A characterization of equivalent martingale measures in a renewal risk model with applications to premium calculation principles

Abstract

Generalizing earlier work of Delbaen & Haezendonck for given compound renewal process S under a probability measure P we characterize all probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound renewal process under Q. As a consequence, we prove that any compound renewal process can be converted into a compound Poisson process through a change of measures and we show how this approach is related to premium calculation principles.

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

“…Since P ae F t " Q ae F t , we have Q Θ " P Θ by σpΘq Ď F t , while by [5], Lemma 2.1, we have Q X 1 ae F S t " P X 1 ae F S t implying Q X 1 " P X 1 by F S t Ď F t . Finally, applying similar arguments to those of the proof of Proposition 2.1 form [18], we get Q Wn ae F S t " P Wn ae F S t , implying Q Wn " P Wn for any n P N by F S t Ď F t . The latter together with Remark 2.1 yields…”

“…In fact, put L ˚:" r L Y Ă M and fix on an arbitrary θ R L ˚. Since pQ θ q W 1 " pP θ q W 1 by (c), and pQ θ q X 1 " pP θ q X 1 by (a) and (b), we can apply [18], Proposition 2.1, to complete the whole proof.…”

“…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.…”

“…e.g. Chen et al [2] and Wang et al [28]) motivated Macheras & Tzaninis [18] to generalize the Delbaen & Haezendonck characterization for the more general renewal risk model (also known as the Sparre Andersen model).…”

“…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.…”