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34
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 98 (9 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 66 (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 ...
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 25 (7 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....
NonColliding Random Walks, Tandem Queues, And Discrete Orthogonal Polynomial Ensembles
, 2001
"... We show that the function h(x) = Q i
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Cited by 21 (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).
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 13 (6 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.
Brownian Motion, Bridge, Excursion, and Meander Characterized by Sampling at Independent Uniform Times
 ELECTRON. J. PROBAB
, 1999
"... For a random process X consider the random vector defined by the values of X at times 0
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Cited by 12 (3 self)
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For a random process X consider the random vector defined by the values of X at times 0 <U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive pairs of these times, where the U n,i are the order statistics of n independent uniform (0, 1) variables, independent of X . The joint law of this random vector is explicitly described when X is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at n independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae a...
Paths in Weyl chambers and random matrices
"... Baryshnikov [3] and Gravner, Tracy & Widom [14] have shown that the largest eigenvalue of a random matrix of the G.U.E. of order d has the same distribution as 1#t1 ##td1#0 +W d (t d1 )] , where W = (W 1 , , W d ) is a ddimensional Brownian motion. We provide a generalization of th ..."
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Cited by 10 (0 self)
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Baryshnikov [3] and Gravner, Tracy & Widom [14] have shown that the largest eigenvalue of a random matrix of the G.U.E. of order d has the same distribution as 1#t1 ##td1#0 +W d (t d1 )] , where W = (W 1 , , W d ) is a ddimensional Brownian motion. We provide a generalization of this formula to all the eigenvalues and give a geometric interpretation. For any Weyl chamber a + of an Euclidean finitedimensional space a, we define a natural continuous path transformation which associates to a path w in a a path w in a+ . This transformation occurs in the description of the asymptotic behaviour of some deterministic dynamical systems on the symmetric space G/K where G is the complex group with chamber a + . When a = R , a + = , x d ); x 1 > x 2 > > x d if W is the Euclidean Brownian motion on a then W is the process of the eigenvalues of the Dyson Brownian motion on the set of Hermitian matrices and (T W )(1) is distributed as the eigenvalues of the G.U.E.
Brownian Motion Reflected On Brownian Motion
 Probab. Theory Related Fields
, 2002
"... Introduction. We will investigate some aspects of the local time, parabolic measure and excursion theory of Brownian motion reected on Brownian motion. Reected Brownian motion in a domain with a timevarying boundary has appeared in several articles (Bass and Burdzy [BB1], Cranston and Le Jan [CLJ], ..."
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Cited by 8 (1 self)
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Introduction. We will investigate some aspects of the local time, parabolic measure and excursion theory of Brownian motion reected on Brownian motion. Reected Brownian motion in a domain with a timevarying boundary has appeared in several articles (Bass and Burdzy [BB1], Cranston and Le Jan [CLJ], El Karoui and Karatzas [EKK1,2], Knight [K] and Soucaliuc, Toth and Werner [STW]) although it had never been the main subject of study until a recent paper of Burdzy, Chen and Sylvester [BCS]. The last article contains a number of theorems on reected Brownian motion and the corresponding heat equation in domains with smooth spacetime boundaries. If the boundary of a spacetime domain is of class C 3 then practically all results on reected Brownian motion in xed domains can be proved in the new setting. [BCS] also shows that various singularities appear in domains which have rough boundaries. The critical shape of the moving boundary seems to be the square root