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
1024Keywords: Markov chains,
Ray compactification,
Local times,
ExcursionsNature: Original
Retrieve article from Numdam
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 formulaNature: Original Retrieve article from Numdam
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 chainsNature: Original Retrieve article from Numdam
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 chainsNature: Original Retrieve article from Numdam
XIII: 43, 490-494, LNM 721 (1979)
WILLIAMS, David
Conditional excursion theory (
Brownian motion,
Markov processes)
To be completed
Keywords: ExcursionsNature: Original Retrieve article from Numdam
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
1437Keywords: Wiener-Hopf factorizations,
Additive functionals,
Changes of time,
Markov chainsNature: Original Retrieve article from Numdam
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 timeNature: Original Retrieve article from Numdam
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) Retrieve article from Numdam
XVI: 05, 91-94, LNM 920 (1982)
WILLIAMS, David
A potential-theoretic note on the quadratic Wiener-Hopf equation for Q-matrices Retrieve article from Numdam
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 representationNature: Original Retrieve article from Numdam
XXIX: 14, 155-161, LNM 1613 (1995)
WILLIAMS, David
Non-linear Wiener-Hopf theory, 1: an appetizer Retrieve article from Numdam
XLI: 16, 349-369, LNM 1934 (2008)
WILLIAMS, David
A new look at `Markovian' Wiener-Hopf theoryNature: 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
TémoignagesNature: Tribute
