A global view of Brownian penalisations

In this monograph, we construct and study a sigma-finite measure on continuous functions from R_+ to R, strongly related to many probability measures obtained by penalisation of Brownian motion, i.e. as limits of probabilities which are absolutely continuous with respect to Wiener measure. This remarkable sigma-finite measure can be generalized in three other cases: one can start from a two-dimensional Brownian motion, from a recurrent diffusion with values in R_+, and from a discrete, recurrent Markov chain

Penalisations of multidimensional Brownian motion, VI

International audienceAs in preceding papers in which we studied the limits of penalized one-dimensional Wiener measures with certain functionals Γt, we obtain here the existence of the limit, as t → ∞, of d-dimensional Wiener measures penalized by a function of the maximum up to time t of the Brownian winding process (for d = 2), or in d ≥ 2 dimensions for Brownian motion prevented to exit a cone before time t. Various extensions of these multidimensional penalisations are studied, and the limit laws are described. Throughout this paper, the skew-product decomposition of d-dimensional Brownian motion plays an important role

Limiting laws associated with Brownian motion perturbated by normalized exponential weights I

We determine the rate of decay of the expectation Z(t) of some multiplicative functional related to Brownian motion up to time t. This permits to prove that the Wiener measure, penalized by this multiplicative functional, converges as t goes to infinity to a probability measure (p.m.) . We obtain the law of the canonical process under this new p.m

Limiting laws associated with Brownian motion perturbed by its maximum, minmum and local time II

We obtain probability measures on the canonical space penalizing the Wiener measure by a function of its maximum (resp. minimum, local time). We study the law of the canonical process under these new probability measures

Generalized Gamma Convolutions, Dirichlet means, Thorin measures, with explicit examples

In Section 1, we present a number of classical results concerning the Generalized Gamma Convolution (:GGC) variables, their Wiener-Gamma representations, and relation with the Dirichlet processes.To a GGC variable, one may associate a unique Thorin measure. Let $G$ a positive r.v. and $\Gamma_t(G)$ (resp. $\Gamma_t(1/G))$ the Generalized Gamma Convolution with Thorin measure $t$-times the law of $G$ (resp. the law of $1/G$). In Section 2, we compare the laws of $\Gamma_t(G)$ and $\Gamma_t(1/G)$.In Section 3, we present some old and some new examples of GGC variables, among which the lengths of excursions of Bessel processes straddling an independent exponential time.Comment: Published in at http://dx.doi.org/10.1214/07-PS118 the Probability Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local limit theorems for Brownian additive functionals and penalisation of Brownian paths, IX

We obtain a local limit theorem for the laws of a class of Brownian additive functionals and we apply this result to a penalisation problem. We study precisely the case of the additive functional : $\Big(A_t^{-} := \int_0^t 1_{X_s < 0}ds, t\geq 0\Big)$. On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges thus completing some similar previous study for standard Brownian motion (see [RVY,I])

Unifying Black-Scholes type formulae which involve Brownian last passage times up to a finite horizon

The authors recently discovered some interesting relations between the Black-Scholes formula and last passage times of the Brownian exponential martingales, which invites one to seek analogous results for last passage times up to a finite horizon. This is achieved in the present paper, where Yuri's formula, as originally presented in Akahori, Imamura and yano (2008), is also derived

Two examples of functional penalisations of Brownian motion, VIII

On one hand, we penalise Brownian paths by a function of one-sided supremum of Brownian motion, considered up to the last, resp. the first, zero, before $t$, resp. after $t$. This study provides some analogy with penalisations by the longest length of Brownian excursions, up to time $t$. On the other hand, we describe Feynman-Kac type penalisation results for long Brownian bridges, thus completing some similar study for free Brownian motion