Given a price process U T : = { U t } t ∈ T , where T : = [ 0 , T ], T > 0, on a probability space ( Ω , Σ , P ), a basic method in Mathematical Finance is to replace the initial probability measure P by an equivalent one Q, which converts U 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 T represents the liabilities of an insurance company, then there exist infinitely many equivalent martingale measures for U 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 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 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 ( Θ ) induced by the initial reserve u, the stochastic premium intensity c ( Θ ) 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 ( Θ ) 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.

N, Q and R stand for the natural, the rational and the real numbers, respectively, while N 0 : = N ∪ { 0 } and R + : = { x ∈ R : x ≥ 0 }. If d ∈ N then R d denotes the Euclidean space of dimension d. For a map f : A → E and for a nonempty set B ⊆ A write f ↾ 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, ( Ω , Σ , P ) is a fixed but arbitrary probability space and Θ : Ω → D ⊆ R d ( d ∈ N) is a d-dimensional random vector. By L ℓ ( P ) we denote the family of all Σ-measurable real-valued functions f on Ω such that ∫ | f | ℓ d P < ∞ ( ℓ ∈ { 1 , 2 }). For any Hausdorff topology T over Ω, by B ( Ω ) is denoted the Borel σ-algebra on Ω, i.e., the σ-algebra generated by T, while B : = B ( R ) and B d : = B ( R d ), where d ∈ N, stand for the Borel σ-algebras of subsets of R and R d , respectively, generated by the Euclidean topology E over R and by the product topology E d over R d , respectively. For any J ⊆ R d denote by B ( J ) the σ-algebra on J generated by the topology E d ∩ J : = { J ∩ G : G ∈ E d }. In particular, write B ( 0 , ∞ ) : = B ( ( 0 , ∞ ) ), 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 ( θ ) : = K ( θ ) to mean that X is distributed according to the law K ( θ ), where θ ∈ D ⊆ R d is the parameter of the distribution. Notation Ga ( b , a ), where a , b ∈ ( 0 , ∞ ), stands for the law of gamma distribution (cf., e.g., [30], page 180). In particular, Ga ( b , 1 ) = 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 σ ( Θ )- B-Markov kernel (see [3], Definition 36.1 for the definition) denoted by P X ∣ Θ : = P X ∣ σ ( Θ ) and satisfying for each B ∈ B the equality P X ∣ Θ ( ∙ , B ) = P ( X − 1 ( B ) ∣ σ ( Θ ) ) ( ∙ ) P ↾ σ ( Θ )-almost surely (written a.s. for short). Clearly, for every B ( R d )- B-Markov kernel k, the map K ( Θ ) from Ω × B into [ 0 , 1 ] defined by K ( Θ ) ( ω , B ) : = ( k ( ∙ , B ) ∘ Θ ) ( ω ) for any ( ω , B ) ∈ Ω × B is a σ ( Θ )- B-Markov kernel. Then for θ = Θ ( ω ) with ω ∈ Ω the probability measures k ( θ , ∙ ) are distributions on B and so we may write K ( θ ) ( ∙ ) instead of k ( θ , ∙ ). Consequently, in this case K ( Θ ) will be denoted by K ( Θ ).

For any real-valued random variables X, Y on Ω we say that P X | Θ and P Y | Θ are P ↾ σ ( Θ )-equivalent and we write P X | Θ = P Y | Θ P ↾ σ ( Θ )-a.s., if there exists a P-null set M ∈ σ ( Θ ) such that for any ω ∉ M and B ∈ B the equality P X | Θ ( ω , B ) = P Y | Θ ( ω , B ) holds true.

A sequence { V n } n ∈ N of real-valued random variables on Ω is said to be: ∙

P-conditionally (stochastically) independent over σ ( Θ ) if, for each n ∈ N with n ≥ 2, we have P ( ⋂ j = 1 n { V i j ≤ v i j } ∣ σ ( Θ ) ) = ∏ j = 1 n P ( { V i j ≤ v i j } ∣ σ ( Θ ) ) P ↾ σ ( Θ ) -a.s. , whenever i 1 , … , i n are distinct members of I ⊆ N and ( v i 1 , … , v i n ) ∈ R n ;

Throughout what follows we write “conditionally” in the place of “conditionally given Θ” whenever conditioning refers to Θ.

For the definitions of a counting (or claim number) process N : = { N t } t ∈ R + , an arrival process T : = { T n } n ∈ N 0 induced by N, an interarrival process W : = { W } n ∈ N induced by T, a claim size process X : = { X n } n ∈ N and an aggregate claims process S : = { S t } t ∈ R + induced by N and X, we refer to [30]. We assume that P ( { X n > 0 } ) = 1 for all n ∈ 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 K ( Θ ) (written P-MRP ( K ( Θ ) ) for short), if the interarrival process W is P-conditionally independent and for all n ∈ N the condition P W n ∣ Θ = K ( Θ ) P ↾ σ ( Θ )-a.s. is valid (see [21], Definition 3.1). In particular, if the distribution of Θ is degenerate at some point θ 0 ∈ D, then the counting process N becomes a P-renewal process with interarrival time distribution K ( θ 0 ) (written P-RP ( K ( θ 0 ) ) for short). If N is a P-MRP ( K ( Θ ) ) then according to [33], Corollary 3.5(ii), it has zero probability of explosion, i.e., P ( { sup n ∈ N 0 T n < ∞ } ) = 0.

Accordingly, an aggregate claims process S induced by a P-risk process ( N , X ) such that N is a P-MRP ( K ( Θ ) ) is called a compound mixed renewal process with parameters K ( Θ ) and P X 1 (P-CMRP ( K ( Θ ) , P X 1 ) for short). If Θ is a real-valued random variable and K ( Θ ) = Exp ( Θ ), we say that S is a compound mixed Poisson process with parameters Θ and P X 1 and write P-CMPP ( Θ , P X 1 ). In particular, if the distribution of Θ is degenerate at θ 0 ∈ D, then S is called a compound renewal process with parameters K ( θ 0 ) and P X 1 (P-CRP ( K ( θ 0 ) , P X 1 ) for short). In the special case θ 0 > 0 and K ( θ 0 ) = Exp ( θ 0 ) we say that S is a compound Poisson process with parameters θ 0 and P X 1 and write P-CPP ( θ 0 , P X 1 ).

Throughout what follows we denote again by K ( Θ ) and K ( θ ) the conditional distribution function and the distribution function induced by the conditional probability distribution K ( Θ ) and the probability distribution K ( θ ), 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 ∈ Z of probability measures on Σ is called a regular conditional probability (rcp for short) of P over R if for any fixed E ∈ Σ the map Z ∋ z ↦ P z ( E ) is H-measurable, and ∫ P z ( E ) R ( d z ) = P ( E ) for every E ∈ Σ. If f : Ω → Z is an inverse-measure-preserving function (i.e., P ( f − 1 ( B ) ) = R ( B ) for each B ∈ H), an rcp { P z } z ∈ Z of P over R is called consistent with f if, for each B ∈ H, the equality P z ( f − 1 ( B ) ) = 1 holds for R-almost every z ∈ B (cf., e.g., [33], Definition 3.3). We say that an rcp { P z } z ∈ Z of P over R consistent with f is essentially unique, if for any other rcp { P ˜ z } z ∈ Z of P over R consistent with f there exists a R-null set M ∈ H such that for any z ∉ M the equality P z = P ˜ z holds true.

From now on ( Z , H , R ) : = ( D , B ( D ) , P Θ ) and the family { P θ } θ ∈ D is an rcp of P over P Θ consistent with Θ.

Let T ⊆ R + . For a process Y T : = { Y t } t ∈ T denote by F T Y : = { F t Y } t ∈ T the canonical filtration of Y T . For T = R + or T = N we simply write F Y instead of F R + Y or F N Y , respectively. Also, we write F : = { F t } t ∈ R + , where F t : = σ ( F t S ∪ σ ( Θ ) ), for the canonical filtration of S and Θ, F ∞ S : = σ ( ⋃ t ∈ R + F t S ) and F ∞ : = σ ( F ∞ S ∪ σ ( Θ ) ).

Let Q be a probability measures on ( Ω , Σ ). We say that P and Q are progressively equivalent, if P and Q are equivalent (in the sense of absolute continuity) on F t (in symbols Q ↾ F t ∼ P ↾ F t ) for any t ∈ R + , see [33], Definition 3.1. If P and Q are equivalent on Σ we write P ∼ 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 , Θ ) or (if no confusion arises) the probability measure P satisfies (a1) or (a2), respectively.

Recall that, for given T ⊆ R + a martingale in L ℓ ( P ) adapted to the filtration Z T : = { Z t } t ∈ T , or else a Z T -martingale in L ℓ ( P ), is a family Z T : = { Z t } t ∈ T of random variables in L ℓ ( P ) such that Z t is Z t -measurable for each t ∈ T, and whenever u ≤ t in T and E ∈ Z u then ∫ E Z u d P = ∫ E Z t d P. For Z R + = F we simply write that Z is a martingale in L ℓ ( P ). A Z T -martingale { Z t } t ∈ T in L ℓ ( P ) is P-a.s. positive, if Z t is P-a.s. positive for each t ∈ T.

Given Ω : = ( 0 , ∞ ) N × ( 0 , ∞ ) N × D and Σ : = B ( Ω ) = B ( 0 , ∞ ) N ⊗ B ( 0 , ∞ ) N ⊗ B ( D ), let μ be a probability measure on B ( D ), and let P n ( θ ) : = K ( θ ) and R n : = R be probability measures on B ( 0 , ∞ ) for any n ∈ N and fixed θ ∈ D. Assume that for any fixed B ∈ B ( 0 , ∞ ) the function θ ↦ K ( θ ) ( B ) is B ( D )-measurable. It then follows by [33], Proposition 4.1, that there exist:

Throughout what follows, unless stated otherwise, ( Ω , Σ , P ), N, W, X, S, Θ and { P θ } θ ∈ D are as above, Σ = F ∞ , ℓ ∈ { 1 , 2 } and S t ( γ ) : = ∑ j = 1 N t γ ( X j ) for any t ∈ R + and for any real-valued B ( 0 , ∞ )-measurable function γ satisfying condition E P [ e γ ( X 1 ) ] = 1.

For the validity of the equality Σ = F ∞ 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. Proposition 1

For given P ∈ M S , K ( Θ ) ℓ with P ( { E P [ W 1 ∣ Θ ] < ∞ } ) = 1 the following hold true: (i)

for every pair ( ρ , Q ) ∈ M + ( D ) × M S , Exp ( ρ ( Θ ) ) ℓ there exists an essentially unique pair ( β , ξ ) ∈ F P , Θ ℓ × R + ( D ), where e γ and ξ are the Radon–Nikodým derivatives of Q X 1 with respect to P X 1 and of Q Θ with respect to P Θ , respectively, such that (∗) α ( Θ ) = ln ρ ( Θ ) + ln E P [ W 1 ∣ Θ ] P ↾ σ ( Θ ) -a.s. , and (RPM_ξ) Q ( A ) = ∫ A M t ( β ) ( Θ ) d P for all 0 ≤ s ≤ t and A ∈ F s , where M t ( β ) ( Θ ) : = ξ ( Θ ) · e S t ( γ ) − ρ ( Θ ) · J t 1 − K ( Θ ) ( J t ) · ∏ j = 1 N t d Exp ( ρ ( Θ ) ) d K ( Θ ) ( W j ) , with J t : = t − T N t , and the family M ( β ) ( Θ ) : = { M t ( β ) ( Θ ) } t ∈ R + is a P- a.s positive martingale in L 1 ( P );

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 ∈ R + on ( Ω , Σ ) a probability measure Q on Σ is called an ℓ-martingale measure for Y, if Y is a martingale in L ℓ ( 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 > 0, T : = [ 0 , T ], F T : = { F t } t ∈ T , Q T : = Q ↾ F T , P T : = P ↾ F T and Y T : = { Y t } t ∈ T . We will say that the process Y T satisfies condition (EMM) if there exists a 2-martingale measure Q T for Y 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). Theorem 1.

If P ∈ M S , K ( Θ ) ∗ , ℓ the following statements hold true: (i)

for every pair ( ρ , Q ) ∈ M + ( D ) × M S , Exp ( ρ ( Θ ) ) ∗ , ℓ there exists an essentially unique pair ( β , ξ ) ∈ F P , Θ ℓ × R + ∗ , ℓ ( D ), where e γ and ξ are the Radon–Nikodým derivatives of Q X 1 with respect to P X 1 and of Q Θ with respect to P Θ , respectively, satisfying conditions (∗) and ( R P M ξ );