Maria Jolis

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:

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).

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.

On Stratonovich and Skorohod stochastic calculus for Gaussian processes

In this article, we derive a Stratonovich and Skorohod type change of variables formula for a multidimensional Gaussian process with low Holder regularity (typically lower than 1/4). To this aim, we combine tools from rough paths theory and stochastic analysis.

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

We give an approximation in law of the d-parameter Wiener process by processes constructed from a Poisson process with parameter in . This approximation is an extension of previous results of Stroock (1982, Topics in Stochastic Differential Equations, Springer, Berlin) and Bardina and Jolis (2000, Bernoulli 4 (6)).

MULTIPLE STRATONOVICH INTEGRAL AND HU―MEYER FORMULA FOR LÉVY PROCESSES

In the framework of vector measures and the combinatorial approach to stochastic multiple integral introduced by Rota and Wallstrom [Ann. Probab. 25 (1997) 1257―1283], we present an Ito multiple integral and a Stratonovich multiple integral with respect to a Levy process with finite moments up to a convenient order. In such a framework, the Stratonovich multiple integral is an integral with respect to a product random measure whereas the Ito multiple integral corresponds to integrate with respect to a random measure that gives zero mass to the diagonal sets. A general Hu-Meyer formula that gives the relationship between both integrals is proved. As particular cases, the classical Hu-Meyer formulas for the Brownian motion and for the Poisson process are deduced. Furthermore, a pathwise interpretation for the multiple integrals with respect to a subordinator is given.

On compact Itô's formulas for martingales of m4c

We prove that the class mc4 of continuous martingales with parameter set [0,1]2, bounded in L4, is included in the class of semi-martingales Sc8(L0(P)) defined by Allain in [A]. As a consequence we obtain a compact Itos formula. Finally we relate this result with the compact Ito formula obtained by Sanz in [S] for martingales of mc4.