Continuous-Time Mean-Variance Portfolio Optimization Under Proportional Borrowing Constraints, Stochastic Interest Rates, and Regime Switching

This paper studies a continuous-time target-return mean-variance portfolio problem with proportional borrowing constraints, stochastic short rates, and finite-state regime switching. To preserve the economic meaning of a wealth-proportional borrowing limit, the main constrained formulation uses a solvency-preserving long-only set, while a net-exposure half-space is retained as a borrowing-only benchmark. The terminal-mean constraint is handled through a primal-dual LQ embedding that states explicitly when an auxiliary optimizer recovers a target-return optimum and when a converse requires strong duality and dual attainment. For the constrained HJB equation, no global quadratic value-function ansatz is imposed. Instead, when the value function is locally convex in wealth, the pointwise Hamiltonian optimizer is the covariance-weighted projection of the unconstrained Hamiltonian minimizer associated with the constrained value function, and a verification theorem supplies the additional step needed for global optimality. Under independence between the Brownian motion driving the short rate and the Brownian motion driving risky-asset returns, the unconstrained quadratic coefficients satisfy linear parabolic equations with direct Feynman–Kac representations. The regime-switching extension leads to coupled coefficient systems and a coupled constrained HJB. Semi-Lagrangian numerical experiments generate constrained efficient frontiers directly from the dynamic program and quantify constraint-binding frequencies.