Just to confirm: Writing pt, the probability of A at time t, as pt=E[1A∣Ft] (here Ft is the sigma-algebra at time t), we see that pt must be a martingale via the tower rule.
The log-odds xt=logpt1−pt are not martingales unless pt≡const because Itô gives us dxtdef=dlogpt1−ptItô=1pt(1−pt)dptmartingale part+12(1(1−pt)2−1p2t)d[p]tdrift part.
So unless pt is continuous and of bounded variation (⇒ d[p]t=0, but this also implies that pt≡const; the integrand of the drift part only vanishes if pt≡12 for all t), the log-odds are not a martingale.
Interesting analysis on log-odds might still be possible (just use dpt=pt+1−pt and d[p]t=(pt+1−pt)2 for discrete-time/jump processes as we naturally get when working with real data), but it’s not obvious to me if this comes with any advantages over just working with pt directly.
Thanks for looking into this. Did you happen to model this in log-odds space?
No—I think probability is the thing supposed to be a martingale, but I might be being dumb here.
Just to confirm: Writing pt, the probability of A at time t, as pt=E[1A∣Ft] (here Ft is the sigma-algebra at time t), we see that pt must be a martingale via the tower rule.
The log-odds xt=logpt1−pt are not martingales unless pt≡const because Itô gives us
dxtdef=dlogpt1−ptItô=1pt(1−pt)dptmartingale part+12(1(1−pt)2−1p2t)d[p]tdrift part.
So unless pt is continuous and of bounded variation (⇒ d[p]t=0, but this also implies that pt≡const; the integrand of the drift part only vanishes if pt≡12 for all t), the log-odds are not a martingale.
Interesting analysis on log-odds might still be possible (just use dpt=pt+1−pt and d[p]t=(pt+1−pt)2 for discrete-time/jump processes as we naturally get when working with real data), but it’s not obvious to me if this comes with any advantages over just working with pt directly.