ON DUALITY AND THE SPITZER-POLLACZEK FACTORIZATION FOR RANDOM-WALKS

A new formulation of duality for pairs of stopping times is given. Thi s formulation is constructive in that it provides a method for generat ing examples of dual times. We also use it to form the basis for a dir ect sample path proof of the Spitzer-Pollaczek factorization associate d with a dual pair. The Spitzer-Pollaczek factorization relates, in a single expression, the distributions of a dual pair of times and the d istribution of a random walk at each of these times. The accepted prob abilistic derivation introduces an independent geometric time. The dir ect approach here omits this step and in doing so allows a separate tr eatment of the stopping time and the stopped random walk distributions and provides clear interpretations for the identities that arise. Thi s novel look at duality makes clear further generalizations of the Spi tzer-Pollaczek factorization which must hold and we conclude by provin g a matrix factorization associated with a Markov-modulated random wal k on R-d. (C) 1998 Elsevier Science B.V. All rights reserved.