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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 (10 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.
The Heat Kernel on Noncompact Symmetric Spaces
"... he origin 0 = eK is identified with p. Let a be a Cartan subspace of p, let m be the centralizer of a in k and let g = a # { # ### g# be the root space decomposition of g with respect to a. Select in a a positive Weyl chamber a , in # the corresponding sets # of positive roots, # 0 of ..."
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Cited by 10 (0 self)
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he origin 0 = eK is identified with p. Let a be a Cartan subspace of p, let m be the centralizer of a in k and let g = a # { # ### g# be the root space decomposition of g with respect to a. Select in a a positive Weyl chamber a , in # the corresponding sets # of positive roots, # 0 of positive indivisible roots, # of simple roots, and in g the corresponding nilpotent subalgebra n = ### +g# . Let # = + m# # be the half sum of positive roots, counted with multiplicities 2000 Mathematics Subject Classification. Primary 22E30, 35B50, 43A85, 58J35; Secondary 22E46, 43A80, 43A90. Key words and phrases. Abel transform, heat kernel, maximum principle, semisimple Lie group, symmetric space, subLaplacian. Both authors partially supported by the European Commission (IHP Network HARP ) Typeset by A M ST E X m# = dim g # , let # = dim a be the rank of X and let n = # + + m# be the dimension of X. Finally A = exp a and N = exp n are closed subgroups of G and M denotes
Some Generalizations of Bessel Processes
, 1997
"... Contents Introduction 1 Main properties of Bessel processes : : : : : : : : : : : : : : : 2 1 First hitting times of radial OrnsteinUhlenbeck processes 7 2 BESQ processes with negative dimensions and extensions 14 3 Time reversal 22 3.1 Doob's htransform : : : : : : : : : : : : : : : : : : : : ..."
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Contents Introduction 1 Main properties of Bessel processes : : : : : : : : : : : : : : : 2 1 First hitting times of radial OrnsteinUhlenbeck processes 7 2 BESQ processes with negative dimensions and extensions 14 3 Time reversal 22 3.1 Doob's htransform : : : : : : : : : : : : : : : : : : : : : : : : 25 3.2 Checking a time reversal theorem in ElworthyLiYor : : : : 28 3.3 The threeparametersfamily of processes with law P ffi;¯ : : : : 31 Bibliography 38 i Introduction Bessel processes are a oneparameter family of diffusion processes that appear in many financial problems and have remarkable properties. Following this introduction we recall the definition of Bessel processes and their properties in detail (see I.1  I.6). One important property of Bessel processes is, that the transition densities ar
Limiting behaviors of the Brownian motions on
, 901
"... brownian motions on hyperbolic spaces 1 ..."
Limiting behaviors of the Brownian
, 901
"... Abstract: By adopting the upper half space realizations of the real, complex and quaternionic hyperbolic spaces and solving the corresponding stochastic differential equations, we can represent the Brownian motions on these classical families of the hyperbolic spaces as explicit Wiener functionals. ..."
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Abstract: By adopting the upper half space realizations of the real, complex and quaternionic hyperbolic spaces and solving the corresponding stochastic differential equations, we can represent the Brownian motions on these classical families of the hyperbolic spaces as explicit Wiener functionals. Using the representations, we show that the almost sure convergence of the Brownian motions and the central limit theorems for the radial components as time tends to infinity are easily obtained. We also give a straightforward strategy to obtain the explicit expressions for the Poisson kernels by combining the representations with some results on the distributions of the random variables which are defined by the perpetual (infinite) integrals of the usual geometric Brownian motions with negative drifts.