Expected exponential utility maximization of insurers with a Linear Gaussian stochastic factor model

Abstract

In this paper, we consider the problem of optimal investment by an insurer. The insurer invests in a market consisting of a bank account and m risky assets. The mean returns and volatilities of the risky assets depend nonlinearly on economic factors that are formulated as the solutions of general stochastic differential equations. The wealth of the insurer is described by a Cramér-Lundberg process, and the insurer preferences are exponential. Adapting a dynamic programming approach, we derive Hamilton-Jacobi-Bellman (HJB) equation. And, we prove the unique solvability of HJB equation. In addition, the optimal strategy is also obtained using the coupled forward and backward stochastic differential equations (FBSDEs). Finally, proving the verification theorem, we construct the optimal strategy.2000 Mathematics Subject Classification. 93E20, 60H30, 91B28, 91B30, 49L20, 90C40, 60J70, 62P05.

An Optimal Investment Strategy for Insurers in Incomplete Markets

“…Following the approach described in Hata and Yasuda (2017), let X t be the wealth process without investment (The Cramér-Lundberg model) and X K * t the wealth process when the insurer decides to follow the optimal strategy K * . The ruin probability under K * is defined as follows:…”

“…A general verification theorem was proved as well as an existence and uniqueness theorem, which led to a closed form of the optimal strategy, when the correlation factor is equal to zero (ρ = 0). Recently Hata and Yasuda (2017) introduced an extension of (Badaoui and Fernández (2013), Fernández et al (2008)) by considering a multidimensional linear Gaussian stochastic-factor model. They produce an explicit optimal strategy by solving the matrix Riccati equation.…”

Optimal investment and reinsurance of insurers with lognormal stochastic factor model

“…Optimal strategies are constructed by the solutions of HJB equations. In the literature on portfolio optimization, Badaoui et al [3,7,12,22,25,26] studied optimal investment problems for maximizing expected exponential utilities. The optimization problems based on Black-Scholes models are employed by [7,22,26].…”

“…in order to forecast the return of risky assets, we realize that stochastic factor models seem to be more realistic than Black-Scholes models. Hence, stochastic factor models are also adopted in [3,4,12].…”

“…In particular, we state Badaoui-Fernández [3], Badaoui et al [4], Fernández et al [7], Hata and Yasuda [12], Liang et al [18] and Xu et al [25]. Fernández et al [7] studied maximization of the expected exponential utility with Black-Scholes model.…”