A Strong Approximation of Subfractional Brownian Motion by Means of Transport Processes

Many years ago, Griego, Heath and Ruiz-Moncayo proved that it is possible to define realizations of a sequence of uniform transform processes that converges almost surely to the standard Brownian motion, uniformly on the unit time interval. In this paper we extend their results to the multi parameter case. We begin constructing a family of processes, starting from a set of independent standard Poisson processes, that has realizations that converge almost surely to the Brownian sheet,

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“…Putting together (6), (7) and (8) and using thatW k (t) := n λ 2 W k (t) and (3), at the end we obtain…”

Section: Proof Of the Main Resultsmentioning

confidence: 99%

Strong approximations of Brownian sheet by uniform transport processes

2019

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“…Garzón, Gorostiza and León [6] defined a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals, for any Hurst parameter H ∈ (0, 1). In [7] and [8] the same authors deal with subfractional Brownian motion and fractional stochastic differential equations. Garzón, Torres and Tudor [5] also studied the Rosenblatt process.…”

Section: Introductionmentioning

confidence: 99%

On the strong convergence of multiple ordinary integrals to multiple Stratonovich integrals

Bardina¹,

Rovira²

2020

Preprint

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Given {W (m) (t), t ∈ [0, T ]} m≥1 a sequence of approximations to a standard Brownian motion W in [0, T ] such that W (m) (t) converges almost surely to W (t) we show that, under regular conditions on the approximations, the multiple ordinary integrals with respect to dW (m) converge to the multiple Stratonovich integral. We are integrating functions of the type f (x1, . . . , xn) = f1(x1) . . . fn(xn)I {x 1 ≤...≤xn} ,where for each i ∈ {1, . . . , n}, fi has continuous derivatives in [0, T ]. We apply this result to approximations obtained from uniform transport processes.

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“…More recently, Garzón, Gorostiza and León [6] defined a sequence of processes that converges strongly to fractional Brownian motion uniformly on bounded intervals, for any Hurst parameter H ∈ (0, 1) and computed the rate of convergence. In [7] and [8] the same authors deal with subfractional Brownian motion and fractional stochastic differential equations. Bardina, Binotto and Rovira [1] proved the strong convergence to a complex Brownian motion and obtained the corresponding rate of convergence.…”

Section: Introductionmentioning

confidence: 99%