Stochastic Area Formula Research Articles

Euler polynomials, Bernoulli polynomials, and Lévyʼs stochastic area formula

In 1951, P. Lévy represented the Euler and Bernoulli numbers in terms of the moments of Lévyʼs stochastic area. Recently the authors extended his result to the case of Eulerian polynomials of types A and B. In this paper, we continue to apply the same method to the Euler and Bernoulli polynomials, and will express these polynomials with the use of Lévyʼs stochastic area. Moreover, a natural problem, arising from such representations, to calculate the expectations of polynomials of the stochastic area and the norm of the Brownian motion will be solved.

On distributions associated with the generalized Levy’S stochastic area formula

A closed-form expression is obtained for the conditional probability distribution of ∫t0R2sds given Rt, where (Rs, s ≧ 0) is a Bessel process of dimension δ > 0 started from 0, in terms of parabolic cylinder functions. This is done by inverting the following Laplace transform also known as the generalized Lévy’s stochastic area formula: . We also examine the joint distribution of (R2t, ∫t0R2sds.

On distributions associated with the generalized Levys stochastic area formula

On distributions associated with the generalized Levys stochastic area formula

REMARKS ON THE SELFDECOMPOSABILITY AND NEW EXAMPLES

The analytic property of the selfdecomposability of characteristic functions is presented from stochastic processes point of view. This provides new examples or proofs, as well as a link between the stochastic analysis and the theory of charactersitic functions. A new interpretation of the famous Levy’s stochastic area formula is given.

Exact formulas of certain functional integrals on wiener spaces

An elementary method was used to derive formulas of the form on an abstract Wiener space (H.B.W). Here heH A is a trace class operator on H and z is a complex number satisfying . Some explicit examples including Levy's stochastic area formula are given to illustrate how to implement this simple method

Lévy's stochastic area formula for gaussian processes

Communications on Pure and Applied MathematicsVolume 47, Issue 3 p. 329-360 Article Lévy's stochastic area formula for gaussian processes Nobuyuki Ikeda, Nobuyuki Ikeda Osaka UniversitySearch for more papers by this authorSigeo Kusuoka, Sigeo Kusuoka Kyoto UniversitySearch for more papers by this authorShojiro Manabe, Shojiro Manabe Osaka UniversitySearch for more papers by this author Nobuyuki Ikeda, Nobuyuki Ikeda Osaka UniversitySearch for more papers by this authorSigeo Kusuoka, Sigeo Kusuoka Kyoto UniversitySearch for more papers by this authorShojiro Manabe, Shojiro Manabe Osaka UniversitySearch for more papers by this author First published: March 1994 https://doi.org/10.1002/cpa.3160470306Citations: 7AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume47, Issue3March 1994Pages 329-360 RelatedInformation

On stochastic areas and averages of planar Brownian motion

Duplantier (1989) has computed the characteristic function of the stochastic area of planar Brownian motion taken with respect to its centre of gravity. In this paper it is shown that this result can be deduced from Levy's stochastic area formula. Moreover, Duplantier's stochastic area is shown to have the same law as the stochastic area of the restriction to the time interval (0, 1) of the Brownian ring defined on the time interval (0, 4/3). The same method also applies to the stochastic area formulae, obtained by Biane and Yor (1986), which are relative to certain planar processes associated with the orthogonal decomposition of Brownian motion along the L2(0, 1) basis of Legendre polynomials.

An extension of lévy's stochastic area formula

We obtain an extension of Levy's formula in which the two dimensional Brownian motion in addition to being conditioned by its value at time t = 1 as in the classical formula, is also conditioned by the values of the integral of each of its components with respect to some given measures on the time interval (0,1).

Levy's stochastic area formula in higher dimensions

Let ( W t ) = ( W 1 t , W 2 t ,…, W d t ), d ⩾ 2, be a d-dimensional standard Brownian motion and let A( t) be a bounded measurable function from R + into the space of d × d skew-symmetric matrices and x( t) such a function into R d . A class of stochastic processes (L t A,x), a particular example of which is Levy's “stochastic area” L t = 1 2 ∝ 0 −t (W 1 s,dW 2 s − W 2 s,dW 1 s) , is dealt with. The joint characteristic function of W t and L 1 A, x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions ( W t , L t A, x ) is given.