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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.
Exponential functionals of Lévy processes
 Probabilty Surveys
, 2005
"... Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0 ..."
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Cited by 34 (4 self)
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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of realvalued Lévy processes ξ = (ξt, t ≥ 0). 0
A Note on the Distribution of Integrals of Geometric Brownian Motion ∗
, 2000
"... The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At: = ∫ t exp{Zs}ds, t ≥ 0, where {Zs: s ≥ 0} is a onedimensional ..."
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Cited by 3 (1 self)
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The purpose of this note is to identify an interesting and surprising duality between the equations governing the probability distribution and expected value functional of the stochastic process defined by At: = ∫ t exp{Zs}ds, t ≥ 0, where {Zs: s ≥ 0} is a onedimensional
A Note on the Ruin Problem with Risky Investments
, 2005
"... We reprove a result concerning certain ruin in the classical problem of the probability of ruin with risky investments and several of it’s generalisations. We also provide the combined transition density of the risk and investment processes in the diffusion case. ..."
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We reprove a result concerning certain ruin in the classical problem of the probability of ruin with risky investments and several of it’s generalisations. We also provide the combined transition density of the risk and investment processes in the diffusion case.
The classical CramérLundberg risk model may be written as
, 2005
"... We reprove a result concerning certain ruin in the classical problem of the probability of ruin with risky investments and several of its generalisations. We also provide the combined transition density of the risk and investment processes in the diffusion case. ..."
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We reprove a result concerning certain ruin in the classical problem of the probability of ruin with risky investments and several of its generalisations. We also provide the combined transition density of the risk and investment processes in the diffusion case.
Electronic addresses:
, 1995
"... The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts such as continuous time finance models and onedimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a ..."
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The paper deals with exponential functionals of the linear Brownian motion which arise in different contexts such as continuous time finance models and onedimensional disordered models. We study some properties of these exponential functionals in relation with the problem of a particle coupled to a heat bath in a Wiener potential. Explicit expressions for the distribution of the free energy are presented.
reality: Hybrid Brownian motion with price
, 2009
"... A model of returns for the postcreditcrunch ..."
A Model of Returns for the PostCreditCrunch environment
, 2009
"... The market events of 20072009 have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the mediumterm logreturn dynamics in a market with both fundamental and technical traders. This is based on a Poisson trade arrival ..."
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The market events of 20072009 have reinvigorated the search for realistic return models that capture greater likelihoods of extreme movements. In this paper we model the mediumterm logreturn dynamics in a market with both fundamental and technical traders. This is based on a Poisson trade arrival model with variable size orders. With simplifications we are led to an SDE mixing both arithmetic and geometric Brownian motions, whose solution is an given by a class of integrals of exponentials of Brownian motions, in forms considered by Yor and collaborators. The reduction of the hybrid SDE to a single Brownian motion leads to an SDE of the form considered by Nagahara, which is a type of “Pearson diffusion”, or equivalently a hyperbolic OU SDE. Various dynamics and equilibria are possible depending on the balance of trades. Under meanreverting circumstances we arrive naturally at an equilibrium fattailed return distribution with a Student or Pearson Type IV form. Under less restrictive assumptions richer dynamics are possible. The phenomenon of variance explosion is identified that gives rise to much larger price movements that might have a priori been expected, so that “25σ ” events are realistic. We exhibit simple example solutions of the FokkerPlanck equation that shows how such variance explosion can hide beneath a standard Gaussian facade. These are elementary members of an extended class of distributions with a rich and varied structure, capable of describing a wide range of market behaviours. Several approaches to the density function are possible, and an example of the computation of a hyperbolic VaR is given. The model also suggests generalizations of the Bougerol identity.