The fourth characteristic of a semimartingale

Schnurr, Alexander

doi:10.3150/19-bej1145

Cited by 3 publications

(3 citation statements)

References 29 publications

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“…Unless stated otherwise, we will assume that a(x) ≡ 0. For more on the case when a(x) ≡ 0, see the paper by Schnurr [28], which discusses such processes satisfying a(x) ≡ 0 and their connection to the symbol. Lemma 3.1.…”

Section: Resultsmentioning

confidence: 99%

On association and other forms of positive dependence for Feller processes

2019

J. Appl. Probab.

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We characterize various forms of positive dependence, such as association, positive supermodular association and dependence, and positive orthant dependence, for jump-Feller processes. Such jump processes can be studied through their statespace dependent Lévy measures. It is through these Lévy measures where we will provide our characterization. Finally, we present applications of these results to stochastically monotone Feller processes, including Lévy processes, the Ornstein-Uhlenbeck process, pseudo-Poisson processes, and subordinated Feller processes.

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Section: Resultsmentioning

confidence: 99%

On association and other forms of positive dependence for Feller processes

2019

J. Appl. Probab.

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“…We expect that Theorem 3.2 can be extended to affine semimartingales with explosion or killing, by adding a 'fourth characteristic' (cf. [37] and also [3]), which possesses an affine decomposition similar to (12). The rigorous formulation of the corresponding results will not be pursued here, and is left for future research.…”

Section: The Characterization Of Affine Semimartingalesmentioning

confidence: 99%

“…By the Kolmogorov existence theorem (see, e.g., [23,Theorem 8.4]), this guarantees the existence of a unique Markov process with transition kernels pp s,t q 0ďsďt . Let p s,t px, .q be the transition kernels of the semimartingale X, defined by (37). Note that by the affine property (3), these kernels satisfy (36) for all x P supppX s q, and it remains to extend the identity to all x P D.…”

Section: Affine Markov Processes and Infinite Divisibilitymentioning

confidence: 99%

Affine processes beyond stochastic continuity

Keller‐Ressel¹,

Schmidt²,

Wardenga³

2019

Ann. Appl. Probab.

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In this paper we study time-inhomogeneous affine processes beyond the common assumption of stochastic continuity. In this setting times of jumps can be both inaccessible and predictable. To this end we develop a general theory of finite dimensional affine semimartingales under very weak assumptions. We show that the corresponding semimartingale characteristics have affine form and that the conditional characteristic function can be represented with solutions to measure differential equations of Riccati type. We prove existence of affine Markov processes and affine semimartingales under mild conditions and elaborate on examples and applications including affine processes in discrete time. arXiv:1804.07556v2 [math.PR] E " e xu,Xty |F s ‰ " exp`φ s pt, uq`xψ s pt, uq, X s y˘(3)for all 0 ď s ď t and u P U. Moreover, X is called time-homogeneous, if φ s pt, uq " φ 0 pt´s, uq and ψ s pt, uq " ψ 0 pt´s, uq, again for all 0 ď s ď t and u P U.Note that the left-hand side of (3) is always well-defined and bounded in absolute value by 1, due to the definition of U.Remark 2.2. Comparing Definition 2.1 with the definition of an affine process in [10] (which treats the time-homogeneous case) and [14] (which treats the time-inhomogeneous case), we have replaced the Markov assumption of [10,14] with a semimartingale assumption. In view of [10, Thm. 2.12] this seems to slightly restrict the scope of the definition, since it excludes non-conservative processes. On the other hand, and this is the central point of our paper, we do not impose a stochastic continuity assumption on X, as has been done in [10,14]. It turns out that omitting this assumption leads to a significantly larger class of stochastic processes and to a substantial extension of the results in [10,14].To continue, we introduce an important condition on the support of the process X. Recall that the support of a generic random variable X, is the smallest closed set C such that P pX P Cq " 1; we denote this set by supppXq. For a set A we write convpAq for its convex hull, i.e. the smallest convex set containing A.Condition 2.3. We say that an affine semimartingale X has support of full convex span, if convpsupppX t qq " D for all t ą 0.Under Condition 2.3, φ and ψ are uniquely specified:Lemma 2.4. Let X be an affine semimartingale satisfying the support condition 2.3. Then φ s pt, uq and ψ s pt, uq are uniquely specified by (3) for all 0 ă s ď t and u P U.Proof. Fix 0 ă s ď t and suppose that r φ s pt, uq and r ψ s pt, uq are also continuous in u P U and satisfy (3). Write p s pt, uq :" r φ s pt, uq´φ s pt, uq and q s pt, uq :" r φ s pt, uq´φ s pt, uq. Due to (3) it must hold that p s pt, uq`xq s pt, uq, X s y takes values in t2πik : k P Nu a.s. @ u P U.However, the set U is simply connected, and hence its image under a continuous function must also be simply connected. It follows that u Þ Ñ p s pt, uq`xq s pt, uq, X s y is constant on U and therefore equal to p s pt, 0q`xq s pt, 0q, X s y " 0. Hence, p s pt, uq`xq s pt, uq, xy " 0, for all x P supppX s q and u P U. T...

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From Markov processes to semimartingales

Schnurr

Rickelhoff

2023

Probab. Surveys

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The fourth characteristic of a semimartingale

Cited by 3 publications

References 29 publications

On association and other forms of positive dependence for Feller processes

On association and other forms of positive dependence for Feller processes

Affine processes beyond stochastic continuity

From Markov processes to semimartingales

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