Affine Processes on Symmetric Cones

Cuchiero, Christa; Keller‐Ressel, Martin; Mayerhofer, Eberhard; Teichmann, Josef

doi:10.1007/s10959-014-0580-x

Cited by 23 publications

(44 citation statements)

References 27 publications

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“…The above proposition already disproves the conjectures stated in [6, Section 2.1.2] and [8,Section 4]. However, we believe that it is important to prove Theorem 5 in its generality due to the following reasons:…”

Section: Exotic Cross-positive Maps On Euclidean Jordan Algebrassupporting

confidence: 75%

“…In particular, in Section 2.1.2 of [6] it was conjectured that s(H n (R) + ) = End(H n (R) + ) + g(H n (R) + ). The same conjecture was stated in Section 4 of [8].…”

Section: Corollary 2 a Linear Map A: V → V Belongs To G(c)supporting

confidence: 74%

“…• Nonnegativity of the scalar product B(X), Y for rank 1 idempotents satisfying X, Y = 0 requires an inequality which is not widely known, together with some determinant computation. • The Lie algebra g(H n (D) + ) is characterized in [23] only for associative algebras D (see also [8] for the characterization in the real and complex case). We give a characterization also for octonions.…”

Section: Exotic Cross-positive Maps On Euclidean Jordan Algebrasmentioning

confidence: 99%

“…This characterization will be useful in Section 5, where we will prove that the linear map B from Theorem 7 does not belong to End(H n (D) + ) + g(H n (D) + ). In the real and complex case the elements of g(H n (D) + ) were characterized in [8], using computational methods. Moreover, Tao and Gowda [23] characterized the elements of g(H n (D) + ) also in the quaternionic case.…”

Section: Proof By Corollary 9 and Lemma 8 We Havementioning

confidence: 99%

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Exotic one-parameter semigroups of endomorphisms of a symmetric cone

Kuzma

Omladič

Šivic

et al. 2015

Linear Algebra and its Applications

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We construct an exotic one-parameter semigroup of endomorphisms of a symmetric cone C, whose generator is not the sum of a Lie group generator and an endomorphism of C. The question is motivated by the theory of affine processes on symmetric cones, which plays an important role in mathematical finance. On the other hand, theoretical question that we solve in this paper seems to have been implicitly open even much longer then this motivation suggests.

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Section: Exotic Cross-positive Maps On Euclidean Jordan Algebrassupporting

confidence: 75%

“…In particular, in Section 2.1.2 of [6] it was conjectured that s(H n (R) + ) = End(H n (R) + ) + g(H n (R) + ). The same conjecture was stated in Section 4 of [8].…”

Section: Corollary 2 a Linear Map A: V → V Belongs To G(c)supporting

confidence: 74%

Section: Exotic Cross-positive Maps On Euclidean Jordan Algebrasmentioning

confidence: 99%

Section: Proof By Corollary 9 and Lemma 8 We Havementioning

confidence: 99%

See 2 more Smart Citations

Exotic one-parameter semigroups of endomorphisms of a symmetric cone

Kuzma

Omladič

Šivic

et al. 2015

Linear Algebra and its Applications

Self Cite

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“…This result plays a crucial part in the construction of suitable test functions for which the generator of affine processes satisfies the Foster-Lyapunov drift condition (Section 3). In Section 2.2 we recall the definition of affine processes on cones, as provided by [18], and summarize some properties. Section 2.3 recalls the definition of Harris recurrence and geometric ergodicity, and sufficient criteria for the latter property.…”

Section: Program Of the Papermentioning

confidence: 99%

Geometric ergodicity of affine processes on cones

Mayerhofer

Stelzer

Vestweber

2020

Stochastic Processes and their Applications

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For affine processes on finite-dimensional cones, we give criteria for geometric ergodicitythat is exponentially fast convergence to a unique stationary distribution. Ergodic results include both the existence of exponential moments of the limiting distribution, where we exploit the crucial affine property, and finite moments, where we invoke the polynomial property of affine semigroups. Furthermore, we elaborate sufficient conditions for aperiodicity and irreducibility. Our results are applicable to Wishart processes with jumps on the positive semidefinite matrices, continuous-time branching processes with immigration in high dimensions, and classical termstructure models for credit and interest rate risk.

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The Gärtner-Ellis Theorem, Homogenization, and Affine Processes

Gulisashvili

Teichmann

2015

Springer Proceedings in Mathematics &Amp; Statistics

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We obtain a first order extension of the large deviation estimates in the Gärtner-Ellis theorem. In addition, for a given family of measures, we find a special family of functions having a similar Laplace principle expansion up to order one to that of the original family of measures. The construction of the special family of functions mentioned above is based on heat kernel expansions. Some of the ideas employed in the paper come from the theory of affine stochastic processes. For instance, we provide an explicit expansion with respect to the homogenization parameter of the rescaled cumulant generating function in the case of a generic continuous affine process. We also compute the coefficients in the homogenization expansion for the Heston model that is one of the most popular stock price models with stochastic volatility.

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Affine Processes on Symmetric Cones

Cited by 23 publications

References 27 publications

Exotic one-parameter semigroups of endomorphisms of a symmetric cone

Exotic one-parameter semigroups of endomorphisms of a symmetric cone

Geometric ergodicity of affine processes on cones

The Gärtner-Ellis Theorem, Homogenization, and Affine Processes

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