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10
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 97 (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 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 20 (7 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.
Random matrices, noncolliding processes and queues
 TO APPEAR IN SÉMINAIRE DE PROBABILITÉS XXXVI
, 2002
"... This is survey of some recent results connecting random matrices, noncolliding processes and queues. ..."
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Cited by 20 (3 self)
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This is survey of some recent results connecting random matrices, noncolliding processes and queues.
Conditioned stochastic differential equations: theory, examples and application to finance
, 2002
"... ..."
On the free energy of a directed polymer in a Brownian environment
, 2006
"... We prove a formula conjectured in [14] for the free energy density of a directed polymer in a Brownian environment in 1 + 1 dimensions. Mathematics Subject Classification (2000): 60K37,82D30,60K25,60J65 1 ..."
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Cited by 5 (0 self)
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We prove a formula conjectured in [14] for the free energy density of a directed polymer in a Brownian environment in 1 + 1 dimensions. Mathematics Subject Classification (2000): 60K37,82D30,60K25,60J65 1
EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION
, 2007
"... Abstract. We find a simple expression for the probability density of R exp(Bs − s/2)ds in terms of its distribution function and the distribution function for the time integral of exp(Bs + s/2). The relation is obtained with a change of measure argument where expectations over events determined by t ..."
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Cited by 1 (1 self)
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Abstract. We find a simple expression for the probability density of R exp(Bs − s/2)ds in terms of its distribution function and the distribution function for the time integral of exp(Bs + s/2). The relation is obtained with a change of measure argument where expectations over events determined by the time integral are replaced by expectations over the entire probability space. We develop precise information concerning the lower tail probabilities for these random variables as well as for time integrals of geometric Brownian motion with arbitrary constant drift. In particular, E [ exp ` θ / R exp(Bs)ds ´ ] is finite iff θ < 2. We present a new formula for the price of an Asian call option.
ELECTRONIC COMMUNICATIONS in PROBABILITY FURTHER EXPONENTIAL GENERALIZATION OF PITMAN’S 2MX THEOREM
, 2001
"... Diffusion processes, Exponential analogue of the 2M − X Pitman’s theorem We present a class of processes which enjoy an exponential analogue of Pitman’s 2MX theorem, improving hence some works of H. Matsumoto and M. Yor. 1 ..."
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Diffusion processes, Exponential analogue of the 2M − X Pitman’s theorem We present a class of processes which enjoy an exponential analogue of Pitman’s 2MX theorem, improving hence some works of H. Matsumoto and M. Yor. 1
Brownian environment
, 2007
"... We prove a formula conjectured in [14] for the free energy density of a directed polymer in a Brownian environment in 1 + 1 dimensions. Mathematics Subject Classification (2000): 60K37,82D30,60K25,60J65 ..."
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We prove a formula conjectured in [14] for the free energy density of a directed polymer in a Brownian environment in 1 + 1 dimensions. Mathematics Subject Classification (2000): 60K37,82D30,60K25,60J65