Optimal Management of a Variable Annuity Invested in a Black–Scholes Market Driven by a Multidimensional Fractional Brownian Motion

This article considers a defined-contribution pension scheme. It focuses in the post-retirement period and investigates the problem of controlling the level of payment of a variable annuity. The general version of the model is solved assuming a vector control variable differentiating the payments for each pensioner according to his age and an enhanced version for the market behavior, modeled via a multidimensional correlated fractional Brownian motion. Then, a reduced version of the basic model is also examined assuming an identical payment rate for all pensioners and a modified version of the typical Black-Scholes model driven by a standard fractional Brownian motion. Finally, a numerical application is developed for investigating the different investment strategies and also exploring the impact of the Hurst exponent in the final formula.