On strong solutions for positive definite jump diffusions

Eberhard Mayerhofer, Oliver Pfaffel, Robert Stelzer

Research output: Contribution to journalArticlepeer-review

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 αth positive semidefinite power of the process itself with 0.5<α<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.

Original languageEnglish
Pages (from-to)2072-2086
Number of pages15
JournalStochastic Processes and their Applications
Volume121
Issue number9
DOIs
Publication statusPublished - Sep 2011
Externally publishedYes

Keywords

  • 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

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