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Some remarks on the martingales satisfying the structure equation \([X,X]_t= t+\int_0^t \beta X_s- dX_s\). (English) Zbl 0982.60033

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXXV. Berlin: Springer. Lect. Notes Math. 1755, 87-97 (2001).
This paper studies sample path and local time properties of martingales obeying the structure equation \[ [X, X]_t= t+ \int^t_0 \beta X_s- dX_s,\tag{\(*\)} \] where \(\beta\) is a real number. Moreover, it is shown that the Bouleau-Yor extension of Itô’s formula to functions with derivatives in \(L^\infty_{\text{loc}}\) also holds for martingales satisfying \((*)\). This fact, in turn, is used to prove inequalities of the Burkholder-Davis-Gundy type for martingales obeying \((*)\).
For the entire collection see [Zbl 0960.00020].

MSC:

60G42 Martingales with discrete parameter
60G44 Martingales with continuous parameter
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