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 language | English |
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Pages (from-to) | 2072-2086 |
Number of pages | 15 |
Journal | Stochastic Processes and their Applications |
Volume | 121 |
Issue number | 9 |
DOIs | |
Publication status | Published - Sep 2011 |
Externally published | Yes |
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