Infinite dimensional Wishart processes (Christa Cuchiero, University of Vienna)

28.01.2025 14:00

We introduce and analyse infinite dimensional Wishart processes taking values in the cone  of positive self-adjoint trace class operators on a separable real Hilbert space. 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. For showing existence we construct these processes as squares of Ornstein-Uhlenbeck processes. By providing explicit solutions to operator valued Riccati equations, we prove that their Fourier-Laplace transform is exponentially affine in the initial value.  We also demonstrate that matrix-valued Volterra-Wishart processes can be interpreted as infinite-dimensional Wishart processes via an appropriate lift.  Applications of our results range from tractable infinite-dimensional covariance modelling to the analysis of the limit spectrum of large random matrices.

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