White noise-based stochastic calculus with respect to multifractional Brownian motion

Lebovits, Joachim; Véhel, Jacques Lévy

doi:10.1080/17442508.2012.758727

Cited by 22 publications

(31 citation statements)

References 36 publications

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“…The definition of the operator

\frac{\partial M_{H}}{\partial H}

is quite similar (Lebovits and Lévy Véhel ). Precisely, define, for H in (0, 1), the space

Γ_{H} (R) : = {u \in {scriptS}^{'} (double-struckR) : \hat{u} = T_{f}; f \in L_{l o c}^{1} {(double-struckR) a n d ∥ u ∥}_{δ_{H} (double-struckR)} < + \infty}

, where the norm

{∥ \cdot ∥}_{δ_{H} (R)}

derives from the inner product

{⟨ \cdot, \cdot ⟩}_{δ_{H}}

defined on

Γ_{H} (double-struckR)

{⟨ u, v ⟩}_{δ_{H}} : = \frac{1}{c_{H}^{2}} \int_{double-struckR} (β_{H} + {ln | ξ |)}^{2} {| ξ |}^{1 - 2 H} \overset{u}{true} (ξ) d ξ

.…”

Section: Stochastic Calculus With Respect To Mbmmentioning

confidence: 99%

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Multifractional Stochastic Volatility Models

Corlay

Lebovits

Véhel

2013

Mathematical Finance

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The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. mBm is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for varying local regularity, and to decouple it from long‐range dependence properties. While we believe that mBm is potentially useful in a variety of applications in finance, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in Comte and Renault. Using the stochastic calculus with respect to mBm developed in Lebovits and Lévy Véhel, we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization‐based cubature methods, to get accurate approximations. This allows us to test the behavior of our model (as well as the one in Comte and Renault) with respect to its parameters, and in particular its ability to explain some features of the implied volatility surface. An advantage of our model is that it is able both to fit smiles at different maturities, and to take volatility persistence into account in a more precise way than Comte and Renault.

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“…The definition of the operator

\frac{\partial M_{H}}{\partial H}

is quite similar (Lebovits and Lévy Véhel ). Precisely, define, for H in (0, 1), the space

Γ_{H} (R) : = {u \in {scriptS}^{'} (double-struckR) : \hat{u} = T_{f}; f \in L_{l o c}^{1} {(double-struckR) a n d ∥ u ∥}_{δ_{H} (double-struckR)} < + \infty}

, where the norm

{∥ \cdot ∥}_{δ_{H} (R)}

derives from the inner product

{⟨ \cdot, \cdot ⟩}_{δ_{H}}

defined on

Γ_{H} (double-struckR)

{⟨ u, v ⟩}_{δ_{H}} : = \frac{1}{c_{H}^{2}} \int_{double-struckR} (β_{H} + {ln | ξ |)}^{2} {| ξ |}^{1 - 2 H} \overset{u}{true} (ξ) d ξ

.…”

Section: Stochastic Calculus With Respect To Mbmmentioning

confidence: 99%

“…(Lebovits and Lévy Véhel , theorem‐definition 5.1) Let

h : R \to (0, 1)

be a C 1 deterministic function such that its derivative function h ′ is bounded. The process W h defined by (5.2) is an

{(scriptS)}^{*}

‐process, which verifies the following equality (in

{(scriptS)}^{*}

): …”

Section: Stochastic Calculus With Respect To Mbmmentioning

confidence: 99%

Multifractional Stochastic Volatility Models

Corlay

Lebovits

Véhel

2013

Mathematical Finance

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“…A multifractional Wick-Itô integral with respect to mBm was defined [14]. It is interesting to check whether it coincides with the one defined in Corollary 5.9.…”

Section: A Comparison Between Multifractional Wick-itô Integral and Lmentioning

confidence: 99%

“…mBm using approximating fBms, and to compare this integral with the one defined in [14]. For definiteness, we will use the field B 1 (which was the one used in [14]), but any other field would lead to similar developments. We first briefly recall some basic facts about white noise theory and the Bochner integral, as well as on the construction of the integral w.r.t.…”

Section: Skohorod Integral With Respect To Mbm Through Approximating mentioning

confidence: 99%

“…It is interesting to check whether it coincides with the one defined in Corollary 5.9. In that view, we need to adapt the definition of [14], which used Pettis integrals, to deal with Bochner integrals. In order to compare our two integrals with respect to mBm when they both exist, it seems natural to assume that Y = (Y t ) t∈[0,1] is a Bochner integrable process of index p 0 ∈ N. The space E and the norm || || H E are defined as in the previous subsection.…”

Section: A Comparison Between Multifractional Wick-itô Integral and Lmentioning

confidence: 99%

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Stochastic integration with respect to multifractional Brownian motion via tangent fractional Brownian motions

Lebovits

Véhel

Herbin

2014

Stochastic Processes and their Applications

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Stochastic integration w.r.t. fractional Brownian motion (fBm) has raised strong interest in recent years, motivated in particular by applications in finance and Internet traffic modelling. Since fBm is not a semi-martingale, stochastic integration requires specific developments. Multifractional Brownian motion (mBm) generalizes fBm by letting the local Hölder exponent vary in time. This is useful in various areas, including financial modelling and biomedicine. The aim of this work is twofold: first, we prove that an mBm may be approximated in law by a sequence of "tangent" fBms. Second, using this approximation, we show how to construct stochastic integrals w.r.t. mBm by "transporting" corresponding integrals w.r.t. fBm. We illustrate our method on examples such as the Hitsuda-Skohorod and Wick-Itô stochastic integrals.

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