Conditional Distributions of Processes Related to Fractional Brownian Motion

Fink, Holger; Klüppelberg, Claudia; Zähle, M.

doi:10.1017/s0021900200013188

Cited by 13 publications

(19 citation statements)

References 18 publications

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“…Such processes appear in finance, hydrology, telecommunication, turbulence and image processing. In particular, various financial applications of the fractional Vasicek model (1.2) can be found in the articles [3][4][5][6][7][8][9]21]. The goal of the paper is to construct maximum likelihood estimators (MLEs) for the unknown parameters α and β and to establish their consistency and asymptotic normality.…”

Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Estimation in the Fractional Vasicek Model

Lohvinenko¹,

Ralchenko²

2017

LJS

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Section: Introductionmentioning

confidence: 99%

Maximum Likelihood Estimation in the Fractional Vasicek Model

Lohvinenko¹,

Ralchenko²

2017

LJS

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“…Grippenberg and Norros [8] provided a technical and difficult approach to calculate the conditional mean of fBm. Fink et al [5] also addressed this problem when studying the price of a zero-coupon bond in a fractional bond market. Since fBm is generally not a Markov process, both authors restricted themselves to calculate conditional expectations given the current value of B H t , and not given the whole path of B H t preceding t.…”

Section: Introductionmentioning

confidence: 99%

Fractional Hida-Malliavan derivatives and series representations of fractional conditional expectations

Jin¹,

Peng²,

Schellhorn³

2015

COSA

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Abstract. We represent fractional conditional expectations of a functional of fractional Brownian motion as a convergent series in L 2 (P H ) space. When the target random variable is some function of a discrete trajectory of fractional Brownian motion, we obtain a backward Taylor series representation; when the target functional is generated by a continuous fractional filtration, the series representation is obtained by applying a "frozen path" operator and an exponential operator to the functional. Three examples are provided to show that our representation gives useful series expansions of ordinary expectations of target random variables.

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“…

d ℳ_{d}^{λ} (t) = - λ ℳ_{d}^{λ} (t) d t + M_{d} (d t), t \in R .

Based on the earlier work of Zähle () and Buchmann & Klüppelberg (), Fink & Klüppelberg () showed that under some conditions on coefficient functions of a general SDE,

d X (t) = μ (X (t)) d t + σ (X (t)) M_{d} (d t), t \in R,

pathwise solutions can be obtained in the form

X (t) = f ((), {scriptℳ}_{d}^{λ} (t)), t \in double-struckR

, where f is an invertible function. Using the well‐known Fourier techniques, like in Theorem 3.8 of Fink et al (), we can straightforwardly calculate the conditional characteristic function of such pathwise solutions under assumption of existing suitable exponential moments of

{scriptℳ}_{d}^{λ}

via

\begin{matrix} double-struckE ([], (|, e^{iuf ((), {scriptℳ}_{d}^{λ} (t))}) {scriptF}_{s}^{{scriptℳ}_{d}^{λ}}) = double-struckE ([], (|, e^{iuf ((), {scriptℳ}_{d}^{λ} (t))}) {scriptF}_{s}^{M_{d}}) \\ = \int_{double-struckR} ((), double-struckE ([], (|, e^{(iξ}))) \end{matrix}

…”

Section: Applications To Fractional Time Series Models and Stochasticmentioning

confidence: 99%

Conditional Distributions of Mandelbrot–van ness Fractional LÉVY Processes and Continuous‐Time ARMA–GARCH‐Type Models with Long Memory

Fink

2015

Journal Time Series Analysis

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Long-memory effects can be found in many data sets from finance to hydrology. Therefore, models that can reflect these properties have become more popular in recent years. Mandelbrot-Van Ness fractional Lévy processes allow for such stationary long-memory effects in their increments and have been used in different settings ranging from fractionally integrated continuous-time ARMA-GARCH-type setups to general stochastic differential equations. However, their conditional distributions have not yet been considered in detail. In this article, we provide a closed formula for their conditional characteristic functions and suggest several applications to continuous-time ARMA-GARCH-type models with long memory.where the integrals are considered in the L 2 . /-sense. Then we call the process M d D .M d .t // t2R an MvN-fLp and L the driving Lévy process of M d . Proposition 2.2. summarizes the main properties of MvN-fLps.

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Conditional Distributions of Processes Related to Fractional Brownian Motion

Cited by 13 publications

References 18 publications

Maximum Likelihood Estimation in the Fractional Vasicek Model

Maximum Likelihood Estimation in the Fractional Vasicek Model

Fractional Hida-Malliavan derivatives and series representations of fractional conditional expectations

Conditional Distributions of Mandelbrot–van ness Fractional LÉVY Processes and Continuous‐Time ARMA–GARCH‐Type Models with Long Memory

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