Exponential functionals of Brownian motion, I: Probability laws at fixed time

Abstract

This paper is the first part of our survey on various results about the distribution of exponential type Brownian functionals defined as an integral over time of geometric Brownian motion. Several related topics are also mentioned.

Published at http://dx.doi.org/10.1214/154957805100000159 in the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• DIFFUSION IN ONE DIMENSIONAL RANDOM MEDIUM AND HYPERBOLIC BROWNIAN MOTION
• Exact Asymptotic Results for Persistence in the Sinai Model with Arbitrary Drift

Cited by in corpus (67)

• Asymptotics for rough stochastic volatility models
• A confidence-based model for asset and derivative prices in the BitCoin market
• A sentiment-based model for the BitCoin: theory, estimation and option pricing
• Markov processes related to the stationary measure for the open KPZ equation
• Exponential functionals of Brownian motion and class-one Whittaker functions
• Law of the absorption time of some positive self-similar Markov processes
• Markov limits of steady states of the KPZ equation on an interval
• The Hunter-Saxton equation with noise
• Matrix Kesten Recursion, Inverse-Wishart Ensemble and Fermions in a Morse Potential
• Relativistic diffusion with friction on a pseudoriemannian manifold
• Hyperbolic normal stochastic volatility model
• O'Connell's process as a vicious Brownian motion
• Far-from-Equilibrium Time Evolution between two Gamma Distributions
• Logistic and $\theta$-logistic models in population dynamics: General analysis and exact results
• Optimal stopping problems for a Brownian motion with a disorder on a finite interval
• Survival probability of mutually killing Brownian motions and the O'Connell process
• Existence of a Fundamental Solution of Partial Differential Equations associated to Asian Options
• Integrability properties and limit theorems for the exit time from a cone of planar Brownian motion
• Law of the exponential functional of a new family of one-sided Levy processes via self-similar continuous state branching processes with immigration and the Wright hypergeometric functions
• Fluctuation Exponents of the KPZ equation on a large torus
• On hyperbolic Bessel processes and beyond
• High temperature behaviors of the directed polymer on a cylinder
• Wigner-Smith matrix, exponential functional of the matrix Brownian motion and matrix Dufresne identity
• Hitting times of Bessel processes
• Hitting spheres on hyperbolic spaces
• Tail asymptotics for diffusion processes, with applications to local volatility and CEV-Heston models
• The equivalent constant-elasticity-of-variance (CEV) volatility of the stochastic-alpha-beta-rho (SABR) model
• On some identities in law involving exponential functionals of Brownian motion and Cauchy variable
• Steady state of the KPZ equation on an interval and Liouville quantum mechanics
• A note on the option price and 'Mass at zero in the uncorrelated SABR model and implied volatility asymptotics'
• Reciprocal Time Relation of Noncolliding Brownian Motion with Drift
• A Free Boundary Characterisation of the Root Barrier for Markov Processes
• On some Brownian functionals and their applications to moments in lognormal and Stein stochastic volatility models
• On the angle between the first and the second Lyapunov vectors in spatio-temporal chaos
• An Economic Bubble Model and Its First Passage Time
• Very rare events for diffusion processes in short time
• Matrix Dufresne Identity
• System of Complex Brownian Motions Associated with the O'Connell Process
• From (Martingale) Schrodinger bridges to a new class of Stochastic Volatility Models
• Invariance of Brownian motion associated with exponential functionals
• Localization length of the $1+1$ continuum directed random polymer
• On the distribution of the time-integral of the geometric Brownian motion
• On the distribution of the integral of the exponential Brownian motion
• Estimates for the Poisson kernel and the evolution kernel on nilpotent meta-abelian groups
• A Girsanov-type formula for a class of anticipative transforms of Brownian motion associated with exponential functionals
• Reciprocal of the First hitting time of the boundary of dihedral wedges by a radial Dunkl process
• Invariance of Brownian motion associated with past and future maxima
• Moments of exponential functionals of Lévy processes on a deterministic horizon -- identities and explicit expressions
• On hitting times of the winding processes of planar Brownian motion and of Ornstein-Uhlenbeck processes, via Bougerol's identity
• Sharp estimates of transition probability density for Bessel process in half-line
• Extensions of Bougerol's identity in law and the associated anticipative path transformations
• On an ordering-dependent generalization of Tutte polynomial
• Exact Distribution of Verhulst process
• Inside the nature of squared Bessel process
• Determinantal Martingales and Interacting Particle Systems
• In the Life Insurance Business Risky Investments are Dangerous
• On two-dimensional extensions of Bougerol's identity in law
• Hitting hyperbolic half-space
• On the exact asymptotics of exit time from a cone of an isotropic $\alpha$-self-similar Markov process with a skew-product structure
• Bivariate Bernstein-gamma functions, potential measures, and asymptotics of exponential functionals of Lévy processes
• Efficient and accurate simulation of the stochastic-alpha-beta-rho model
• Exact probability distribution function for the volatility of cumulative production
• The Brownian Motion on Aff(R) and Quasi-Local Theorems
• Linear stochastic volatility models
• A model of returns for the post-credit-crunch reality: Hybrid Brownian motion with price feedback
• Success probability for selectively neutral invading species in the line model with a random fitness landscape
• First hitting time of the boundary of a wedge of angle $\pi /4$ by a radial Dunkl process