Book•
Levy Processes and Infinitely Divisible Distributions
13 Nov 1999-
About: The article was published on 1999-11-13 and is currently open access. It has received 3839 citations till now.
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TL;DR: The authors construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive Ornstein-Uhlenbeck (OU) processes, and study these models in relation to financial data and theory.
Abstract: Non-Gaussian processes of Ornstein–Uhlenbeck (OU) type offer the possibility of capturing important distributional deviations from Gaussianity and for flexible modelling of dependence structures. This paper develops this potential, drawing on and extending powerful results from probability theory for applications in statistical analysis. Their power is illustrated by a sustained application of OU processes within the context of finance and econometrics. We construct continuous time stochastic volatility models for financial assets where the volatility processes are superpositions of positive OU processes, and we study these models in relation to financial data and theory.
1,991 citations
TL;DR: Regular affine processes as discussed by the authors unify the concepts of continuous state branching processes with immigration and Ornstein-Uhlenbeck type processes, and provide foundations for a wide range of financial applications.
Abstract: We provide the definition and a complete characterization of regular affine processes. This type of process unifies the concepts of continuousstate branching processes with immigration and Ornstein-Uhlenbeck type processes. We show, and provide foundations for, a wide range of financial applications for regular affine processes.
1,082 citations
Book•
01 Jan 2008
TL;DR: In this paper, the authors present decompositions of the paths of Levy processes in terms of their local maxima and an understanding of their short-and long-term behaviour.
Abstract: Levy processes are the natural continuous-time analogue of random walks and form a rich class of stochastic processes around which a robust mathematical theory exists. Their mathematical significance is justified by their application in many areas of classical and modern stochastic models.
This textbook forms the basis of a graduate course on the theory and applications of Levy processes, from the perspective of their path fluctuations. Central to the presentation are decompositions of the paths of Levy processes in terms of their local maxima and an understanding of their short- and long-term behaviour.
The book aims to be mathematically rigorous while still providing an intuitive feel for underlying principles. The results and applications often focus on the case of Levy processes with jumps in only one direction, for which recent theoretical advances have yielded a higher degree of mathematical transparency and explicitness. Each chapter has a comprehensive set of exercises with complete solutions.
1,058 citations
01 Jan 2014
TL;DR: In this paper, Dzherbashian [Dzh60] defined a function with positive α 1 > 0, α 2 > 0 and real α 1, β 2, β 3, β 4, β 5, β 6, β 7, β 8, β 9, β 10, β 11, β 12, β 13, β 14, β 15, β 16, β 17, β 18, β 20, β 21, β 22, β 24
Abstract: Consider the function defined for \(\alpha _{1},\ \alpha _{2} \in \mathbb{R}\) (α 1 2 +α 2 2 ≠ 0) and \(\beta _{1},\beta _{2} \in \mathbb{C}\) by the series
$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$
(6.1.1)
Such a function with positive α 1 > 0, α 2 > 0 and real \(\beta _{1},\beta _{2} \in \mathbb{R}\) was introduced by Dzherbashian [Dzh60].
919 citations
Posted Content•
TL;DR: In this paper, the Cauchy problem for the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation.
Abstract: We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order � ∈ (0,2] and skewness � (|�| ≤ min {�,2 − �}), and the first-order time derivative with a Caputo derivative of order � ∈ (0,2]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We review the particular cases of space-fractional diffusion {0 < � ≤ 2, � = 1}, time-fractional diffusion {� = 2, 0 < � ≤ 2}, and neutral-fractional diffusion {0 < � = � ≤ 2}, for which the fundamental solution can be interpreted as a spatial probability density function evolving
793 citations
Cites methods from "Levy Processes and Infinitely Divis..."
...For more details on Lévy stable densities we refer the reader to specialistic treatises, as Feller [14], Zolotarev [49], Samorodnitsky and Taqqu [40], Janicki and Weron [25], Sato[41], Uchaikin and Zolotarev[48], where different notations are adopted....
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