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686
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.
On The Distribution And Asymptotic Results For Exponential Functionals Of Lévy Processes
, 1997
"... The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, ..."
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Cited by 121 (11 self)
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The aim of this note is to study the distribution and the asymptotic behavior of the exponential functional A t := R t 0 e s ds, where ( s ; s 0) denotes a L'evy process. When A1 ! 1, we show that in most cases, the law of A1 is a solution of an integrodifferential equation ; moreover, this law is characterized by its integral moments. When the process is asymptotically ffstable, we prove that t \Gamma1=ff log A t converges in law, as t !1, to the supremum of an ffstable L'evy process ; in particular, if E [ 1 ] ? 0, then ff = 1 and (1=t) log A t converges almost surely to E [ 1 ]. Eventually, we use Girsanov's transform to give the explicit behavior of E \Theta (a +A t ()) \Gamma1 as t ! 1, where a is a constant, and deduce from this the rate of decay of the tail of the distribution of the maximum of a diffusion process in a random L'evy environment.
New upper bounds on sphere packings
, 2001
"... Abstract. We develop an analogue for sphere packing of the linear programming bounds for errorcorrecting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to s ..."
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Cited by 68 (7 self)
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Abstract. We develop an analogue for sphere packing of the linear programming bounds for errorcorrecting codes, and use it to prove upper bounds for the density of sphere packings, which are the best bounds known at least for dimensions 4 through 36. We conjecture that our approach can be used to solve the sphere packing problem in dimensions 8 and 24.
A Survey and Some Generalizations of Bessel Processes
 Bernoulli
, 1999
"... Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. ..."
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Cited by 66 (1 self)
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Bessel processes play an important role in financial mathematics because of their strong relation to financial processes like geometric Brownian motion or CIR processes. We are interested in the first time Bessel processes and more generally, radial OrnsteinUhlenbeck processes hit a given barrier. We give explicit expressions of the Laplace transforms of first hitting times by (squared) radial OrnsteinUhlenbeck processes, i. e., CIR processes. As a natural extension we study squared Bessel processes and squared OrnsteinUhlenbeck processes with negative dimensions or negative starting points and derive their properties. Keywords: First hitting times; CIR processes; Bessel processes; radial Ornstein Uhlenbeck processes; Bessel processes with negative dimensions 1 Introduction Bessel processes have come to play a distinguished role in financial mathematics for at least two reasons, which have a lot to do with the models being usually considered. One of these models is the CoxI...
Optimality and Uniqueness of the Leech Lattice Among Lattices
 arXiv:math.MG/04 03263v1 16
, 2004
"... Abstract. We prove that the Leech lattice is the unique densest lattice in R 24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R 24 can exceed the Leech lattice’s density by a factor of ..."
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Cited by 53 (5 self)
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Abstract. We prove that the Leech lattice is the unique densest lattice in R 24. The proof combines human reasoning with computer verification of the properties of certain explicit polynomials. We furthermore prove that no sphere packing in R 24 can exceed the Leech lattice’s density by a factor of more than 1 + 1.65 · 10 −30, and we give a new proof that E8 is the unique densest lattice in R 8. 1.
ESTIMATING FUNCTIONS OF PROBABILITY DISTRIBUTIONS FROM A FINITE SET OF SAMPLES Part II: Bayes Estimators for Mutual Information, ChiSquared, Covariance, and other Statistics.
"... This paper is the second in a series of two on the problem of estimating a function of a probability distribution from a finite set of samples of that distribution. In the first paper, the Bayes estimator for a function of a probability distribution was introduced, the optimal properties of the Baye ..."
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Cited by 52 (4 self)
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This paper is the second in a series of two on the problem of estimating a function of a probability distribution from a finite set of samples of that distribution. In the first paper, the Bayes estimator for a function of a probability distribution was introduced, the optimal properties of the Bayes estimator were discussed, and the Bayes and frequencycounts estimators for the Shannon entropy were derived and graphically contrasted. In the current paper the analysis of the first paper is extended by the derivation of Bayes estimators for several other functions of interest in statistics and information theory. These functions are (powers of) the mutual information, chisquared for tests of independence, variance, covariance, and average. Finding Bayes estimators for several of these functions requires extensions to the analytical techniques developed in the first paper, and these extensions form the main body of this paper. This paper extends the analysis in other ways as well, for example by enlarging the class of potential priors beyond the uniform prior assumed in the first paper. In particular, the use of the entropic and Dirichlet priors is considered.
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.
Extension problem and Harnack’s inequality for some fractional operators
 Comm. Partial Differential Equations
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A.: Ground states of Nonlinear Schrödinger Equations with Potentials Vanishing at
 Infinity, J. Eur. Math. Soc
"... We considered the nonlinear Schrödinger equation (1) in the case that V (x) ∼ (1 + x)−α and K is bounded and positive. If 0 ≤ α ≤ 2 and 1 < p < n+2 n−2 it is shown that (1) has, for all ε ¿ 1, a bound state concentrating at some point x0, provided x0 is a stable stationary point of the auxi ..."
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Cited by 40 (2 self)
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We considered the nonlinear Schrödinger equation (1) in the case that V (x) ∼ (1 + x)−α and K is bounded and positive. If 0 ≤ α ≤ 2 and 1 < p < n+2 n−2 it is shown that (1) has, for all ε ¿ 1, a bound state concentrating at some point x0, provided x0 is a stable stationary point of the auxiliary potential Q defined in (Q). 1
Brownian excursion area, Wright’s constants in graph enumeration, and other Brownian areas
 PROBABILITY SURVEYS
, 2007
"... This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider bot ..."
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Cited by 35 (9 self)
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This survey is a collection of various results and formulas by different authors on the areas (integrals) of five related processes, viz. Brownian motion, bridge, excursion, meander and double meander; for the Brownian motion and bridge, which take both positive and negative values, we consider both the integral of the absolute value and the integral of the positive (or negative) part. This gives us seven related positive random variables, for which we study, in particular, formulas for moments and Laplace transforms; we also give (in many cases) series representations and asymptotics for density functions and distribution functions. We further study Wright’s constants arising in the asymptotic enumeration of connected graphs; these are known to be closely connected to the moments of the Brownian excursion area. The main purpose is to compare the results for these seven Brownian areas by stating the results in parallel forms; thus emphasizing both the similarities and the differences. A recurring theme is the Airy function which