A note on optimal investment–consumption–insurance in a Lévy market

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

“…Duarte et al [6] considered a problem of a wage earner who invests and buys a life insurance in a financial market with n diffusion risky shares. Similar works include (Guambe and Kufakunesu [8], Huang et al [9], Liang and Guo [10], Shen and Wei [18], among others). In all the above-mentioned papers, a single life insurance contract was considered.…”

Application of Generalized Geometric Itô-Lévy Process to Investment-Consumption-Insurance Optimization Problem under Inflation Risk

“…Huang, Milevsky & Wang [6] [7] also considered optimal portfolio, consumption and life insurance premium choice problem of a family under hyperbolic absolute Risk aversion (HARA) using a continuous time Markov chain approach. Guambe & Kufakunesu [8], studied optimal investment-consumption and life insurance selection problem under inflation using backward stochastic differential equation approach. Recently, Wang, et al [9] discussed optimal investment-consumption-insurance problem in a continuous-time economy using both martingale approach and dynamic programming approach.…”

Optimal Portfolio and Consumption for a Markovian Regime-Switching Jump-Diffusion Process

“…To tackle jumps in stock price, Merton [24] extended the GBM model to a jump-diffusion model for option pricing. Under this model, Aase [1] studied the optimal portfolio-consumption problem on a finite-time horizon; Framstad et al [10] considered the problem on an infinite-time horizon in the presence of proportional transaction costs with constant relative risk aversion utility; Ruan et al [27] investigated the optimal problem with habit formulation in an incomplete market using the maximum principle; Guambe and Kufakunesu [14] extended the work of Shen and Wei [29] to a geometric Itô-Lévy jump process [7], and solved the problem by combining the Hamilton-Jacobi-Bellman (HJB) equation [32] and a backward stochastic differential equation (SDE); Nguyen [25] examined the problem with downside risk constraint. Apart from the diffusion and jump-diffusion models, there are some other models to describe financial markets in the literature.…”

“…The results show that the optimal strategies derived by both methods are the same. In the case of a power utility function [14], under the constraints of no-short-selling and nonnegative consumption, we investigate the existence and uniqueness of the optimal solution and obtain closed-form expressions for the optimal strategy and the value function. Moreover, in order to illustrate the effects of jumps on the optimal results, we carry out some comparisons between the results for the risk model with and that without jumps.…”

“…It generalizes the model of a risky asset from geometric Brownian motion to Markov regime switching jump-diffusion processes, which makes the analysis more complicated. Secondly, this paper considers the constraints of no-short-selling and nonnegative consumption and extends the interval of the optimal portfolio solution from (0, 1) in Framstad et al [9] or Guambe and Kufakunesu [14] to (0, π 0 ]. We also prove the existence and uniqueness of the optimal strategy for the jump-diffusion model, and closed-form expressions of the optimal strategy and the value function are derived.…”