On some universal sigma-finite measures related to a remarkable class of submartingales

Najnudel, J; Nikeghbali, Ashkan (2012). On some universal sigma-finite measures related to a remarkable class of submartingales. Stochastic Processes and their Applications, 122(4):1582-1600.

Abstract

In this paper, for any submartingale of class (Sigma) defined on a filtered probability space (Omega, F, P, (F-t)(t >= 0)) satisfying some technical conditions, we associate a sigma-finite measure Q on (Omega, F), such that for all t >= 0, and for all events Lambda(t) is an element of F-t:

Q[Lambda(t), g <= t ] = Ep[1(Lambda t)X(t)].

where g is the last time for which the process X hits zero. The existence of Q has already been proven in several particular cases, some of them are related with Brownian penalization, and others are involved with problems in mathematical finance. More precisely, the existence of Q in the general case gives an answer to a problem stated by Madan, Roynette and Yor, in a paper about the link between the Black-Scholes formula and the last passage times of some particular submartingales. Moreover, the equality defining Q still holds if the fixed time t is replaced by any bounded stopping time. This generalization can be considered as an extension of Doob's optional stopping theorem. (c) 2012 Elsevier B.V. All rights reserved.

Abstract

In this paper, for any submartingale of class (Sigma) defined on a filtered probability space (Omega, F, P, (F-t)(t >= 0)) satisfying some technical conditions, we associate a sigma-finite measure Q on (Omega, F), such that for all t >= 0, and for all events Lambda(t) is an element of F-t:

Q[Lambda(t), g <= t ] = Ep[1(Lambda t)X(t)].

where g is the last time for which the process X hits zero. The existence of Q has already been proven in several particular cases, some of them are related with Brownian penalization, and others are involved with problems in mathematical finance. More precisely, the existence of Q in the general case gives an answer to a problem stated by Madan, Roynette and Yor, in a paper about the link between the Black-Scholes formula and the last passage times of some particular submartingales. Moreover, the equality defining Q still holds if the fixed time t is replaced by any bounded stopping time. This generalization can be considered as an extension of Doob's optional stopping theorem. (c) 2012 Elsevier B.V. All rights reserved.

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