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On strong solutions for positive definite jump diffusions

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  • Mayerhofer, Eberhard
  • Pfaffel, Oliver
  • Stelzer, Robert

Abstract

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

Suggested Citation

  • Mayerhofer, Eberhard & Pfaffel, Oliver & Stelzer, Robert, 2011. "On strong solutions for positive definite jump diffusions," Stochastic Processes and their Applications, Elsevier, vol. 121(9), pages 2072-2086, September.
    • Handle: RePEc:eee:spapps:v:121:y:2011:i:9:p:2072-2086
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    References listed on IDEAS

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    Cited by:

    1. Mayerhofer, Eberhard, 2012. "Affine processes on positive semidefinite d×d matrices have jumps of finite variation in dimension d>1," Stochastic Processes and their Applications, Elsevier, vol. 122(10), pages 3445-3459.
    2. Abi Jaber, Eduardo & Bouchard, Bruno & Illand, Camille, 2019. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1726-1748.
    3. Hiroaki Hata & Jun Sekine, 2017. "Risk-Sensitive Asset Management in a Wishart-Autoregressive Factor Model with Jumps," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 24(3), pages 221-252, September.
    4. Da Fonseca, José, 2016. "On moment non-explosions for Wishart-based stochastic volatility models," European Journal of Operational Research, Elsevier, vol. 254(3), pages 889-894.
    5. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Prömel, 2023. "Model‐free portfolio theory: A rough path approach," Mathematical Finance, Wiley Blackwell, vol. 33(3), pages 709-765, July.
    6. Eduardo Abi Jaber & Bruno Bouchard & Camille Illand & Eduardo Abi Jaber, 2018. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Post-Print hal-01349639, HAL.
    7. He, Yunhao & Leippold, Markus, 2020. "Short-run risk, business cycle, and the value premium," Journal of Economic Dynamics and Control, Elsevier, vol. 120(C).
    8. Christa Cuchiero, 2017. "Polynomial processes in stochastic portfolio theory," Papers 1705.03647, arXiv.org.
    9. Song, Jian & Yao, Jianfeng & Yuan, Wangjun, 2022. "Recent advances on eigenvalues of matrix-valued stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 188(C).
    10. Eduardo Abi Jaber & Bruno Bouchard & Camille Illand & Eduardo Jaber, 2018. "Stochastic invariance of closed sets with non-Lipschitz coefficients," Working Papers hal-01349639, HAL.
    11. Mayerhofer, Eberhard & Stelzer, Robert & Vestweber, Johanna, 2020. "Geometric ergodicity of affine processes on cones," Stochastic Processes and their Applications, Elsevier, vol. 130(7), pages 4141-4173.
    12. Cuchiero, Christa, 2019. "Polynomial processes in stochastic portfolio theory," Stochastic Processes and their Applications, Elsevier, vol. 129(5), pages 1829-1872.
    13. Jan Baldeaux & Eckhard Platen, 2012. "Computing Functionals of Multidimensional Diffusions via Monte Carlo Methods," Papers 1204.1126, arXiv.org.
    14. Andrew L. Allan & Christa Cuchiero & Chong Liu & David J. Promel, 2021. "Model-free Portfolio Theory: A Rough Path Approach," Papers 2109.01843, arXiv.org, revised Oct 2022.
    15. Scott Robertson & Hao Xing, 2014. "Long Term Optimal Investment in Matrix Valued Factor Models," Papers 1408.7010, arXiv.org.
    16. Carlos G. Pacheco, 2016. "Picard Iterations for Diffusions on Symmetric Matrices," Journal of Theoretical Probability, Springer, vol. 29(4), pages 1444-1457, December.
    17. Damir Filipović & Martin Larsson, 2016. "Polynomial diffusions and applications in finance," Finance and Stochastics, Springer, vol. 20(4), pages 931-972, October.

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