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VI: 06, 98-100, LNM 258 (1972)

**KAZAMAKI, Norihiko**

Examples on local martingales (Martingale theory)

Two simple examples are given, the first one concerning the filtration generated by an exponential stopping time, the second one showing that local martingales are not preserved under time changes (Kazamaki,*Zeit. für W-theorie,* **22**, 1972)

Keywords: Changes of time, Local martingales, Weak martingales

Nature: Original

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VII: 12, 118-121, LNM 321 (1973)

**KAZAMAKI, Norihiko**

Une note sur les martingales faibles (Martingale theory)

Métivier has distinguished in the general theory of processes localization from prelocalization: a process $X$ is a local martingale if there exist stopping times $T_n$ increasing to infinity and martingales $M_n$ such that $X=M_n$ on the closed interval $[0,T_n]$ (omitting for simplicity the convention about time $0$). Replacing the closed intervals by open intervals $[0,T_n[$ defines prelocal martingales or*weak martingales.* It is shown that in the filtration generated by one single stopping time, processes which are prelocally martingales (square integrable martingales) are so globally. It follows that prelocal martingales may not be prelocally square integrable

Comment: The interest of weak martingales arises from their invariance by (possibly discontinuous) changes of time, see Kazamaki,*Zeit. für W-theorie,* **22**, 1972

Keywords: Weak martingales

Nature: Original

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XII: 02, 20-21, LNM 649 (1978)

**STRICKER, Christophe**

Une remarque sur les changements de temps et les martingales locales (Martingale theory)

It is well known (see 606) that in general the class of local martingales is not invariant under changes of time. Here it is shown that, if ${\cal F}_0$ is trivial, a process which remains a local martingale under all changes of time (with bounded stopping times) is a true martingale (in full generality, it is so conditionally to ${\cal F}_0$)

Keywords: Changes of time, Weak martingales

Nature: Original

Retrieve article from Numdam

Examples on local martingales (Martingale theory)

Two simple examples are given, the first one concerning the filtration generated by an exponential stopping time, the second one showing that local martingales are not preserved under time changes (Kazamaki,

Keywords: Changes of time, Local martingales, Weak martingales

Nature: Original

Retrieve article from Numdam

VII: 12, 118-121, LNM 321 (1973)

Une note sur les martingales faibles (Martingale theory)

Métivier has distinguished in the general theory of processes localization from prelocalization: a process $X$ is a local martingale if there exist stopping times $T_n$ increasing to infinity and martingales $M_n$ such that $X=M_n$ on the closed interval $[0,T_n]$ (omitting for simplicity the convention about time $0$). Replacing the closed intervals by open intervals $[0,T_n[$ defines prelocal martingales or

Comment: The interest of weak martingales arises from their invariance by (possibly discontinuous) changes of time, see Kazamaki,

Keywords: Weak martingales

Nature: Original

Retrieve article from Numdam

XII: 02, 20-21, LNM 649 (1978)

Une remarque sur les changements de temps et les martingales locales (Martingale theory)

It is well known (see 606) that in general the class of local martingales is not invariant under changes of time. Here it is shown that, if ${\cal F}_0$ is trivial, a process which remains a local martingale under all changes of time (with bounded stopping times) is a true martingale (in full generality, it is so conditionally to ${\cal F}_0$)

Keywords: Changes of time, Weak martingales

Nature: Original

Retrieve article from Numdam