For a large class of ℝ+ valued, continuous local martingales (Mtt ≥ 0), with M0 = 1 and M∞ = 0, the put quantity: ΠM (K,t) = E ((K - Mt)+) turns out to be the distribution function in both variables K and t, for K ≤ 1 and t ≥ 0, of a probability γM on [0,1] × [0, ∞[. In this...

Four distribution functions are associated with call and put prices seen as functions of their strike and maturity. The random variables associated with these distributions are identified when the process for moneyness defined as the stock price relative to the forward price is a positive local...

Let us consider the following stochastic differential equation:where (Bt)t[greater-or-equal, slanted]0 is a d-dimensional brownian motion starting at 0 and b a function from to which is a gradient field. We aim at studying the convergence rate of the semi-group associated to (E) to its invariant...

We now analyze the asymptotic behaviour of Xt, as t approaches infinity, X being solution of where [beta] is a given odd and increasing Lipschitz-continuous function with polynomial growth. We prove with additional assumptions on [beta] that Xt converges in distribution to the invariant...

Taking an odd, non-decreasing function [beta], we consider the (nonlinear) stochastic differential equation and we prove the existence and uniqueness of solution of Eq. E , where and (Bt; t[greater-or-equal, slanted]0) is a one-dimensional Brownian motion, B0=0. We show that Eq. E admits a...

We use a Stochastic Differential Equation satisfied by Brownian motion taking values in the unit sphere Sn−1⊂Rn and we obtain a Central Limit Theorem for a sequence of such Brownian motions. We also generalize the results to the case of the n-dimensional Ornstein–Uhlenbeck processes.

Wolfe (Stochastic Process. Appl. 12(3) (1982) 301) and Sato (Probab. Theory Related Fields 89(3) (1991) 285) gave two different representations of a random variable X1 with a self-decomposable distribution in terms of processes with independent increments. This paper shows how either of these...

We prove that the law of the euclidean norm of an n-dimensional Brownian bridge is, in general, only equivalent and not equal to the law of a n-dimensional Bessel bridge and we compute explicitly the mutual density. Relations with Bessel processes with drifts are also discussed.