ${1_{\{{T_{u}}(\varTheta )\lt \infty \}}}$. Thus, we have to carefully choose an appropriate pair $(\rho ,Q)\in {\mathfrak{M}_{+}}(D)\times {\mathcal{M}_{S,\mathbf{Exp}(\rho (\varTheta ))}^{\ast ,\ell }}$, or equivalently an appropriate pair $(\beta ,\xi )\in {\mathcal{F}_{P,\varTheta }^{\ell }}\times {\mathcal{R}_{+}^{\ast ,\ell }}(D)$, in order to eliminate such a dependence. This motivates us to formulate and prove the following result.

The following example shows that the classical approach to the ruin problem via a change of measures technique in the case of CPPs (cf., e.g., [28], Section 8.2) appears as a special instance of Theorem 3.

One of the main drawbacks of the method used in Example 1 is the assumption that ${M_{{X_{1}}}}(r)$ exists for some $r\gt 0$, since it excludes heavy-tailed distributions. In the following example we consider again $P\in {\mathcal{M}_{S,\mathbf{Exp}(\varTheta )}^{\ast ,\ell }}$ and we demonstrate how Theorem 3 can be used to handle such cases.

Note that the arguments appearing in the above example, remain true for any claim size distribution ${P_{{X_{1}}}}$ with finite expectations.

In this section, we discuss implications of our results for the computation of premium calculation principles in a model of an insurance market possessing the property (NFLVR). In this context, the financial pricing of insurance (FPI for short) approach proposed by Delbaen and Haezendonck [8] plays a key role.

Let $T\gt 0$. According to the FPI approach, the liabilities of an insurance company over a fixed period of time $\mathbb{T}:=[0,T]$ can be represented as a price process ${U_{\mathbb{T}}}:={\{{U_{t}}\}_{t\in \mathbb{T}}}$ defined by ${U_{t}}:={p_{t}}+{S_{t}}$ for any $t\in \mathbb{T}$, where ${S_{t}}$ represents the total amount of claims paid up to time t and ${p_{t}}$ represents the total premium for the remaining risk ${S_{T}}-{S_{t}}$. Under the assumption that the random behavior of the price process ${U_{\mathbb{T}}}$ is described by the given probability measure P on Σ, and that the insurance market is liquid enough (see [8], Section 1, for more details) by applying the Harrison–Kreps theory (see Harrison and Kreps [15]) it follows that the existence of a 2-martingale measure Q on Σ for ${U_{\mathbb{T}}}$ which is equivalent to P implies the elimination of arbitrage opportunities in the insurance market and vice versa (see [8], Section 1). However, as the insurance market is not, in general, complete (see, e.g., [11], page 20, or [31], Section 4) the measure Q is not unique; hence the next step should be the selection of such a measure Q.

Under the assumption that the aggregate process S is a P-CPP, Delbaen and Haezendonck [8] were interested in all those measures Q with linear premiums of the form where $p(Q)$ is the premium density under Q. But as ${U_{\mathbb{T}}}$ is a martingale under Q if and only if ${U_{\mathbb{T}}}-{p_{0}}={\{{S_{t}}-p(Q)\cdot t\}_{t\in \mathbb{T}}}$ is such, Delbaen and Haezendonck faced the problem of characterizing all those risk-neutral measures Q on Σ under which the compensator of ${S_{\mathbb{T}}}$ is a linear deterministic function of $t\in \mathbb{T}$. As the linearity of the premiums implies that ${S_{\mathbb{T}}}$ is a CPP under Q and vice versa (see [8], Section 1), the above problem is equivalent to the following one:

Recall that under the FPI framework, a premium calculation principle (written PCP for short) is a probability measure Q on Σ which is progressively equivalent to P, the process S is a Q-CPP and ${X_{1}}\in {\mathcal{L}^{1}}(Q)$, see [8], Definition 3.1. If the distribution ${Q_{\varTheta }}$ is not degenerate, then the probability measure $Q\in {\mathcal{M}_{S,\mathbf{Exp}(\rho (\varTheta ))}^{\ast ,1}}$ constructed in Theorem 1(ii) fails to be a PCP. Nevertheless, by virtue of Theorem 1(iii) there exists an essentially unique rcp ${\{{Q_{\theta }}\}_{\theta \in D}}$ of Q over ${Q_{\varTheta }}$ consistent with Θ such that for ${P_{\varTheta }}$-a.a. $\theta \in D$ the probability measures ${Q_{\theta }}$ are PCPs. Thus, it seems natural to call every probability measure $Q\in {\mathcal{M}_{S,\mathbf{Exp}(\rho (\varTheta ))}^{\ast ,1}}$ a mixed PCP.

In order for a mixed PCP to provide a realistic and viable pricing framework it should give more weight to unfavorable events in a risk-averse environment, i.e., conditions and must hold true.

However, the existence of a mixed PCP does not, in general, guarantee the validity of conditions (10) and (11) as the next two examples demonstrate. In the first example, we construct a mixed PCP that does not satisfy condition (10) and leads to a P-a.s. ruin.

In the next example, we construct a mixed PCP for which condition (10) holds but condition (11) fails.

Examples 3 and 4 raise the question when a mixed PCP satisfies conditions (10) and (11) or the implication (10)⇒(11). In the next proposition we find sufficient conditions for the validity of the implication (10)⇒(11). To prove it, we need the following lemma, which is a consequence of Schmidt [29], Theorem 2.2, but we write it exactly in the form needed for our purposes.

In this section, applying our results, we provide some examples to show how to construct mixed PCPs Q satisfying conditions (10) and (11) and such that for any $T\gt 0$ the processes ${V_{\mathbb{T}}}(\varTheta )$ and ${R_{\mathbb{T}}^{u}}(\varTheta )$, for $u\in {\mathbb{R}_{+}}$, appearing in Theorem 2 and Remark 8, respectively, have the property (NFLVR). Moreover, we provide explicit formulas for the ruin probability for the reserve process ${R^{u}}(\varTheta )$ with respect to the original measure P.

In our next example we rediscover Shaun Wang’s risk-adjusted premium principle (see [35], for the definition and its properties).

The following counter-example shows that the assumption of the same monotonicity of the functions $\theta \mapsto p({Q_{\theta }})$ and $\theta \mapsto \xi (\theta )$ for all $\theta \notin {\widetilde{L}_{\ast \ast }}$ is essential for the validity of the conclusion $p(P)\le p(Q)$ in Proposition 3.

Notations: Assumptions (see Section 2, page 5): Classes of functions: Classes of measures: Conditions: