On Sharp Rate of Convergence for Discretization of Integrals Driven by Fractional Brownian Motions and Related Processes with Discontinuous Integrands (original) (raw)
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arXiv (Cornell University), 2022
We consider equidistant approximations of stochastic integrals driven by Hölder continuous Gaussian processes of order H > 1 2 with discontinuous integrands involving bounded variation functions. We give exact rate of convergence in the L 1-distance and provide examples with different drivers. It turns out that the exact rate of convergence is proportional to n 1−2H that is twice better compared to the best known results in the case of discontinuous integrands, and corresponds to the known rate in the case of smooth integrands. The novelty of our approach is that, instead of using multiplicative estimates for the integrals involved, we apply change of variables formula together with some facts on convex functions allowing us to compute expectations explicitly.
Rate of Convergence for Discretization of Integrals with Respect to Fractional Brownian Motion
Journal of Theoretical Probability, 2013
In this article, an uniform discretization of stochastic integrals 1 0 f ′ − (Bt)dBt, with respect to fractional Brownian motion with Hurst parameter H ∈ (1 2 , 1), for a large class of convex functions f is considered. In [1], Statistics & Decisions, 27, 129-143 , for any convex function f , the almost sure convergence of uniform discretization to such stochastic integral is proved. Here we prove L r-convergence of uniform discretization to stochastic integral. In addition, we obtain a rate of convergence. It turns out that the rate of convergence can be brought as closely as possible to H − 1 2 .
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Statistics & Probability Letters, 2009
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Annales de l'Institut Henri Poincare (B) Probability and Statistics, 2003
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Unilateral small deviations for the integral of fractional Brownian motion
Arxiv preprint math/0310413, 2003
We consider the paths of a Gaussian random process x(t), x(0) = 0 not exceeding a fixed positive level over a large time interval (0, T ), T ≫ 1. The probability p(T ) of such event is frequently a regularly varying function at ∞ with exponent θ. In applications this parameter can provide information on fractal properties of processes that are subordinate to x(·). For this reason the estimation of θ is an important theoretical problem. Here, we consider the process x(t) whose derivative is fractional Brownian motion with self-similarity parameter 0 < H < 1. For this case we produce new computational evidence in favor of the relations log p(T ) = −θ log T (1 + o(1)) and θ = H(1 − H). The estimates of θ are to within 0.01 in the range 0.1 ≤ H ≤ 0.9. An analytical result for the problem in hand is known for the markovian case alone, i.e., for H = 1/2. We point out other statistics of x(t) whose small values have probabilities of the same order as p(T ) in the log scale.
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Econometric Theory, 2009
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Methodology and Computing in Applied Probability, 2012
We study the problem of optimal approximation of a fractional Brownian motion by martingales. We prove that there exists a unique martingale closest to The second and third authors are grateful to European commission for the support within Marie Curie Actions program, grant PIRSES-GA-2008-230804.
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Using recent criteria for the convergence of sequences of multiple stochastic integrals based on the Malliavin calculus, we analyze the asymptotic behavior of quadratic variations for the fractional Brownian motion (fBm) and we apply our results to the design of a strongly consistent statistical estimators for the fBm's self-similarity parameter H.