On the distribution of the integral of the exponential Brownian motion (original) (raw)

A note on the distribution of integrals of geometric Brownian motion

Statistics & Probability Letters, 2001

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 0 exp{Zs}ds, t ≥ 0, where {Zs : s ≥ 0} is a one-dimensional Brownian motion with drift coefficient µ and diffusion coefficient σ 2. In particular, both expected values of the form v(t, x) := Ef (x + At), f homogeneous, as well as the probability density a(t, y)dy := P (At ∈ dy) are shown to be governed by a pair of linear parabolic partial differential equations. Although the equations are not the backward/forward adjoint pairs one would naturally have in the general theory of Markov processes, unifying and remarkably simple derivations of these equations are provided.

Exponential Martingales and Time integrals of Brownian Motion

Arxiv preprint math/0612034, 2006

Abstract. We find a simple expression for the probability density of ∫ exp(Bs − s/2)ds in terms of its distribution function and the distribution function for the time integral of exp(Bs + s/2). The relation is obtained with a change of measure argument where expectations over events ...

On the Distribution of the Square Integral of the Brownian Bridge

The Annals of Probability, 2002

Smirnov obtained the distribution F for his ω 2-test in the form of a certain series. F is identical to the distribution of the the Brownian bridge in the L 2 norm. Smirnov, Kac and Shepp determined the Laplace-Stieltjes transform of F. Anderson and Darling expressed F in terms of Bessel functions. In the present paper we compute the moments of F and their asymptotics, obtain expansions of F and its density f in terms of the parabolic cylinder functions and Laguerre functions, and determine their asymptotics for the small and large values of the argument. A novel derivation of expansions of Smirnov and of Anderson and Darling is obtained.

On the distribution of Brownian areas

The Annals of Applied Probability, 1996

We study the distributions of the areas under the positive parts of a Brownian motion process B and a Brownian bridge process U: with q 1 q Ž .

Density of Skew Brownian Motion and Its Functionals with Application in Finance

Mathematical Finance

We derive the joint density of a Skew Brownian motion, its last visit to the origin, its local and occupation times. The result allows to obtain explicit analytical formulas for pricing European options under both a two valued local volatility model and a displaced diffusion model with constrained volatility.

Small ball probabilities for integrals of weighted Brownian motion

Statistics & Probability Letters, 2000

Let X (t) := t 0 (s)W (s) ds; t¿0, where W (t); t¿0, is a standard Brownian motion and is a weight function. We determine the rate of −log P(sup t∈[0; 1] |X (t)| ¡ ); as → 0, for a large class of weight functions. The methods of our proofs are general and can be applied to many other problems. As an application, a Chung-type law of the iterated logarithm is given for X (t) with (t) = t − ; ¡ 3 2 .

A Note on Lévy’s Brownian motion

Nagoya Mathematical Journal, 1987

The Lévy Brownian motion with multidimensional parameter was introduced and discussed in his book [1] and it is known as the most important random field. Many approaches have been made to the investigation of the Lévy Brownian motion by H.P. McKean [7], Yu. A. Rozanov and others, by using various techniques.

Functionals of exponential Brownian motion and divided differences

Fuel and Energy Abstracts, 2011

We calculate an analytic value for the correlation coefficient beween a geometric, or exponential, Brownian motion and its time-average, a novelty being our use of divided differences to elucidate formulae. This provides a simple approximation for the value of certain Asian options regarding them as exchange options. We also illustrate that the higher moments of the time-average can be expressed neatly as divided differences of the exponential function via the Hermite-Genocchi integral relation, as well as demonstrating that these expressions agree with those obtained by Oshanin and Yor when the drift term vanishes.