  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.
