Infinite-dimensional Wishart processes

We introduce and analyse infinite-dimensional Wishart processes taking values in the cone S⁺₁(H) of positive self-adjoint trace class operators on a separable real Hilbert space H. Our main result gives necessary and sufficient conditions for their existence, showing that these processes are necessarily of fixed finite rank almost surely, but they are not confined to a finite-dimensional subspace of S⁺₁(H). By providing explicit solutions to operator valued Riccati equations, we prove that their Fourier-Laplace transform is exponentially affine in the initial value. As a corollary, we obtain uniqueness in law as well as the Markov property. The formulas we provide for the Fourier-Laplace transform of a Wishart process cover a wide parameter regime, thereby also extending what is known in the finite-dimensional setting. We also demonstrate that matrix-valued Volterra-Wishart processes can be interpreted as infinite-dimensional Wishart processes via an appropriate lift. Finally, under minor conditions on the parameters we prove the Feller property with respect to a slight refinement of the weak-∗-topology on S⁺₁(H). Applications of our results range from tractable infinite-dimensional covariance modelling to the analysis of the limit spectrum of large random matrices.