A generalization of Doob's maximal identity

Abstract

In this paper, using martingale techniques, we prove a generalization of Doob's maximal identity in the setting of continuous nonnegative local submartingales $(X_t)$ of the form: $X_t=N_t+A_t$, where the measure $(d{A}_t)$ is carried by the set $\left\{t:X_t=0\right\}$. In particular, we give a multiplicative decomposition for the Azéma supermartingale associated with some last passage times related to such processes and we prove that these non-stopping times contain very useful information. As a consequence, we obtain the law of the maximum of a continuous nonnegative local martingale $(M_t)$ which satisfies $M_{\infty }=\psi ({\sup }_{t\ge 0}{M}_t)$ for some measurable function $\psi$ as well as the law of the last time this maximum is reached.

References in corpus (2)

• The Skorokhod embedding problem and its offspring
• An essay on the general theory of stochastic processes

Cited by in corpus (1)

• From Black-Scholes and Dupire formulae to last passage times of local martingales. Part A : The infinite time horizon