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

Source abstract

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.