2019
DOI: 10.1016/j.spa.2018.10.005
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Statistical inference for Vasicek-type model driven by Hermite processes

Abstract: Let Z denote a Hermite process of order q 1 and self-similarity parameter H ∈ ( 1 2 , 1). This process is H-self-similar, has stationary increments and exhibits long-range dependence. When q = 1, it corresponds to the fractional Brownian motion, whereas it is not Gaussian as soon as q 2. In this paper, we deal with a Vasicek-type model driven by Z, of the form dX t = a(b − X t )dt + dZ t . Here, a > 0 and b ∈ R are considered as unknown drift parameters. We provide estimators for a and b based on continuous-ti… Show more

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Cited by 40 publications
(33 citation statements)
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“…Since |h(t) −h(t)| = e −αt | 0 −∞ e αs L(s)ds|, we conclude that the first integral converges to 1 0h 2 (t)dt. For the second integral note that due to boundedness of 1 n n 0 Y s ds (shown in [14]) and of | 1 by applying Proposition 1. The almost sure limit of the third integral equals α −2H HΓ(2H), as demonstrated in [14].…”
Section: Strong Consistencymentioning
confidence: 99%
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“…Since |h(t) −h(t)| = e −αt | 0 −∞ e αs L(s)ds|, we conclude that the first integral converges to 1 0h 2 (t)dt. For the second integral note that due to boundedness of 1 n n 0 Y s ds (shown in [14]) and of | 1 by applying Proposition 1. The almost sure limit of the third integral equals α −2H HΓ(2H), as demonstrated in [14].…”
Section: Strong Consistencymentioning
confidence: 99%
“…Therefore, we will define some alternative estimators that can be expressed only in terms of Wiener and Lebesque integrals and consequently they can be simulated. One of these new estimators represents an extended version of the estimators proposed in [6] or [14] as it reduces to them when the periodic drift L reduces to a constant.…”
Section: Different Estimatorsmentioning
confidence: 99%
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“…This kind of generalized Ornstein-Uhlenbeck processes are applied e.g., in mathematical finance to describe mean-reverting systems under the influence of shocks, and they are a highly active topic of research. Equations of type (1) with varying driving forces, and estimation in such models have been concerned e.g., in [20][21][22][23][24][25][26][27][28][29][30][31][32], to mention but a few. Furthermore, in [33] we showed that a generalized version of (1) characterizes all multivariate strictly stationary processes with continuous paths.…”
Section: Introductionmentioning
confidence: 99%
“…Recently, various types of stochastic integral and stochastic equations driven by Hermite noises have been considered by many authors. We refer, among others, to [3], [10], [11], [12], [17], [25], [8], [13], [21], [22]. Our purpose is to analyze the asymptotic behavior in distribution of the solution to the stochastic heat equation with additive Hermite noise, when the Hurst parameter (which is also the self-similarity index of the Hermite process) converges to the extreme values of its interval of definition, i.e when it tends to one and to one half.…”
Section: Introductionmentioning
confidence: 99%