Generalized Merkle Tree Representation for Actuarial Applications

The Merkle Tree (MT) was introduced by Merkle (1987) in his PhD thesis and studied in depth in further papers – Merkle (1987-1991). The MT is a tree where the values associated with each node are one-way functions of the values of the born-nodes and therefore strictly path-dependent. The MT finds wide applications within cryptography – Merkle(1987-1991) –and particularly in authentication protocols and transmission – Indrajit (2003), Merkle (1982), Perrig (2002) – in coding and decoding of digital images – Deng(2003). The present paper aims at proposing a study, finalized to financial (optin pricing) and insurance (determination of premium) applications, of a discrete Bayes-process as described through the Merkle Tree with K states and N stages. We hereby suggested three algorithms – two of them are generalized, as already described in Addessi (2002), and aim at calculating the average current value of a random performance, whose discreet variable below may take up K separate determinations at the most (stages of the system). Such algorithms are based on the use of vectors and matrices: according to the first algorithm, a vector having KN components, where N stands for the number of the stages, is associated with each time unit (stage of the system); according to the second one, the node values are calculated through the matrices and at a fixed stage, based on all the nodes that have been generated; according to the third one, a recursive function is proposed, based on the use of fractal and attimes diagonal matrices. The algorithms we propose are applicable to a discreet stochastic path dependent process: in particular, a simple application within the actuarial field is proposed.