OPTIMAL INVESTMENT AND CONSUMPTION STRATEGIES WITH DEBT RATIO IN A STOCHASTIC VOLATILITY MARKET

This paper investigates an optimal investment-consumption problem with endogenous leverage in a continuous-time market characterized by stochastic volatility and housing risk. We develop a unified framework in which an investor allocates wealth among a risk-free asset, a risky financial asset with Heston-type stochastic volatility, and a non-financial housing asset financed partly through debt. The investor simultaneously chooses the portfolio allocation, consumption rate, and debt ratio to maximize expected discounted utility under constant relative risk aversion preferences. Using stochastic dynamic programming, we derive the associated Hamilton-Jacobi-Bellman equation and obtain explicit closed-form solutions for the optimal portfolio policy, optimal debt ratio, and optimal consumption rule. The analytical results reveal that stochastic volatility affects investment decisions not only directly through the financial asset but also indirectly through leverage and housing exposure. Numerical illustrations demonstrate the sensitivity of the optimal policies to changes in volatility persistence, correlation structures, borrowing costs, and risk preferences.