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12
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 200 (15 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Pinching and twisting Markov processes
, 2001
"... Abstract. We develop a technique for \partially collapsing " one Markovprocesses to produce another. The state space of the new Markov process is obtained by a pinching operation that identi es points of the original state space via an equivalence relation. To ensure that the new process is ..."
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Cited by 6 (2 self)
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Abstract. We develop a technique for \partially collapsing &quot; one Markovprocesses to produce another. The state space of the new Markov process is obtained by a pinching operation that identi es points of the original state space via an equivalence relation. To ensure that the new process is Markovian we need to introduce a randomised twist according to an appropriate probability kernel. Informally, this twist randomises over the uncollapsed region of the state space when the process leaves the collapsed region. The Markovianity of the new process is ensured by suitable intertwining relations between the semigroup of the original process and the pinching and twising operations. We construct the new Markov process, identify its resolvent and transition function and, under some natural assumptions, exhibit a core for its generator. We also investigate its excursion decomposition. We apply our theory to a number of examples, including Walsh's spider and a process similar to one introduced by Sowers in studying stochastic averaging. Short Title: Pinching and twisting 1.
HITTING TIMES AND INTERLACING EIGENVALUES: A STOCHASTIC APPROACH USING INTERTWININGS
"... Abstract. We develop a systematic matrixanalytic approach, based on intertwinings of Markov semigroups, for proving theorems about hittingtime distributions for finitestate Markov chains—an approach that (sometimes) deepens understanding of the theorems by providing corresponding samplepathbys ..."
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Abstract. We develop a systematic matrixanalytic approach, based on intertwinings of Markov semigroups, for proving theorems about hittingtime distributions for finitestate Markov chains—an approach that (sometimes) deepens understanding of the theorems by providing corresponding samplepathbysamplepath stochastic constructions. We employ our approach to give new proofs and constructions for two theorems due to Mark Brown, theorems giving two quite different representations of hittingtime distributions for finitestate Markov chains started in stationarity. The proof, and corresponding construction, for one of the two theorems elucidates an intriguing connection between hittingtime distributions and the interlacing eigenvalues theorem for bordered symmetric matrices. 1. Introduction and Outline
Fractional intertwinings between two Markov semigroups
"... Abstract We define the notion of αintertwining between two Markov Feller semigroups on R+ and we give some examples. The 1intertwining, in particular, is merely the intertwining via the first derivative operator. It can be used in the study of the existence of pseudoinverses, a notion recently i ..."
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Abstract We define the notion of αintertwining between two Markov Feller semigroups on R+ and we give some examples. The 1intertwining, in particular, is merely the intertwining via the first derivative operator. It can be used in the study of the existence of pseudoinverses, a notion recently introduced by MadanRoynetteYor [12] and RoynetteYor [15]. Key words Markov semigroup, intertwining, fractional derivative, fractional integration, pseudoinverse 1
Stochastic duality of Markov processes: a study via generators. arXiv:1304.1688
, 2013
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A PATHTRANSFORMATION FOR RANDOM WALKS AND THE ROBINSONSCHENSTED CORRESPONDENCE
"... Abstract. The author and Marc Yor recently introduced a pathtransformation G (k) with the property that, for X belonging to a certain class of random walks on Z k +, the transformed walk G(k) (X) has the same law as the original walk conditioned never to exit the Weyl chamber {x: x1 ≤···≤xk}. In th ..."
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Abstract. The author and Marc Yor recently introduced a pathtransformation G (k) with the property that, for X belonging to a certain class of random walks on Z k +, the transformed walk G(k) (X) has the same law as the original walk conditioned never to exit the Weyl chamber {x: x1 ≤···≤xk}. In this paper, we show that G (k) is closely related to the RobinsonSchensted algorithm, and use this connection to give a new proof of the above representation theorem. The new proof is valid for a larger class of random walks and yields additional information about the joint law of X and G (k) (X). The corresponding results for the Brownian model are recovered by Donsker’s theorem. These are connected with Hermitian Brownian motion and the Gaussian Unitary Ensemble of random matrix theory. The connection we make between the pathtransformation G (k) and the RobinsonSchensted algorithm also provides a new formula and interpretation for the latter. This can be used to study properties of the RobinsonSchensted algorithm and, moreover, extends easily to a continuous setting. 1. Introduction and