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13 matches found
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
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