A Numerical Scheme for Multisignal Weight Constrained Conditioned Portfolio Optimisation Problems

In this paper, we consider optimal control problems involving a multidimensional objective function integral. We propose a direct collocation discretisation scheme suitable for the numerical solution of problems of this type. A convergence result is established to show that the scheme is consistent with multidimensional Pontryagin Principle relations in several important respects. Whilst the discussion focuses on the two-dimensional case, the simplicity of the scheme allows for easy generalisation. As an application taken from the domain of finance, we then introduce conditioned portfolio optimisation. The optimal control transcription of the two-dimensional mean-variance problem is given and solved using the scheme under consideration. We carry out a backtest using real-world data and confirm that its results validate our proposed numerical scheme.