Xavier Bardina

Weak approximation of the Brownian sheet from a Poisson process in the plane

A motivation for proving results of this type is that they provide examples of processes of finite variation that can be approximated in law by the Wiener process. These processes have very different properties from the classical examples constructed from sums of independent random variables and from stationary processes, which also converge in law to the Wiener process. Another point of interest is that they give a nice relationship between the two more important processes. Our result is the following:

Weak convergence towards two independent Gaussian processes from a unique Poisson process

We consider two independent Gaussian processes that admit a representation in terms of a stochastic integral of a deterministic kernel with respect to a standard Wiener process. In this paper we construct two families of processes, from a unique Poisson process, the finite dimensional distributions of which converge in law towards the finite dimensional distributions of the two independent Gaussian processes.As an application of this result we obtain families of processes that converge in law towards fractional Brownian motion and subfractional Brownian motion.

Convergence in law to the multiple fractional integral

We study the convergence in law in , as [epsilon]-->0, of the family of continuous processes {I[eta][epsilon](f)}[epsilon]>0 defined by the multiple integralswhere f is a deterministic function and {[eta][epsilon]}[epsilon]>0 is a family of processes, with absolutely continuous paths, converging in law in to the fractional Brownian motion with Hurst parameter . When f is given by a multimeasure and for any family {[eta][epsilon]} with trajectories absolutely continuous whose derivatives are in L2([0,1]), we prove that {I[eta][epsilon](f)} converges in law to the multiple fractional integral of f. This last integral is a multiple Stratonovich-type integral defined by Dasgupta and Kallianpur (Probab. Theory Relat. Fields 115 (1999) 505) on the space , where is a measure on [0,1]n. Finally, we have shown that, for two natural families {[eta][epsilon]} converging in law in to the fractional Brownian motion, the family {I[eta][epsilon](f)} converges in law to the multiple fractional integral for any . In order to prove the convergence, we have shown that the integral introduced by Dasgupta and Kallianpur (1999a) can be seen as an integral in the sense of Sole and Utzet (Stochastics Stochastics Rep. 29(2) (1990) 203).

Approximation in law to the d-parameter fractional Brownian sheet based on the functional invariance principle

We show a result of approximation in law of the d-parameter fractional Brownian sheet in the space of the continuous functions on [0,T]d. The construction of these approximations is based on the functional invariance principle.

Weak convergence to the multiple Stratonovich integral

We have considered the problem of the weak convergence, as [var epsilon] tends to zero, of the multiple integral processesin the space , where f[set membership, variant]L2([0,T]n) is a given function, and {[eta][var epsilon](t)}[var epsilon]>0 is a family of stochastic processes with absolutely continuous paths that converges weakly to the Brownian motion. In view of the known results when n[greater-or-equal, slanted]2 and f(t1,...,tn)=1{t1

An extension of Ito's formula for elliptic diffusion processes

We prove an extension of Itos formula for F(Xt, t), where F(x, t) has a locally square integrable derivative in x that satisfies a mild continuity condition in t, and X is a one-dimensional diffusion process such that the law of Xt has a density satisfying some properties. Following the ideas of Follmer, et al. (1995), where they prove an analogous extension when X is the Brownian motion, the proof is based on the existence of a backward integral with respect to X. For this, conditions to ensure the reversibility of the diffusion property are needed. In a second part of this paper we show, using techniques of Malliavin calculus, that, under certain regularity on the coefficients, the extended Itos formula can be applied to strongly elliptic and elliptic diffusions.

The complex Brownian motion as a strong limit of processes constructed from a Poisson process

We construct a family of processes, from a single Poisson process, that converges in law to a complex Brownian motion. Moreover, we find realizations of these processes that converge almost surely to the complex Brownian motion, uniformly on the unit time interval. Finally the rate of convergence is derived.

On Itô's formula for elliptic diffusion processes

Bardina and Jolis [Stochastic process. Appl. 69 (1997) 83--109] prove an extension of It\^{o}s formula for

Weak approximation of the Wiener process from a Poisson process: the multidimensional parameter set case

F(X_t,t)

An analysis of a stochastic model for bacteriophage systems.

, where