On strong solutions for positive definite jump diffusions

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:121

Article

Mayerhofer, Eberhard^2,3  Pfaffel, Oliver^4,5  Stelzer, Robert¹ 

[1] Univ Ulm, Inst Math Finance, D-89081 Ulm, Germany

[2] Univ Vienna, Vienna Inst Finance, A-1190 Vienna, Austria

[3] Vienna Univ Econ & Business Adm, A-1190 Vienna, Austria

[4] Tech Univ Munich, TUM Inst Adv Study, D-85747 Garching, Germany

[5] Tech Univ Munich, Zentrum Math, D-85747 Garching, Germany

关键词: Affine diffusions;    Jump diffusion processes on positive definite matrices;    Local martingales on stochastic intervals;    Matrix subordinators;    Stochastic differential equations on open sets;    Strong solutions;    Wishart processes;   

DOI  :  10.1016/j.spa.2011.05.006

来源: Elsevier

【 摘 要 】

We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics. Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the alpha th positive semidefinite power of the process itself with 0.5 < alpha < 1 and obtain existence conditions for them. In the case of a diffusion coefficient which is linear in the process we likewise get a positive definite analogue of the univariate GARCH diffusions. (C) 2011 Elsevier B.V. All rights reserved.

【 授权许可】

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