On a subclass of square integrable martingales

Dean Isaacson
1972 Proceedings of the American Mathematical Society  
Let Jl* denote the class of continuous, nowhere constant, square integrable martingales, M{t)=X{(M)t), for which {M)t is a time change on the cr-fields generated by the Brownian motion X{t). It is shown that if M{t)eJl2, then the family of crfields generated by M{t) is a right continuous family. If M{t) e Jl* and if o{M{s):s^t} = a{X{s):s^t} for some Brownian motion X{t), then M(t) = ft®(s)dX(s) and X(í)=J0 {ll®{s))dM{s) for some process $>{s) with <¡>{s)¿¿0 a.e. dtxdP.
doi:10.1090/s0002-9939-1972-0295432-1 fatcat:mppl6jcx6nf67fdk73rdcdst6m