Convergence of local supermartingales (Q2028957)

The theme of this paper is a characterisation of the event of the almost sure convergence of a local supermartingale. This is known through the Dambis-Dubins-Schwarz theorem which covers the case of continuous local supermartingales (the event of convergence coincides with that of having a finite quadratic variation). This paper considers local martingales which may not be continuous, but are restricted on a stochastic interval. In particular, the latter is defined as the interval between time 0 and a stopping time \(\tau\) which is \textit{foretellable}, that is, it can be approximated from below by a non-decreasing sequence of stopping times. The main theorem gives a number of characterisations for the event of convergence within a measurable subset \(D\) of such a local supermartingale \((X_t)_{0\leq t < \tau}\) as \(t\to \tau\). This characterisation uses the notion of \textit{stationarily local integrability} on \(D\). This property is defined through a non-decreasing sequence of stopping times \((\rho_n)_{n\in \mathbb{N}}\) such that the stopped process \((X_t^{\rho_n})_{t\geq 0}\) is uniformly bounded by an integrable random variable \(\Theta_n\), for each \(n\in \mathbb{N}\) and moreover \(D\) contains the event that one of these stopping times exceeds \(\tau\).