Resumen de Contributions to stochastic analysis
The aim of this dissertation is to present some new results on stochastic analysis. It consists on three works that deal with two Gaussian processes: the Brownian motion and the fractional Brownian motion with Hurst parameter .
In the first work we construct a family of processes, from a single Poisson process and a sequence of independent random variables with common Bernoulli distribution Ber( ), that converges in law to a complex Brownian motion. We find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly on the unit time interval, and we derive the rate of convergence.
In the second work, we establish the weak convergence, in the topology of the Skorohod space, of the ν-symmetric Riemann sums for functionals of the fractional Brownian motion when the Hurst parameter takes the critical value H = (4` + 2)−1, where ` = `(ν) ≥ 1 is the largest natural number satisfying for all j = 0,...,` − 1. As a consequence, we derive a change-of-variable formula in distribution, where the correction term is a stochastic integral with respect to a Brownian motion that is independent of the fractional Brownian motion.
The last work is devoted to prove that, when the delay goes to zero, the solution of delay differential equations driven by a Ho¨lder continuous function of order ) converges with the supremum norm to the solution of the equation without delay.
