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V: 18, 191-195, LNM 191 (1971)

**MEYER, Paul-André**

Démonstration simplifiée d'un théorème de Knight (Martingale theory)

A well known theorem (Dambis, Dubins) asserts that a continuous martingale reduces to Brownian motion when time-changed by its own increasing process. Knight's theorem (LN in M**190**) asserts that this operation performed on $n$ orthogonal martingales yields $n$ independent Brownian motions. The result is extended to Poisson processes

Comment: Still simpler proofs can be given, see 1448 (included in Revuz-Yor*Continuous Martingales and Brownian Motion,* Chapter V)

Keywords: Continuous martingales, Changes of time

Nature: Exposition, Original additions

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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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XI: 05, 47-50, LNM 581 (1977)

**DELLACHERIE, Claude**

Deux remarques sur la séparabilité optionnelle (General theory of processes)

Optional separability was defined by Doob,*Ann. Inst. Fourier,* **25**, 1975. See also Benveniste, 1025. The main remark in this paper is the following: given any optional set $H$ with countable dense sections, there exists a continuous change of time $(T_t)$ indexed by $[0,1[$ such that $H$ is the union of all graphs $T_t$ for $t$ dyadic. Thus Doob's theorem amounts to the fact that every optional process becomes separable in the ordinary sense once a suitable continuous change of time has been performed

Keywords: Optional processes, Separability, Changes of time

Nature: Original

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XI: 08, 65-78, LNM 581 (1977)

**EL KAROUI, Nicole**; **MEYER, Paul-André**

Les changements de temps en théorie générale des processus (General theory of processes)

Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care

Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many ``I''s remained in spite of the final joint autorship. See the next paper 1109 for the discontinuous case

Keywords: Changes of time

Nature: Original

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XI: 09, 79-108, LNM 581 (1977)

**EL KAROUI, Nicole**; **WEIDENFELD, Gérard**

Théorie générale et changement de temps (General theory of processes)

The results of the preceding paper 1108 are extended to arbitrary changes of times, i.e., without the continuity assumption on the increasing process. They require even more care

Comment: Unfortunately, the material presentation of this paper is rather poor. For related results, see 1333

Keywords: Changes of time

Nature: Original

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XI: 23, 365-375, LNM 581 (1977)

**DELLACHERIE, Claude**; **STRICKER, Christophe**

Changements de temps et intégrales stochastiques (Martingale theory)

A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see 925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also 1135

Comment: The last section states an interesting open problem

Keywords: Changes of time, Spectral representation

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

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XIII: 33, 385-399, LNM 721 (1979)

**LE JAN, Yves**

Martingales et changement de temps (Martingale theory, Markov processes)

The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to 1108 and 1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary

Keywords: Changes of time, Energy, Douglas formula

Nature: Original

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XIV: 36, 324-331, LNM 784 (1980)

**BARLOW, Martin T.**; **ROGERS, L.C.G.**; **WILLIAMS, David**

Wiener-Hopf factorization for matrices (Markov processes)

Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix

Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in 1437

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time, Markov chains

Nature: Original

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XIV: 37, 332-342, LNM 784 (1980)

**ROGERS, L.C.G.**; **WILLIAMS, David**

Time-substitution based on fluctuating additive functionals (Wiener-Hopf factorization for infinitesimal generators) (Markov processes)

This is a first step towards the extension of 1436 to Markov processes with a general state space

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time

Nature: Original

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XIV: 48, 496-499, LNM 784 (1980)

**COCOZZA-THIVENT, Christiane**; **YOR, Marc**

Démonstration d'un théorème de F. Knight à l'aide de martingales exponentielles (Martingale theory)

This is a new proof of Knight's theorem that (roughly) finitely many orthogonal continuous local martingales, when separately time-changed into Brownian motions, become independent. A similar theorem for the Poisson case is proved in the same way

Comment: See 518 for an earlier proof

Keywords: Changes of time

Nature: Original

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Démonstration simplifiée d'un théorème de Knight (Martingale theory)

A well known theorem (Dambis, Dubins) asserts that a continuous martingale reduces to Brownian motion when time-changed by its own increasing process. Knight's theorem (LN in M

Comment: Still simpler proofs can be given, see 1448 (included in Revuz-Yor

Keywords: Continuous martingales, Changes of time

Nature: Exposition, Original additions

Retrieve article from Numdam

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

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

XI: 05, 47-50, LNM 581 (1977)

Deux remarques sur la séparabilité optionnelle (General theory of processes)

Optional separability was defined by Doob,

Keywords: Optional processes, Separability, Changes of time

Nature: Original

Retrieve article from Numdam

XI: 08, 65-78, LNM 581 (1977)

Les changements de temps en théorie générale des processus (General theory of processes)

Given a filtration $({\cal F}_t)$ and a continuous adapted increasing process $(C_t)$, consider its right inverse $(j_t)$ and left inverse $(i_t)$, and the time-changed filtration $\overline{\cal F}_t={\cal F}_{j_t}$. The problem is to study the relation between optional/previsible processes of the time-changed filtration and time-changed optional/previsible processes of the original filtration, to see how the projections or dual projections are related, etc. The results are satisfactory, and require a lot of care

Comment: This paper was originally an exposition by the second author of an unpublished paper of the first author, and many ``I''s remained in spite of the final joint autorship. See the next paper 1109 for the discontinuous case

Keywords: Changes of time

Nature: Original

Retrieve article from Numdam

XI: 09, 79-108, LNM 581 (1977)

Théorie générale et changement de temps (General theory of processes)

The results of the preceding paper 1108 are extended to arbitrary changes of times, i.e., without the continuity assumption on the increasing process. They require even more care

Comment: Unfortunately, the material presentation of this paper is rather poor. For related results, see 1333

Keywords: Changes of time

Nature: Original

Retrieve article from Numdam

XI: 23, 365-375, LNM 581 (1977)

Changements de temps et intégrales stochastiques (Martingale theory)

A probability space $(\Omega, {\cal F}, P)$ such that $L^1(P)$ is separable (a condition which is often fulfilled) is endowed with a filtration $({\cal F}_t)$ satisfying the usual conditions. Then (extending ideas of Yan, see 925) it is shown that there exists a right continuous strictly increasing process $(O_t)$ such that every optional process is indistinguishable from a deterministic function $f(0_t)$, every previsible process from a deterministic function of $(0_{t-})$. Using the change of time associated with this process, previsible processes of the original filtration are time changed into deterministic processes, and the theory of stochastic integration is reduced to spectral integrals (as Stieltjes integration on the line can be reduced to Lebesgue's). A bounded previsible process $(u_t)$ define a bounded operator $U$ on $L^2$ as follows: starting from $h\in L^2$, construct the closed martingale $E[h|{\cal F}_t] =H_t$, and then $Uh=\int_0^\infty u_s dH_s$. Using the preceding results it is shown that the von Neumann algebra generated by the conditional expectation operators $E[\sc |{\cal F}_T]$ where $T$ is a stopping time consists exactly of these stochastic integral operators. On this point see also 1135

Comment: The last section states an interesting open problem

Keywords: Changes of time, Spectral representation

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

XIII: 33, 385-399, LNM 721 (1979)

Martingales et changement de temps (Martingale theory, Markov processes)

The first part of the paper concerns changes of time by a continuous (not strictly increasing) process, with a detailed computation, for instance, of the continuous martingale part of a time-changed martingale. This is a useful addition to 1108 and 1109. The second part is an application to classical potential theory: the martingale is a harmonic function along Brownian motion in a domain, stopped at the boundary; the change of time is defined by a boundary local time. Then the time-changed Brownian motion is a Markov process on the boundary, the time-changed martingale is purely discontinuous, and the computation of its quadratic norm leads to the Douglas formula, which expresses the Dirichlet integral of the harmonic function by a quadratic double integral of its restriction to the boundary

Keywords: Changes of time, Energy, Douglas formula

Nature: Original

Retrieve article from Numdam

XIV: 36, 324-331, LNM 784 (1980)

Wiener-Hopf factorization for matrices (Markov processes)

Let $(X_t)$ be a continuous-time Markov chain with a finite state space $E$, and a transition semigroup $\exp(tQ)$. Consider the fluctuating additive functional $\phi_t=\int_0^t v(X_s)\,ds$ ($v$ is a function on $E$ which may assume negative values) and the corresponding change of time $\tau_t= \inf\{s:\phi_s>t\}$. The problem is to find the joint distribution of $\tau_t$ and $X(\tau_t)$. This is solved using martingale methods, and implies a purely algebraic result on the structure of the Q-matrix

Comment: A mistake is pointed out by the authors at the end of the paper, and is corrected in 1437

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time, Markov chains

Nature: Original

Retrieve article from Numdam

XIV: 37, 332-342, LNM 784 (1980)

Time-substitution based on fluctuating additive functionals (Wiener-Hopf factorization for infinitesimal generators) (Markov processes)

This is a first step towards the extension of 1436 to Markov processes with a general state space

Keywords: Wiener-Hopf factorizations, Additive functionals, Changes of time

Nature: Original

Retrieve article from Numdam

XIV: 48, 496-499, LNM 784 (1980)

Démonstration d'un théorème de F. Knight à l'aide de martingales exponentielles (Martingale theory)

This is a new proof of Knight's theorem that (roughly) finitely many orthogonal continuous local martingales, when separately time-changed into Brownian motions, become independent. A similar theorem for the Poisson case is proved in the same way

Comment: See 518 for an earlier proof

Keywords: Changes of time

Nature: Original

Retrieve article from Numdam