This article provides the mathematical foundation for stochastically continuous affine processes on the cone of positive semidefinite symmetric matrices. This analysis has been motivated by a large and growing use of matrix-valued affine processes in finance, including multi-asset option pricing with stochastic volatility and correlation structures, and fixed-income models with stochastically correlated risk factors and default intensities.

We investigate the maximal domain of the moment generating function of affine processes in the sense of Duffie, Filipović and Schachermayer [Ann. Appl. Probab. 13 (2003) 984–1053], and we show the validity of the affine transform formula that connects exponential moments with the solution of a generalized Riccati differential equation. Our result extends and unifies those preceding it (e.g., Glasserman and Kim [Math. Finance 20 (2010) 1–33], Filipović and Mayerhofer [Radon Ser. Comput. Appl. Math. 8 (2009) 1–40] and Kallsen and Muhle-Karbe [Stochastic Process Appl. 120 (2010) 163–181]) in that it allows processes with very general jump behavior, applies to any convex state space and provides both sufficient and necessary conditions for finiteness of exponential moments.