Une application du calcul du nombre de montées et de descentes aux fonctions de martingales locales continues. (An application of the calculus of up- and down-crossing numbers to functions of continuous local martingales) (Q1103964)

Let \(M=(M_ t)\), \(t\in R_+\), be a local, continuous martingale, \(M_ 0=0\). Denote \(a=-\| \sup_{t}M\) \(-_ t\|_{\infty}\), \(b=\| \sup_{t}M\) \(+_ t\|_{\infty}\). If f(M) is a semimartingale for a function f then the restriction of f'' (in the Schwartz sense) on (a,b) is a Radon measure. If \(P\{\exists t:M_ t=a\) or \(M_ t=b\}=0\) then the above condition is sufficient. In the case when \(M=W\) is a Wiener process, f(W) is a semimartingale iff f is the difference of two convex functions.