Distributions of jumps in a continuous-state branching process with immigration
01 Dec 2016-Journal of Applied Probability (Applied Probability Trust)-Vol. 53, Iss: 4, pp 1166-1177
TL;DR: A representation is given for the distribution of the first jump time of the process with jump size in a given Borel set and the equivalence of this distribution and the total Lévy measure is studied.
Abstract: We study the distributional properties of jumps in a continuous-state branching process with immigration. In particular, a representation is given for the distribution of the first jump time of the process with jump size in a given Borel set. From this result we derive a characterization for the distribution of the local maximal jump of the process. The equivalence of this distribution and the total Levy measure is then studied. For the continuous-state branching process without immigration, we also study similar problems for its global maximal jump.
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TL;DR: This introduction to stochastic calculus applied to finance will help people to understand better how to deal with malicious downloads and how to protect themselves from malicious downloads.
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TL;DR: In this paper, a brief introduction to continuous-state branching processes with or without immigration is given, where the processes are constructed by taking rescaling limits of classical discrete state branching models and given characterizations of the local and global maximal jumps of the processes.
Abstract: This work provides a brief introduction to continuous-state branching processes with or without immigration. The processes are constructed by taking rescaling limits of classical discrete-state branching models. We give quick developments of the martingale problems and stochastic equations of the continuous-state processes. The proofs here are more elementary than those appearing in the literature before. We have made them readable without requiring too much preliminary knowledge on branching processes and stochastic analysis. Using the stochastic equations, we give characterizations of the local and global maximal jumps of the processes. Under suitable conditions, their strong Feller property and exponential ergodicity are studied by a coupling method based on one of the stochastic equations.
43 citations
Cites background or methods from "Distributions of jumps in a continu..."
...−) for t≥ 0. Then for any r>0 we have Px ˆ max 0<s≤t ∆y(s) ≤ r ˙ = exp ˆ − xur(t)− ν(r,∞)t− Z t 0 ψr(ur(s))ds ˙ , where ur(t) = ur(t,m(r,∞)). The results given in this section were adopted from He and Li (2016). We refer the reader to Bernis and Scotti (2018+) and Jiao et al. (2017) for more careful analysis of the jumps of CBI-processes. In particular, the distributions of the numbers of large jumps in int...
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...The results given in this section were adopted from He and Li (2016)....
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TL;DR: A market model for power prices is investigated, including most basic features exhibited by previous models and taking into account self-exciting properties, and a Random Field approach is proposed, extending Hawkes-type models by introducing a twofold integral representation property.
34 citations
TL;DR: The α-CIR model as mentioned in this paper is an extension of the standard CIR model by adopting the α-stable Levy process and preserving the branching property, which allows to describe in a unified and parsimonious way several recent observations on the sovereign bond market together with the presence of large jumps at local extent.
Abstract: We introduce a class of interest rate models, called the α-CIR model, which gives a natural extension of the standard CIR model by adopting the α-stable Levy process and preserving the branching property. This model allows to describe in a unified and parsimonious way several recent observations on the sovereign bond market such as the persistency of low interest rate together with the presence of large jumps at local extent. We emphasize on a general integral representation of the model by using random fields, with which we establish the link to the CBI processes and the affine models. Finally we analyze the jump behaviors and in particular the large jumps, and we provide numerical illustrations.
33 citations
TL;DR: In this paper, an affine extension of the Heston model is introduced where the instantaneous variance process contains a jump part driven by α-stable processes with α ∆ in(1,2]$ in(α ∆)-stable processes.
Abstract: We introduce an affine extension of the Heston model where the instantaneous variance process contains a jump part driven by $\alpha$-stable processes with $\alpha\in(1,2]$ In this framework, we examine the implied volatility and its asymptotic behaviors for both asset and variance options Furthermore, we examine the jump clustering phenomenon observed on the variance market and provide a jump cluster decomposition which allows to analyse the cluster processes
19 citations
Cites background from "Distributions of jumps in a continu..."
...We wish to mention that the distribution of τ1 has been studied in [28] and [32]....
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...We wish to mention that the distribution of 𝜏1 has been studied in He and Li (2016) and Jiao et al. (2017)....
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References
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"Distributions of jumps in a continu..." refers background or methods in this paper
...A special form of the process is known in the financial world as the Cox–Ingersoll–Ross model; see, e.g. Brigo and Mercurio (2006) and Lamberton and Lapeyre (1996)....
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...Brigo and Mercurio (2006) and Lamberton and Lapeyre (1996). The CBI-process is a Feller process, so it has a càdlàg realization X = (Xt : t ≥ 0)....
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TL;DR: The Black-Scholes model as mentioned in this paper is a discrete-time formalism for estimating martingales and arbitrage opportunities in the stock market with continuous-time processes, and it has been applied to American options.
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658 citations
Journal Article•
TL;DR: In this article, the genealogical structure of general critical or subcritical continuous-state branching processes is investigated, and it is shown that whenever a sequence of rescaled Galton-Watson processes converges in distribution, their genealogies also converge to the continuous branching structure coded by the appropriate height process.
Abstract: We investigate the genealogical structure of general critical or subcritical continuous-state branching processes. Analogously to the coding of a discrete tree by its contour function, this genealogical structure is coded by a real-valued stochastic process called the height process, which is itself constructed as a local time functional of a Levy process with no negative jumps. We present a detailed study of the height process and of an associated measure-valued process called the exploration process, which plays a key role in most applications. Under suitable assumptions, we prove that whenever a sequence of rescaled Galton-Watson processes converges in distribution, their genealogies also converge to the continuous branching structure coded by the appropriate height process. We apply this invariance principle to various asymptotics for Galton-Watson trees. We then use the duality properties of the exploration process to compute explicitly the distribution of the reduced tree associated with Poissonnian marks in the height process, and the finite-dimensional marginals of the so-called stable continuous tree. This last calculation generalizes to the stable case a result of Aldous for the Brownian continuum random tree. Finally, we combine the genealogical structure with an independent spatial motion to develop a new approach to superprocesses with a general branching mechanism. In this setting, we derive certain explicit distributions, such as the law of the spatial reduced tree in a domain, consisting of the collection of all historical paths that hit the boundary.
404 citations
"Distributions of jumps in a continu..." refers background in this paper
...See also Duquesne and Le Gall (2002), Kyprianou (2014), and Li (2011) for up-to-date treatments of those processes....
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01 Jan 2014
TL;DR: In this article, Kloeden, P., Ombach, J., Cyganowski, S., Kostrikin, A. J., Reddy, J.A., Pokrovskii, A., Shafarevich, I.A.
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401 citations
"Distributions of jumps in a continu..." refers background in this paper
...See also Duquesne and Le Gall (2002), Kyprianou (2014), and Li (2011) for up-to-date treatments of those processes....
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Book•
08 Apr 2011
TL;DR: In this paper, the authors present a generalization of the Martingale problems of superprocesses to Branching Particle Systems (BPPSs) and one-dimensional Branching Processes.
Abstract: Preface.- 1. Random Measures on Metric Spaces.- 2. Measure-valued Branching Processes.- 3. One-dimensional Branching Processes.- 4. Branching Particle Systems.- 5. Basic Regularities of Superprocesses.- 6. Constructions by Transformations.- 7. Martingale Problems of Superprocesses.- 8. Entrance Laws and Excursion Laws.- 9. Structures of Independent Immigration.- 10. State-dependent Immigration Structures.- 11. Generalized Ornstein-Uhlenbeck Processes.- 12. Small Branching Fluctuation Limits.- 13. Appendix: Markov Processes.- Bibliography.- Index.
327 citations
"Distributions of jumps in a continu..." refers background in this paper
...See also Duquesne and Le Gall (2002), Kyprianou (2014), and Li (2011) for up-to-date treatments of those processes....
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