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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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XV: 16, 227-250, LNM 850 (1981)

**ROGERS, L.C.G.**

Williams' characterization of the Brownian excursion law: proof and applications (Brownian motion)

In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams*Diffusions, Markov Processes and Martingales,* Wiley 1979, II.67), and it collects a number of applications, including the Azéma-Yor approach to Skorohod's imbedding theorem (1306)

Keywords: Excursions, Explicit laws, Bessel processes, Skorohod imbedding

Nature: Original

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XVI: 04, 41-90, LNM 920 (1982)

**LONDON, R.R.**; **McKEAN, Henry P.**; **ROGERS, L.C.G.**; **WILLIAMS, David**

A martingale approach to some Wiener-Hopf problems (two parts)

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XVIII: 03, 42-55, LNM 1059 (1984)

**ROGERS, L.C.G.**

Brownian local times and branching processes

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XXIII: 17, 186-197, LNM 1372 (1989)

**ROGERS, L.C.G.**

Multiple points of Markov processes in a complete metric space

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XLVII: 17, 321-338, LNM 2137 (2015)

**DUEMBGEN, Moritz**; **ROGERS, L. C. G.**

The Joint Law of the Extrema, Final Value and Signature of a Stopped Random Walk

Nature: Original

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

XV: 16, 227-250, LNM 850 (1981)

Williams' characterization of the Brownian excursion law: proof and applications (Brownian motion)

In the early eighties, Ito's rigorous approach to Lévy's ideas on excursions, aroused much enthusiasm, as people discovered it led to simple and conceptual proofs of most classical results on Brownian motion, and of many new ones. This paper contains the first published proof of the celebrated description of the Ito measure discovered by Williams (Williams

Keywords: Excursions, Explicit laws, Bessel processes, Skorohod imbedding

Nature: Original

Retrieve article from Numdam

XVI: 04, 41-90, LNM 920 (1982)

A martingale approach to some Wiener-Hopf problems (two parts)

Retrieve article from Numdam

XVIII: 03, 42-55, LNM 1059 (1984)

Brownian local times and branching processes

Retrieve article from Numdam

XXIII: 17, 186-197, LNM 1372 (1989)

Multiple points of Markov processes in a complete metric space

Retrieve article from Numdam

XLVII: 17, 321-338, LNM 2137 (2015)

The Joint Law of the Extrema, Final Value and Signature of a Stopped Random Walk

Nature: Original