Selfsimilar Processes

Embrechts, Paul; Maejima, Makoto

doi:10.2307/j.ctt7t1hk

Cited by 192 publications

(305 citation statements)

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“…Proposition 2.3. The market defined by (2),(3) and (6) is incomplete Proof : Here we use an integral representation for the fractional Brownian motion [22], [23] …”

Section: No-arbitrage and Incompletenessmentioning

confidence: 99%

No-arbitrage, leverage and completeness in a fractional volatility model

Mendes

Oliveira

Rodrigues

2015

Physica A: Statistical Mechanics and its Applications

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Based on a criterion of mathematical simplicity and consistency with empirical market data, a stochastic volatility model has been obtained with the volatility process driven by fractional noise. Depending on whether the stochasticity generators of log-price and volatility are independent or are the same, two versions of the model are obtained with different leverage behavior. Here, the no-arbitrage and completeness properties of the models are studied.

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“…Proposition 2.3. The market defined by (2),(3) and (6) is incomplete Proof : Here we use an integral representation for the fractional Brownian motion [22], [23] …”

Section: No-arbitrage and Incompletenessmentioning

confidence: 99%

No-arbitrage, leverage and completeness in a fractional volatility model

Mendes

Oliveira

Rodrigues

2015

Physica A: Statistical Mechanics and its Applications

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“…The probability tails of d j=1 θ j U (a, b; t j ) behave like c|x| −α , c > 0 as |x| → ∞. In (1.1), the index of stability, α, is restricted to (1,2]. This is due to the relation x −α dx < ∞ if and only if α > 1.…”

Section: The Ulog-fsm Processmentioning

confidence: 99%

“…Thus, a self-similar process involves a rescaling between the time domain and the spatial domain. We refer to the monograph [2] and [6,Chapter 7] about self-similar processes in general and to [6,Section 7.6] about log-FSM in particular.…”

Section: The Log-fsm Processmentioning

confidence: 99%

The long-range dependence of linear log-fractional stable motion

Levy¹,

Taqqu²

2011

COSA

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Abstract. We study the dependence structure of the α-stable random process linear log-fractional stable motion (Ulog-FSM). It is defined for α ∈ (1, 2] and real numbers (a, b) = (0, 0). Ulog-FSM is actually a collection of processes parametrized by α, a, and b. All of its moments of order p ≥ α are infinite, including the variance. It also has stationary increments. In the "well-balanced" case a = b, it reduces to log-fractional stable motion (log FSM), a self-similar process. Unlike log-FSM, it is not self-similar in the "unbalanced" case a = b. Since the covariance does not exist, other measures are necessary to analyze the dependence structure of Ulog-FSM. We use the codifference and the covariation. Ulog-FSM exhibits long-range dependence because over long lags of time, the codifference and the covariation decay "slowly" to zero.

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“…They are of considerable interest in practice since aspects of the selfsimilarity appear in different phenomena like telecommunications, economics, hydrology or turbulence. We refer to the work of Taqqu [44] for a guide on the appearance of the selfsimilarity in many applications and to the monographs by Samorodnitsky and Taqqu [40] and by Embrechts and Maejima [13] for complete expositions on selfsimilar processes.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Rosenblatt process

2008

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Abstract. We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.Mathematics Subject Classification. 60G12, 60G15, 60H05, 60H07.

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Selfsimilar Processes

Cited by 192 publications

References 0 publications

No-arbitrage, leverage and completeness in a fractional volatility model

No-arbitrage, leverage and completeness in a fractional volatility model

The long-range dependence of linear log-fractional stable motion

Analysis of the Rosenblatt process

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