International Journal For Multidisciplinary Research

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A Hamilton–Jacobi–Bellman Framework for Investment, Transaction Costs, and Debt Repayment in an Incomplete Market

Author(s) Mr. Mwanda Bumali, Dr. Joseph Eyang'an Esekon, Dr. Richard Otieno Opiyo
Country Kenya
Abstract We develop a mathematically rigorous Hamilton–Jacobi–Bellman (HJB) framework for the joint problem of portfolio selection, consumption, debt repayment, and proportional transaction costs in an incomplete financial market. We worked on a fixed filtered probability space with a d-dimensional Brownian motion and introduced investor wealth and outstanding debt as two state variables whose dynamics are governed by explicitly stated Itô stochastic differential equations. Admissible controls are defined through precise measurability, integrability, non-negativity, and solvency conditions. Portfolio rebalancing incurs proportional transaction costs, giving rise to an impulse control problem. Our main contributions are: (i) a complete derivation of the HJB variational inequality via Itô’s lemma under explicitly stated regularity assumptions; (ii) a rigorous characterization of the no-trade region as the open set on which the value function strictly dominates the intervention operator; and (iii) a verification theorem establishing that any classical solution to the HJB system satisfying polynomial growth and transversality conditions equals the unique value function. This is the first complete verification theorem for the combined investment–debt–transaction-cost problem in an incomplete market. Our framework explains corporate underinvestment, large cash holdings, and debt renegotiation as direct implications of optimizing under these fundamental market frictions.
Keywords Hamilton–Jacobi–Bellman equation; impulse control; trans action costs; debt amortisation; incomplete markets; no-trade region; verification theorem; optimal capital structure.
Field Mathematics
Published In Volume 8, Issue 3, May-June 2026
Published On 2026-05-27
DOI https://doi.org/10.36948/ijfmr.2026.v08i03.78081
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