On a generalization of the theorem of p. levy

Abstract

Consider A Closed E of the Real Line R and a (Measureble) Function Defined on R and Strictly Concave on Every Component of R/E. The Present Paper Deals with the Following Marticgale Problem: Find a Continuous Local Martingale (X,F), Given on a Family () of Probability Spaces With for every such that x is stopped at the debut of e, and the process is a local martingale up to We show that this martingale problem possesses a unique solution. This solution is a strong Markov continuous local martingale. Furthermore, it is a pure continuous local martingale and therefore satisfies the previsible representation property. Conversely, for every strong Markov continuous local martingale (X,F) we can find a closed subset E of R and a function f strictly concave on every component of R\E such that (X, F) is the unique solution of the above martingale problem. In the particular case that E and f(X) we recover the theorem of P. Levy on the martingale characterization of the Brownian motion.