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64
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 205 (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.
Brownian Motion in a Weyl Chamber, NonColliding Particles, and Random Matrices
, 1997
"... . Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting posit ..."
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Cited by 91 (2 self)
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. Let n particles move in standard Brownian motion in one dimension, with the process terminating if two particles collide. This is a specific case of Brownian motion constrained to stay inside a Weyl chamber; the Weyl group for this chamber is An\Gamma1 , the symmetric group. For any starting positions, we compute a determinant formula for the density function for the particles to be at specified positions at time t without having collided by time t. We show that the probability that there will be no collision up to time t is asymptotic to a constant multiple of t \Gamman(n\Gamma1)=4 as t goes to infinity, and compute the constant as a polynomial of the starting positions. We have analogous results for the other classical Weyl groups; for example, the hyperoctahedral group Bn gives a model of n independent particles with a wall at x = 0. We can define Brownian motion on a Lie algebra, viewing it as a vector space; the eigenvalues of a point in the Lie algebra correspond to a point ...
Littelmann paths and Brownian paths
, 2004
"... We study some path transformations related to Littelmann path model and their applications to representation theory and Brownian motion in a Weyl chamber. ..."
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Cited by 54 (2 self)
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We study some path transformations related to Littelmann path model and their applications to representation theory and Brownian motion in a Weyl chamber.
Brownian analogues of Burke’s theorem
, 2001
"... We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of a (stable, stationary) M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burke’s theorem for the Brownian queue was stated and proved by Harrison (Brownian Mo ..."
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Cited by 50 (8 self)
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We discuss Brownian analogues of a celebrated theorem, due to Burke, which states that the output of a (stable, stationary) M/M/1 queue is Poisson, and the related notion of quasireversibility. A direct analogue of Burke’s theorem for the Brownian queue was stated and proved by Harrison (Brownian Motion and Stochastic Flow Systems, Wiley, New York, 1985). We present several different proofs of this and related results. We also present an analogous result for geometric functionals of Brownian motion. By considering series of queues in tandem, these theorems can be applied to a certain class of directed percolation and directed polymer models. It was recently discovered that there is a connection between this directed percolation model and the GUE random matrix ensemble. We extend and give a direct proof of this connection in the twodimensional case. In all of the above, reversibility plays a key role.
Directed polymers and the quantum Toda lattice
, 2009
"... We give a characterization of the law of the partition function of a Brownian directed polymer model in terms of the eigenfunctions of the quantum Toda lattice. This is obtained via a multidimensional generalization of theorem of Matsumoto and Yor concerning exponential functionals of Brownian motio ..."
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Cited by 47 (2 self)
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We give a characterization of the law of the partition function of a Brownian directed polymer model in terms of the eigenfunctions of the quantum Toda lattice. This is obtained via a multidimensional generalization of theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion.
NonColliding Random Walks, Tandem Queues, And Discrete Orthogonal Polynomial Ensembles
, 2001
"... We show that the function h(x) = Q i<j (x j x i ) is harmonic for any random walk in R k with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber W = fx : x 1 < x 2 < < x k g onto a point where h vanishes, we ..."
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Cited by 34 (4 self)
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We show that the function h(x) = Q i<j (x j x i ) is harmonic for any random walk in R k with exchangeable increments, provided the required moments exist. For the subclass of random walks which can only exit the Weyl chamber W = fx : x 1 < x 2 < < x k g onto a point where h vanishes, we define the corresponding Doob htransform. For certain special cases, we show that the marginal distribution of the conditioned process at a fixed time is given by a familiar discrete orthogonal polynomial ensemble. These include the Krawtchouk and Charlier ensembles, where the underlying walks are binomial and Poisson, respectively. We refer to the corresponding conditioned processes in these cases as the Krawtchouk and Charlier processes. In [O'Connell and Yor (2001b)], a representation was obtained for the Charlier process by considering a sequence of M/M/1 queues in tandem. We present the analogue of this representation theorem for the Krawtchouk process, by considering a sequence of discretetime M/M/1 queues in tandem. We also present related results for random walks on the circle, and relate a system of noncolliding walks in this case to the discrete analogue of the circular unitary ensemble (CUE).
A REPRESENTATION FOR NONCOLLIDING RANDOM WALKS
 ELECTRONIC COMMUNICATIONS IN PROBABILITY
, 2002
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The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest
, 1997
"... Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solu ..."
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Cited by 32 (8 self)
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Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solution X of the Ito SDE dXv = n 4 \Gamma X 2 v \Gamma t \Gamma R v 0 Xudu \Delta \Gamma1 o dv + 2 p XvdBv on the interval [0; V t (X)), where V t (X) := inffv : R v 0 Xudu = tg, and Xv = 0 for all v V t (X). This conditioned form of the RayKnight description of Brownian local times arises from study of the asymptotic distribution as n !1 and 2k= p n ! ` of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points. The SDE is the continuous analog of a simple description of a GaltonWatson branching process conditioned on its total progeny....
Selfsimilar processes with independent increments associated with Lévy and Bessel processes
, 2001
"... Wolfe [28] and Sato [24] gave two different representations of a random variable X 1 with a selfdecomposable distribution in terms of processes with independent increments. This paper shows how either of these representations follows easily from the other, and makes these representations more e ..."
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Cited by 25 (8 self)
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Wolfe [28] and Sato [24] gave two different representations of a random variable X 1 with a selfdecomposable distribution in terms of processes with independent increments. This paper shows how either of these representations follows easily from the other, and makes these representations more explicit when X 1 is either a first or last passage time for a Bessel process.