A Stochastic Multi-Objective Optimisation Framework for Business Portfolio Management: Hamilton–Jacobi–Bellman Approach with Geometric Convergence

Abstract

Abstract: Dynamic resource allocation under persistent uncertainty is a central challenge in business portfolio management. This paper develops a rigorous continuous-time stochastic multi-objective optimisation framework (SMOF) grounded in Hamilton–Jacobi–Bellman (HJB) theory. Asset values evolve as controlled stochastic differential equations (SDEs) whose drift and diffusion coefficients depend explicitly on allocation decisions—a structural feature absent from prior HJB-based multi-objective treatments. Three managerial objectives—return on investment (ROI) maximisation, portfolio risk minimisation, and resource-cost efficiency—are unified via weighted scalarisation to trace the Pareto-efficient frontier. We establish existence and uniqueness of a continuous viscosity solution to the HJB equation, promote it to a classical C¹² solution under an explicit non-degeneracy condition, derive closed-form feedback controls for the linear-quadratic and exponential-utility special cases, and prove geometric convergence of a policy-iteration algorithm with explicitly computable contraction factor κ. The convergence condition L√T < √2 minᵢ cᵢλ₃ is verified against the calibrated parameter set. Distinct from Pham (2009) and Yong and Zhou (1999), the framework simultaneously handles state-and-control-dependent multiplicative diffusion, a three-criterion Pareto scalarisation, and a policy-iteration scheme with a closed-form, a priori convergence bound—three properties not jointly present in any prior reference. We also present four synthetic figures (Pareto frontier, convergence curve, ROI surface, Monte Carlo confidence bands) generated by the numerical implementation, and identify CRSP NYSE daily data as the immediate empirical test bed.