Financial Mean-Variance Problems and Stochastic LQ Problems: Linear Stochastic Hamilton Systems and Backward Stochastic Riccati Equations

Financial mean-variance problems, including the mean-variance hedging, the mean-variance portfolio selection and the variance-optimal martingale measure, have obvious importance in modern finance. They are one-dimensional singular non-homogeneous stochastic linear-quadratic control problems (LQ), and can be solved in terms of the associated Riccati equation. However, the solution of the Riccati equation associated with the general stochastic LQ problem with random coefficients presents a new problem, which in fact has been open since Bismut (1978). Recently, the general one-dimensional case—which is the right case in the financial mean-variance problems—has been solved by Kohlmann and the author. More recently, the general regular case has been solved by the author with the theory of stochastic Hamilton system. In this article, the extension of the latter work is described to the singular case, which therefore provides an alternative approach to financial mean-variance problems with the theory of stochastic Hamilton system.