Life-cycle planning model with inflation and time-varying consumption constraints

This paper investigates the optimal consumption, investment and insurance strategies for a wage
earner operating within an inflationary environment and subject to time-varying consumption constraints over a finite, continuous time horizon. We assume the financial market comprises a risk-free
asset, a stock, and an index bond, with the wage earner’s preference represented by the Constant Relative Risk Aversion (CRRA) utility function. The primary objective of the wage earner is to devise
an optimal strategy for consumption, investment, and insurance allocation, aimed at maximizing the
expected discounted utilities. By employing the martingale duality method and Feynman-Kac formula, we derive the partial differential equations governing the dual value function in the context of
the Cauchy problem. Subsequently, we obtain the specific expression of the dual value function and
the optimal strategies by employing integral transform methods. The impact of various model parameters on optimal strategies is further elucidated through numerical simulations, utilizing predefined
parameter values.