The Complex Brownian Motion as a Weak Limit of Processes Constructed from a Poisson Process

Approximations of a Complex Brownian Motion by Processes Constructed from a Lévy Process

“…Eight years later, Stroock [9] showed the weak convergence of this solution to a Brownian motion. More precisely, given {N t , t ≥ 0} a standard Poisson process, the laws of the processes x ε {x ε (t) = ε This result have been extended in order to obtain approximations of other processes as, among others: m-dimensional Brownian process [5], SPDE driven by Gaussian white noise [2], fractional SDE [4], multiple Wiener integrals [3] or complex Brownian motion [1].…”

Large deviations for integrals of telegraph processes type

“…32-46] it has been proved that the measures generated by the sequence of the random processes ∫ 0 (−1) ( ) on the space of continuous functions [0, 1] weakly converge to a measure generated by a Wiener process. From [1], it also follows the weak convergence of the following sequences of random processes to a Wiener process: Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 12/8/14 9:54 AM In this paper, we justify the large deviation principle for the sequence (1.1). For a metric space ( , ), by B( , ) we denote the Borel -algebra of its sets.…”

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

“…In order to prove Theorem 2.1 we have to check that the family P θ ε is tight and that the law of all possible limits of P θ ε is the law of a complex Brownian motion. Following the same method that in [1], the proof is based on the following lemma:…”

“…To prove the matingale property with respect to the natural filtration {F t }, following the Section 3.1 in [1], it is enough to see that for any s 1 ≤ s 2 ≤ · · · ≤ s n ≤ s < t and for any bounded continuous function ϕ :…”

“…In this paper we present an extension of the Kac-Stroock result in the directions (ii) and (iii). Our aim is to define a modification of the processesx ε used by Stroock, similar to (3) proposed in [1]. These complex processes, that we will denote by x θ ε = z θ ε + iy θ ε , will depend on a parameter θ ∈ (0, π) ∪ (π, 2π) and will be defined from an unique standard Poisson process and a sequence of independent random variables with common distribution Bernoulli( 1 2 ).…”