On the anisotropic stable JCIR process

Friesen, Martin; Jin, Peng

doi:10.30757/alea.v17-25

Cited by 13 publications

(10 citation statements)

References 34 publications

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“…More precisely, based on the representation by strong solutions of stochastic differential equations, ergodicity was studied in [30] for different Wasserstein distances. By using regularity of transition densities with respect to the Lebesgue measure combined with the Meyn-and-Tweedie stability theory the ergodicity in total variation distances has been studied in [3,27,38,37,53,29]. Finally, coupling techniques for affine processes are studied in [59,52].…”

Section: Below Then the Following Holds Truementioning

confidence: 99%

Stationary Covariance Regime for Affine Stochastic Covariance Models in Hilbert Spaces

Friesen¹,

Karbach²

2022

Preprint

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We study the long-time behavior of affine processes on positive self-adjoiont Hilbert-Schmidt operators which are of pure-jump type, conservative and have finite second moment. For subcritical processes we prove the existence of a unique limit distribution and construct the corresponding stationary affine process. Moreover, we obtain an explicit convergence rate of the underlying transition kernels to the limit distribution in the Wasserstein distance of order p ∈ [1, 2] and provide explicit formulas for the first two moments of the limit distribution. We apply our results to the study of infinite-dimensional affine stochastic covariance models in the stationary covariance regime, where the stationary affine process models the instantaneous covariance process. In this context we investigate the behavior of the implied forward volatility smile for large forward dates in a geometric affine forward curve model used for the modeling of forward curve dynamics in fixed income or commodity markets formulated in the Heath-Jarrow-Morton-Musiela framework.

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Section: Below Then the Following Holds Truementioning

confidence: 99%

Stationary Covariance Regime for Affine Stochastic Covariance Models in Hilbert Spaces

Friesen¹,

Karbach²

2022

Preprint

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“…Further, suppose that 𝑚 < min 𝑖∈𝐼 𝑏 𝑖 𝛼 −1 𝑖,𝑖𝑖 . Then, for every 𝑀 > 0 there exist ℎ > 0 and 𝛿 ∈ (0, 2) such that ‖𝑄 ℎ (𝑥, ⋅) − 𝑄 ℎ (𝑦, ⋅)‖ 𝑇𝑉 ≤ 2 − 𝛿, for all 𝑥, 𝑦 ∈ 𝐷 with ‖𝑥‖, ‖𝑦‖ ≤ 𝑀.The proof of Proposition 4.2 goes along the lines of proof of[5, Proposition 5.3, part (ii)]. We are ready to prove our second main result.Proof of Theorem 1.5.…”

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confidence: 91%

Regularity of transition densities and ergodicity for affine jump‐diffusions

Friesen

Jin

Kremer

et al. 2022

Mathematische Nachrichten

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This paper studies the transition density and exponential ergodicity for affine processes on the canonical state space ℝ 𝑚 ≥0 × ℝ 𝑛 . Under a Hörmander-type condition for diffusion components as well as a boundary nonattainment condition, we derive the existence and regularity of the transition densities and then prove the strong Feller property of the associated semigroup. Moreover, we also show that, under an additional subcriticality condition on the drift, the corresponding affine process is exponentially ergodic in the total variation distance.

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“…The existence of densities for solutions to stochastic differential equations with Hölder continuous coefficients driven by Lévy processes with anisotropic jumps has been proved in [30]. Such type of anisotropies also appear in the study of anisotropic stable JCIR process, see [29].…”

Section: Introductionmentioning

confidence: 98%