Correlation Matrices Driven by Stochastic Isospectral Flows

In many important areas of finance and risk management, time-dependent correlation matrices must be specified. We create valid correlation matrices by extending the idea of correlation flows based on isospectral flows. To incorporate the stochastic behavior of correlations, we adapt this approach by modeling the isospectral flow as a stochastic differential equation (SDE) instead of an ordinary differential equation (ODE). The solution of this SDE lies on the manifold of symmetric and positive semi-definite matrices, so structure-preserving schemes are needed for its numerical approximation. We apply stochastic Lie group methods based on Runge-Kutta–Munthe-Kaas schemes for ODEs to guarantee that the numerical solution evolves on the correct manifold. We also present an application example to illustrate our methodology.