A characterization of martingale-equivalent compound mixed Poisson process

Abstract

If a given aggregate process S is a compound mixed Poisson process under a probability measure P , a characterization of all probability measures Q on the domain of P , such that P and Q are progressively equivalent and S remains a compound mixed Poisson process with improved properties, is provided. This result generalizes earlier work of Delbaen & Haezendonck (1989). Implications related to the computation of premium calculation principles in an insurance market possessing the property of no free lunch with vanishing risk are also discussed.

A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

“…The latter result along with Proposition 3.9, is required for the desired characterization, in terms of regular conditional probabilities, of all those measures Q which are progressively equivalent to an original probability measure P , such that a CMRP under P remains a CMRP under Q, see Theorem 4.5. Note that the main results of [18], Theorem 3.1, and of [16], Theorem 4.3, follow as special instances of Theorem 4.5, see Remarks 4.6 and 4.9, respectively. Another consequence of Theorem 4.5 is that any CMRP can be converted into a compound mixed Poisson one through a change of measures technique, by choosing the "correct" Radon-Nikodým derivative, see Corollary 4.8.…”

“…Assertion (i) is equivalent to the fact that pN, Xq is a risk process and that N is a P -MRP(KpΘq). But according to [16], Lemma 2.3, and [15], Proposition 3.8, we equivalently obtain two P Θ -null sets L P,1 , L P,2 P BpDq such that the pair pN, Xq is a P θ -risk process for any θ R L P,1 , and for any θ R L P,2 the process N is a P θ -RPpKpθqq.…”

“…Next, we prove the one direction of the desired characterization, see Proposition 3.9, which provides a characterization and an explicit calculation of Radon-Nikodým derivatives dQ{dP for well known cases in insurance mathematics, see Example 3.12. The arguments used in the proof of Proposition 3.9 differ from those in the proofs of [5], Proposition 2.1, and [16], Proposition 3.4, respectively, as a (mixed) renewal process does not have in general (conditionally) stationary and independent increments. Furthermore, it is worth noticing that we prove our results avoiding a well-known backward Markovisation technique appearing in e.g.…”

“…The latter interpretation leads to the notion of mixed counting processes. Based on that Meister [19], and latter on (in a more general setup) Lyberopoulos & Macheras [16], generalized the Delbaen & Haezendonck characterization for the more general class of compound mixed Poisson processes, and applied their results to pricing CAT futures and to the theory of premium calculation principles, respectively.…”

“…In an effort to overcome these deficiencies, to allow more fluctuation and to step away from a Markovian environment, we consider the mixed renewal risk model. Since the mixed renewal risk model is strictly more comprising than the mixed Poisson one, the question whether the corresponding characterizations provided in Delbaen & Haezendonck [5], Lyberopoulos & Macheras [16] and [18] can be extended to the more general mixed renewal risk model naturally arises, and it is precisely this problem the paper deals with. In particular, if the process S is a compound mixed renewal process under the probability measure P , it would be interesting to characterize all probability measures Q being progressively equivalent to P and converting S into a compound mixed Poisson process under Q.…”