Formula For Brownian Motion Research Articles

Towards the Validity of Itô's Formula for Discontinuous Functions

Two generalizations of the classical Ito's formula for Brownian motion are derived in this paper. The first is valid for functions with a discontinuous derivative along a smooth curve, and the second formula holds for functions with a gap along a smooth monotonic curve.

A proof of Itô’s formula using a discrete Itô’s formula

In this paper we will prove Itô’s formula for Brownian motion in the case of f ∈ C2 (ℝ), using a discrete Itô’s formula.

Distribution of functionals of Brownian motion stopped at the moment inverse to a linear combination of local times

We consider linear combinations of coefficients of local times with different signs. A new version of the Feynman-Kac formula for Brownian motion stopped at the moment inverse to this linear combination is proved. Bibliography: 6 titles.

QUASI-INVARIANCE FORMULAS FOR COMPONENTS OF QUANTUM LÉVY PROCESSES

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,15,16 and the Meixner process.

Bandwidth statistics from the eigenvalue moments for the Harper-Hofstadter problem

I propose a method for studying the product of bandwidths for the Harper-Hofstader model. This method requires knowledge of the moments of the midband energies. I conjectured a general formula for these moments. I computed the asymptotic representation for the product of bandwidths in the limit of a weak magnetic flux using Szego's theorem for Hankel matrices. I then give a first approximation for the edge of the butterfly spectrum and discuss its connection with P. Levy's formula for Brownian motion .

Versions of the Feynman-Kac formula

Some new versions of the Feynman-Kac formula for Brownian motion are considered. An interesting generalization of the formula is related to solutions of systems of linear differential equations. Bibliography: 7 titles.

Martingale Representation and a Simple Proof of Logarithmic Sobolev Inequalities on Path Spaces

We show how the Clark-Ocone-Haussmann formula for Brownian motion on a compact Riemannian manifold put forward by S. Fang in his proof of the spectral gap inequality for the Ornstein-Uhlenbeck operator on the path space can yield in a very simple way the logarithmic Sobolev inequality on the same space. By an appropriate integration by parts formula the proof also yields in the same way a logarithmic Sobolev inequality for the path space equipped with a general diffusion measure as long as the torsion of the corresponding Riemannian connection satisfies Driver's total antisymmetry condition.

Some variations on Kac's formula for Brownian motion

We show that under suitable conditions $$\begin{gathered} E_x f\left\{ {a + \int_0^t \beta \left[ {b + \int_0^s {\alpha \left( {X_r } \right)dr, c + s, X_s } } \right]ds, b + \int_0^t {\alpha \left( {X_s } \right)ds, c + t, X_t } } \right\} \hfill \\ = e^{tG} f\left[ {a, b, c, x} \right] \hfill \\ \end{gathered} $$ whereXt is a Brownian motion andG is the generator of a (C0) contraction semigroupetG.