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X: 14, 216-234, LNM 511 (1976)

**WILLIAMS, David**

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams*Diffusions, Markov Processes and Martingales,* vol. 1 (second edition), Wiley 1994. See also 1024

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

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X: 15, 235-239, LNM 511 (1976)

**WILLIAMS, David**

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see*Ann. Prob.* **3**, 1975. For modern references, we should ask Yor

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

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X: 24, 505-520, LNM 511 (1976)

**WILLIAMS, David**

The Q-matrix problem 2: Kolmogorov backward equations (Markov processes)

This is an addition to 1014, the problem being now of constructing a chain whose transition probabilities satisfy the Kolmogorov backward equations, as defined in a precise way in the paper. A different construction is required

Keywords: Markov chains

Nature: Original

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XII: 22, 310-331, LNM 649 (1978)

**WILLIAMS, David**

The Q-matrix problem 3: The Lévy-kernel problem for chains (Markov processes)

After solving the Q-matrix problem in 1014, the author constructs here a Markov chain from a Q-matrix on a countable space $I$ which satisfies several desirable conditions. Among them, the following: though the process is defined on a (Ray) compactification of $I$, the Q-matrix should describe the full Lévy kernel. Otherwise stated, whenever the process jumps, it does so from a point of $I$ to a point of $I$. The construction is extremely delicate

Keywords: Markov chains

Nature: Original

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XIII: 43, 490-494, LNM 721 (1979)

**WILLIAMS, David**

Conditional excursion theory (Brownian motion, Markov processes)

To be completed

Keywords: Excursions

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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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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XVI: 05, 91-94, LNM 920 (1982)

**WILLIAMS, David**

A potential-theoretic note on the quadratic Wiener-Hopf equation for Q-matrices

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XVII: 21, 194-197, LNM 986 (1983)

**PRICE, Gareth C.**; **WILLIAMS, David**

Rolling with `slipping': I (Stochastic calculus, Stochastic differential geometry)

If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$

Keywords: Brownian motion in a manifold, Previsible representation

Nature: Original

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XXIX: 14, 155-161, LNM 1613 (1995)

**WILLIAMS, David**

Non-linear Wiener-Hopf theory, 1: an appetizer

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XLI: 16, 349-369, LNM 1934 (2008)

**WILLIAMS, David**

A new look at `Markovian' Wiener-Hopf theory

Nature: Original

XLVII: 01, xi-xxxi, LNM 2137 (2015)

**AZÉMA, Jacques**; **BARRIEU, Pauline**; **BERTOIN, Jean**; **CABALLERO, Maria Emilia**; **DONATI-MARTIN, Catherine**; **ÉMERY, Michel**; **HIRSCH, Francis**; **HU, Yueyun**; **LEDOUX, Michel**; **NAJNUDEL, Joseph**; **MANSUY, Roger**; **MICLO, Laurent**; **SHI, Zhan**; **WILLIAMS, David**

Témoignages

Nature: Tribute

The Q-matrix problem (Markov processes)

This paper completely solves the Q-matrix problem (find necessary and sufficient conditions for an infinite matrix $q_{ij}$ to be the pointwise derivative at $0$ of a transition matrix) in the case when all states are instantaneous. Though the statement of the problem and the two conditions given are elementary and simple, the proof uses sophisticated ``modern'' methods. The necessity of the conditions is proved using the Ray-Knight compactification method, the converse is a clever construction which is merely sketched

Comment: This paper crowns nearly 20 years of investigations of this problem by the English school. It contains a promise of a detailed proof which apparently was never published. See the section of Markov chains in Rogers-Williams

Keywords: Markov chains, Ray compactification, Local times, Excursions

Nature: Original

Retrieve article from Numdam

X: 15, 235-239, LNM 511 (1976)

On a stopped Brownian motion formula of H.M.~Taylor (Brownian motion)

This formula gives the joint distribution of $X_T$ and $T$, where $X$ is standard Brownian motion and $T$ is the first time $M_T-X_T=a$, $M_t$ denoting the supremum of $X$ up to time $t$. Two different new proofs are given

Comment: For the original proof of Taylor see

Keywords: Stopping times, Local times, Ray-Knight theorems, Cameron-Martin formula

Nature: Original

Retrieve article from Numdam

X: 24, 505-520, LNM 511 (1976)

The Q-matrix problem 2: Kolmogorov backward equations (Markov processes)

This is an addition to 1014, the problem being now of constructing a chain whose transition probabilities satisfy the Kolmogorov backward equations, as defined in a precise way in the paper. A different construction is required

Keywords: Markov chains

Nature: Original

Retrieve article from Numdam

XII: 22, 310-331, LNM 649 (1978)

The Q-matrix problem 3: The Lévy-kernel problem for chains (Markov processes)

After solving the Q-matrix problem in 1014, the author constructs here a Markov chain from a Q-matrix on a countable space $I$ which satisfies several desirable conditions. Among them, the following: though the process is defined on a (Ray) compactification of $I$, the Q-matrix should describe the full Lévy kernel. Otherwise stated, whenever the process jumps, it does so from a point of $I$ to a point of $I$. The construction is extremely delicate

Keywords: Markov chains

Nature: Original

Retrieve article from Numdam

XIII: 43, 490-494, LNM 721 (1979)

Conditional excursion theory (Brownian motion, Markov processes)

To be completed

Keywords: Excursions

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

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

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

Retrieve article from Numdam

XVI: 05, 91-94, LNM 920 (1982)

A potential-theoretic note on the quadratic Wiener-Hopf equation for Q-matrices

Retrieve article from Numdam

XVII: 21, 194-197, LNM 986 (1983)

Rolling with `slipping': I (Stochastic calculus, Stochastic differential geometry)

If $Z$ and $\tilde Z$ are two Brownian motions on the unit sphere for the filtration of $Z$, there differentials $\partial Y=(\partial Z) \times Z$ (Stratonovich differentials and vector product) and $\partial\tilde Y$ (similarly defined) are related by $d\tilde Y = H dY$, where $H$ is a previsible, orthogonal transformation such that $HZ=\tilde Z$

Keywords: Brownian motion in a manifold, Previsible representation

Nature: Original

Retrieve article from Numdam

XXIX: 14, 155-161, LNM 1613 (1995)

Non-linear Wiener-Hopf theory, 1: an appetizer

Retrieve article from Numdam

XLI: 16, 349-369, LNM 1934 (2008)

A new look at `Markovian' Wiener-Hopf theory

Nature: Original

XLVII: 01, xi-xxxi, LNM 2137 (2015)

Témoignages

Nature: Tribute