STOCHASTIC STRATONOVICH CALCULUS fBm FOR FRACTIONAL BROWNIAN MOTION WITH HURST PARAMETER LESS THAN 1/2

Alòs, Elisa; León, Jorge A.; Nualart, David

doi:10.11650/twjm/1500574954

Cited by 51 publications

(155 citation statements)

References 12 publications

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“…[20]), has been revealed to be a good tool for considering Gaussian integrators, and in particular fractional Brownian motion. For illustration we quote [6,1] and [21] for the case of X being itself a Skorohod integral.…”

Section: Introductionmentioning

confidence: 99%

m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index

Gradinaru¹,

Nourdin²,

Russo³

et al. 2005

Annales de l'Institut Henri Poincare (B) Probability and Statis

149

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Section: Introductionmentioning

confidence: 99%

m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index

Gradinaru¹,

Nourdin²,

Russo³

et al. 2005

Annales de l'Institut Henri Poincare (B) Probability and Statis

149

View full text Add to dashboard Cite

show abstract

“…Alòs et al [1] (resp. León and Tudor [10]) work with the stochastic Stratonovich integral in the Russo and Vallois sense [18] when H ∈ (1/4, 1/2) (resp.…”

Section: Introductionmentioning

confidence: 93%

Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter Less than 1/2

León

Martı́n

2007

Stochastic Analysis and Applications

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In this paper we use the chaos decomposition approach to establish the existence of a unique continuous solution to linear fractional differential equations of the Skorohod type. Here the coefficients are deterministic, the inital condition is anticipating and the underlying fractional Brownian motion has Hurst parameter less than 1/2. We provide an explicit expression for the chaos decomposition of the solution in order to show our results. IntroductionThe fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1) is a Gaussian process with useful properties. In particular, the stationary of its increments, and the self-similarity and the long-range dependence of this process (see Mandelbrot and Van Ness [13]) become the fBm a suitable driven noise for the construction of stochastic models and the analysis of phenomena that exhibit scale-invariant and long-range correlated force. However the fBm is not a semimartingale when H = 1 2 . Hence we cannot apply the techniques of the stochastic calculus in the Itô sense to define a stochastic integral with respect to the fBm.Different interpretations of stochastic integral with respect to the fBm B have been used by several authors to study the fractional stochastic differential equation of the form(1.1)In the case that H ∈ (1/2, 1), it is reasonable to consider equation (1.1) as a path-by-path ordinary differential equation since B has Hölder-continuous paths with all exponents less than H and T 0 Y s dB s exists as a pathwise Riemann-Stieltjes integral for any λ-Hölder continuous process Y with λ > 1 − H (see Young [22]). This pathwise equation has been studied by several authors (see for instance [7, 8, 11, 19] and [23]). Zähle [23] has improved this pathwise approach * Partially supported by CONACyT

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“…The price to pay for this simplicity is to work with a process, the square root one, the diffusion coefficient of which is not Lipschitz with respect to the state. This leads us to set in this paper a self-contained theory of fractional integration of the square root process which is not a direct application of CR or of the more comprehensive theory which has been developed in the last 5 years (Alos et al 2000;Coutin 2000, Hu et al 2003). However, the discretization scheme we propose for practical implementation of this continuous time model with discrete time data heavily rests upon the recent works of and Coutin and Pontier (2007).…”

Section: Introductionmentioning

confidence: 99%

Affine fractional stochastic volatility models

Comte¹,

Coutin²,

Renault³

2010

Ann Finance

144

117

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By fractional integration of a square root volatility process, we propose in this paper a long memory extension of the Heston (Rev Financ Stud 6:327-343, 1993) option pricing model. Long memory in the volatility process allows us to explain some option pricing puzzles as steep volatility smiles in long term options and co-movements between implied and realized volatility. Moreover, we take advantage of the analytical tractability of affine diffusion models to clearly disentangle long term components and short term variations in the term structure of volatility smiles. In addition, we provide a recursive algorithm of discretization of fractional integrals in order to be able to implement a method of moments based estimation procedure from the high frequency observation of realized volatilities.

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STOCHASTIC STRATONOVICH CALCULUS fBm FOR FRACTIONAL BROWNIAN MOTION WITH HURST PARAMETER LESS THAN 1/2

Cited by 51 publications

References 12 publications

m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index

m-order integrals and generalized Itô's formula; the case of a fractional Brownian motion with any Hurst index

Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter Less than 1/2

Affine fractional stochastic volatility models

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