Martingale - ισοδύναμες κατανομές πιθανότητας με εφαρμογές υπολογισμού ασφαλίστρου
Martingale - equivalent probability distributions with applications in premium calculation principles
Λυμπερόπουλος, Δημήτριος Π.
KeywordsΚανονικές δεσμευμένες πιθανότητες ; Μεμειγμένη διαδικασία Poisson ; Σύνθετη μεμειγμένη διαδικασία Poisson ; Μεμειγμένη ανανεωτική διαδικασία ; Αρχές υπολογισμού ασφαλίστρου ; Χαρακτηρισμοί ; Disintegrations ; Martingales ; Regular conditional probabilities ; Mixed Poisson process ; Compound mixed Poisson process ; Premium calculation principles ; Characterizations
A positive answer to the problem of characterizing martingale-equivalent compound mixed Poisson processes becomes here possible, generalizing in this way a corresponding result of Delbaen & Haezendonk (1989) for compound Poisson processes. Some applications to the theory of premium calculation principles are then obtained. For solving the above problem, the structural role of disintegrations in mixed stochastic processes is studied first. As a result, the reduction of conditional independence to the ordinary one follows for a wide class of stochastic processes, under a proper change of measure. The reduction of conditional identically distributed processes to ordinary ones is obtained in a similar way. As a consequence, some characterizations of mixed Poisson processes as well as of mixed renewal processes are derived. In particular, further characterizations in terms of exchangeability and of different types of disintegrations are given for mixed renewal processes, providing among others an extension of de Finetti’s Theorem. In addition, a new method of constructing mixed renewal processes, including the construction of mixed Poisson processes as a special case, is presented. Based on the latter results, some concrete examples of constructing such processes are given and the corresponding disintegrating measures are computed. Finally, possible applications of our main characterization to premium calculation principles as well as to other problems related to insurance and finance are investigated.