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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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VI: 07, 101-104, LNM 258 (1972)

**KAZAMAKI, Norihiko**

Krickeberg's decomposition for local martingales (Martingale theory)

It is shown that a local martingale bounded in $L^1$ is a difference of two (minimal) positive local martingales

Keywords: Local martingales, Krickeberg decomposition

Nature: Original

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VI: 08, 105-108, LNM 258 (1972)

**KAZAMAKI, Norihiko**

Note on a stochastic integral equation (Stochastic calculus)

Though this paper has been completely superseded by the theory of stochastic differential equations with respect to semimartingales (see 1124), it has a great historical importance as the first step in this direction: the semimartingale involved is the sum of a locally square integrable martingale and a continuous increasing process

Comment: The author developed the subject further in*Tôhoku Math. J.* **26**, 1974

Keywords: Stochastic differential equations

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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X: 04, 40-43, LNM 511 (1976)

**KAZAMAKI, Norihiko**

A simple remark on the conditioned square functions for martingale transforms (Martingale theory)

This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time

Comment: The author has published a paper on a related subject in*Tôhoku Math. J.*, **28**, 1976

Keywords: Angle bracket, Inequalities

Nature: Original

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X: 27, 536-539, LNM 511 (1976)

**KAZAMAKI, Norihiko**

A characterization of $BMO$ martingales (Martingale theory)

A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$

Comment: For an extension to non-continuous martingales, see 1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN**1579**, 1994

Keywords: $BMO$

Nature: Original

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XII: 05, 47-50, LNM 649 (1978)

**KAZAMAKI, Norihiko**

A remark on a problem of Girsanov (Martingale theory)

It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale

Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see 1832

Keywords: Stochastic exponentials, $BMO$

Nature: Original

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XIX: 23, 275-277, LNM 1123 (1985)

**KAZAMAKI, Norihiko**

A counterexample related to $A_p$-weights in martingale theory

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XXIII: 03, 47-51, LNM 1372 (1989)

**KAZAMAKI, Norihiko**

A remark on the class of martingales with bounded quadratic variation

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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

VI: 07, 101-104, LNM 258 (1972)

Krickeberg's decomposition for local martingales (Martingale theory)

It is shown that a local martingale bounded in $L^1$ is a difference of two (minimal) positive local martingales

Keywords: Local martingales, Krickeberg decomposition

Nature: Original

Retrieve article from Numdam

VI: 08, 105-108, LNM 258 (1972)

Note on a stochastic integral equation (Stochastic calculus)

Though this paper has been completely superseded by the theory of stochastic differential equations with respect to semimartingales (see 1124), it has a great historical importance as the first step in this direction: the semimartingale involved is the sum of a locally square integrable martingale and a continuous increasing process

Comment: The author developed the subject further in

Keywords: Stochastic differential equations

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

X: 04, 40-43, LNM 511 (1976)

A simple remark on the conditioned square functions for martingale transforms (Martingale theory)

This is a problem of discrete martingale theory, giving inequalities between the conditioned square funtions (discrete angle brackets) of martingale transforms of two martingales related through a change of time

Comment: The author has published a paper on a related subject in

Keywords: Angle bracket, Inequalities

Nature: Original

Retrieve article from Numdam

X: 27, 536-539, LNM 511 (1976)

A characterization of $BMO$ martingales (Martingale theory)

A $L^2$ bounded continuous martingale belongs to $BMO$ if and only if its stochastic exponential satisfies some (Muckenhoupt) condition $A_p$ for $p>1$

Comment: For an extension to non-continuous martingales, see 1125. For a recent survey see the monograph of Kazamaki on exponential martingales and $BMO$, LN

Keywords: $BMO$

Nature: Original

Retrieve article from Numdam

XII: 05, 47-50, LNM 649 (1978)

A remark on a problem of Girsanov (Martingale theory)

It is shown that, if $M$ is a continuous local martingale which belongs to $BMO$, its stochastic exponential is a uniformly integrable martingale

Comment: This has become a well-known result. It is false for complex valued martingales, even bounded ones: see 1832

Keywords: Stochastic exponentials, $BMO$

Nature: Original

Retrieve article from Numdam

XIX: 23, 275-277, LNM 1123 (1985)

A counterexample related to $A_p$-weights in martingale theory

Retrieve article from Numdam

XXIII: 03, 47-51, LNM 1372 (1989)

A remark on the class of martingales with bounded quadratic variation

Retrieve article from Numdam