Exponential functionals of Brownian motion, II: Some related diffusion processes

By Hiroyuki Matsumoto, Marc Yor. 2005

387 citations recorded by the source corpus.

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 rst part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diusion processes in random environments and extensions of L evy's and Pitman's theorems are discussed. AMS 2000 subject classications: Primary 60J65; secondary 60J60,

Subfields: Finance, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/f524682dacf6eae012cdae2de20a11b1ce0e661f

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

In Mathematical Physics

In Molecular Biology

Citations in the corpus

Exponential functionals of Brownian motion, I: Probability laws at fixed time (2005, background): “The Brownian motions on hyperbolic spaces will be discussed in the second part of the survey [51].”
Fitting combinations of exponentials to probability distributions (2007, background): “The distribution of a stochastic annuity with an exponential lifetime is known in closed form; the mathematical result was first derived by Yor (1992), but see Dufresne (2004a) for the financial interpretation.”
The Lamperti representation of real-valued self-similar Markov processes (2011, methodology): “Since all Bessel processes are obtained as the images by the Lamperti representation of the exponential of Brownian motion with drift, see, for example, Carmona, Petit and Yor [6] or Yor [15], we obtain the following for x ∈ R∗, X0 t = x exp { ξ0 τ(t |x|−α) } , X t = x exp { ξ τ(t |x|−α) } , t ≥ 0, ”
Asymptotic results for exponential functionals of Levy processes (2016, background): “The main strategy of Böinghoff and Hutzenthaler (2012) and Palau and Pardo (2015a) is the formula of Yor (1992) for the distribution of the exponential functional of the Brownian motion with drift; see also Matsumoto and Yor (2003).”
Law of the absorption time of some positive self-similar Markov processes (2009, neutral): “12 03 v1 [ m at h. PR ] 6 N ov 2 00”
The Hazard Potential (2006, methodology): “For convenience, we denote H(t) by Ht, and, following the notation of Yor (1992), note that”
A survey and some generalizations of Bessel processes (2003, background): “We know from Dufresne (1990) – see also Pollack and Siegmund (1985) and Yor (1992b), Théorème 1) – thatð1 0 exp 2(Bs þ s)ds ¼ (d) 1 2Z( ) , where Z , .”
Analytical Calculation of Risk Measures for Variable Annuity Guaranteed Benefits (2012, background): “Many other applications of these techniques can be seen in Geman and Yor [12].”
The log-normal approximation in financial and other computations (2004, methodology): “Yor (2001, p. 54) recently found that the formula (2.4) for the moments of M t had previously appeared in Ramakrishnan (1954), in relation to an astronomical model.”
Bessel processes, the integral of geometric Brownian motion, and Asian options (2003, methodology): “The first of these, as discussed in Part IV, is inspired by [29], and we think Yor could have given this argument had he been aware of the financial motivation for letting the parameter ν be negative.”