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XII: 15, 134-137, LNM 649 (1978)

**LÉPINGLE, Dominique**

Une inégalité de martingales (Martingale theory)

The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein

Keywords: Inequalities, Quadratic variation

Nature: Original

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XV: 09, 143-150, LNM 850 (1981)

**FÖLLMER, Hans**

Calcul d'Ito sans probabilités (Stochastic calculus)

It is shown that if a deterministic continuous curve has a ``quadratic variation'' in a suitable sense (which however depends explicitly on a nested sequence of time subdivisions, for example the standard dyadic one), then it satisfies a deterministic ``Ito formula'' when composed with a twice differentiable function. Thus the only place where probability really appears in the derivation of Ito's formula is in the fact that, given any sequence of subdivisions, almost every path of a semimartingale admits a quadratic variation relative to this sequence (though no path may exist which has a quadratic variation relative to all sequences)

Comment: This subject is developed by T. Lyons' work on differential equations driven by non-smooth functions (in*Rev. Math. Iberoamericana* 14, 1998)

Keywords: Stochastic integrals, Change of variable formula, Quadratic variation

Nature: Original

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XVI: 07, 133-133, LNM 920 (1982)

**MEYER, Paul-André**

Appendice : Un résultat de D. Williams (Malliavin's calculus)

This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as ``quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process

Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis

Nature: Exposition

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XVI: 18, 219-220, LNM 920 (1982)

**STRICKER, Christophe**

Les intervalles de constance de $\langle X,X\rangle$ (Martingale theory, Stochastic calculus)

For a continuous (local) martingale $X$, the constancy intervals of $X$ and $<X,X>$ are exactly the same. What about general local martingales? It is proved that $X$ is constant on the constancy intervals of $<X,X>$, and the converse holds if $X$ has the previsible representation property

Keywords: Quadratic variation, Previsible representation

Nature: Original

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XXXI: 12, 113-125, LNM 1655 (1997)

**ELWORTHY, Kenneth David**; **LI, Xu-Mei**; **YOR, Marc**

On the tails of the supremum and the quadratic variation of strictly local martingales (Martingale theory)

The asymptotic tails of the current maximum and the quadratic variation of a positive continuous local martingale are compared. Applications to strict local martingales associated with transient diffusions, such as Bessel processes, and remarkable identities for Bessel functions are given

Comment: In discrete time, see the following article 3113. Related results are due to Takaoka 3313

Keywords: Continuous martingales, Local martingales, Quadratic variation, Maximal process

Nature: Original

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XLIII: 10, 241-268, LNM 2006 (2011)

**BÉRARD BERGERY, Blandine**; **VALLOIS, Pierre**

Convergence at first and second order of some approximations of stochastic integrals (Theory of Brownian motion, Theory of stochastic integrals)

Keywords: Stochastic integration by regularization, Quadratic variation, First and second order convergence, Stochastic Fubini's theorem

Nature: Original

Une inégalité de martingales (Martingale theory)

The following inequality for a discrete time adapted process $(a_n)$ and its conditional expectations $b_n=E[a_n\,|\,{\cal F}_{n-1}]$ is proved: $$\|(\sum_n b_n^2)^{1/2}\|_1\le 2\|(\sum_n a_n^2)^{1/2}\|_1\ .$$ A similar inequality in $L^p$, $1\!<\!p\!<\!\infty$, does not require adaptedness, and is due to Stein

Keywords: Inequalities, Quadratic variation

Nature: Original

Retrieve article from Numdam

XV: 09, 143-150, LNM 850 (1981)

Calcul d'Ito sans probabilités (Stochastic calculus)

It is shown that if a deterministic continuous curve has a ``quadratic variation'' in a suitable sense (which however depends explicitly on a nested sequence of time subdivisions, for example the standard dyadic one), then it satisfies a deterministic ``Ito formula'' when composed with a twice differentiable function. Thus the only place where probability really appears in the derivation of Ito's formula is in the fact that, given any sequence of subdivisions, almost every path of a semimartingale admits a quadratic variation relative to this sequence (though no path may exist which has a quadratic variation relative to all sequences)

Comment: This subject is developed by T. Lyons' work on differential equations driven by non-smooth functions (in

Keywords: Stochastic integrals, Change of variable formula, Quadratic variation

Nature: Original

Retrieve article from Numdam

XVI: 07, 133-133, LNM 920 (1982)

Appendice : Un résultat de D. Williams (Malliavin's calculus)

This result of Williams (never published as such) can be seen in retrospect as the first example of what came to be known as ``quasi-sure analysis''. It is well known that Wiener measure on the space of continuous functions is carried by the set $Q$ of all sample functions whose quadratic variation (along dyadic subdivisions) is equal to $t$ on each interval $[0,t]$. It is shown here that the complement $Q^c$ is not only a set of Wiener measure $0$, but is a polar set for the Ornstein-Uhlenbeck process

Keywords: Ornstein-Uhlenbeck process, Quadratic variation, Polar sets, Quasi-sure analysis

Nature: Exposition

Retrieve article from Numdam

XVI: 18, 219-220, LNM 920 (1982)

Les intervalles de constance de $\langle X,X\rangle$ (Martingale theory, Stochastic calculus)

For a continuous (local) martingale $X$, the constancy intervals of $X$ and $<X,X>$ are exactly the same. What about general local martingales? It is proved that $X$ is constant on the constancy intervals of $<X,X>$, and the converse holds if $X$ has the previsible representation property

Keywords: Quadratic variation, Previsible representation

Nature: Original

Retrieve article from Numdam

XXXI: 12, 113-125, LNM 1655 (1997)

On the tails of the supremum and the quadratic variation of strictly local martingales (Martingale theory)

The asymptotic tails of the current maximum and the quadratic variation of a positive continuous local martingale are compared. Applications to strict local martingales associated with transient diffusions, such as Bessel processes, and remarkable identities for Bessel functions are given

Comment: In discrete time, see the following article 3113. Related results are due to Takaoka 3313

Keywords: Continuous martingales, Local martingales, Quadratic variation, Maximal process

Nature: Original

Retrieve article from Numdam

XLIII: 10, 241-268, LNM 2006 (2011)

Convergence at first and second order of some approximations of stochastic integrals (Theory of Brownian motion, Theory of stochastic integrals)

Keywords: Stochastic integration by regularization, Quadratic variation, First and second order convergence, Stochastic Fubini's theorem

Nature: Original