On martingale transformations of multidimensional Brownian Motion

Penalisations of multidimensional Brownian motion, VI

On It^ O's Formula for Multidimensional Brownian Motion

Consider a d-dimensional Brownian motion X = (X 1 ; : : : ; X d) and a function F which belongs locally to the Sobolev space W 1;2. We prove an extension of It^ o's formula where the usual second order terms are replaced by the quadratic covariations f k (X); X k ] involving the weak rst partial derivatives f k of F. In particular we show that for any locally square-integrable function f the qu...

A Brownian sheet martingale with the same marginals as the arithmetic average of geometric Brownian motion

We construct a martingale which has the same marginals as the arithmetic average of geometric Brownian motion. This provides a short proof of the recent result due to P. Carr et al [7] that the arithmetic average of geometric Brownian motion is increasing in the convex order. The Brownian sheet plays an essential role in the construction. Our method may also be applied when the Brownian motion ...

Coupling all the Lévy stochastic areas of multidimensional Brownian motion

It is shown how to construct a successful co-adapted coupling of two copies of an n-dimensional Brownian motion (B1, . . . ,Bn) while simultaneously coupling all corresponding copies of Lévy stochastic areas ∫ Bi dBj − ∫ Bj dBi. It is conjectured that successful coadapted couplings still exist when the Lévy stochastic areas are replaced by a finite set of multiply iterated pathand time-integral...

Ergodic Properties of Multidimensional Brownian Motion with Rebirth

In a bounded open region of the d dimensional space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. The evolution is invariant with respect to a density equal, modulo a constant, to the Green function of the Dirichlet Laplacian centered at the point of return. We calculate the resolvent in closed form, study its spectral properties and...