Optimal investment-consumption and life insurance selection problem under inflation. A BSDE approach

TL;DR

This paper models an optimal investment, consumption, and insurance strategy for a wage earner under inflation using BSDEs with jumps, deriving explicit solutions for exponential and power utilities.

Contribution

It introduces a BSDE approach to solve the complex problem of investment, consumption, and insurance under inflation with jump-diffusion asset prices, providing explicit solutions.

Findings

Explicit solutions for optimal strategies under exponential utility.

Explicit solutions for optimal strategies under power utility.

Characterization of the value function via BSDE with jumps.

Abstract

We discuss an optimal investment, consumption and insurance problem of a wage earner under inflation. Assume a wage earner investing in a real money account and three asset prices, namely: a real zero coupon bond, the inflation-linked real money account and a risky share described by jump-diffusion processes. Using the theory of quadratic-exponential backward stochastic differential equation (BSDE) with jumps approach, we derive the optimal strategy for the two typical utilities (exponential and power) and the value function is characterized as a solution of BSDE with jumps. Finally, we derive the explicit solutions for the optimal investment in both cases of exponential and power utility functions for a diffusion case.