Before revision:  Moreover, to eliminate the ruin possibilities and having a minimum guarantee for the final annuity, we consider a safety level for the wealth process which consequently yields a Hamilton-Jacobi-Bellman (HJB) equation on a bounded domain.
After revision:  Moreover, to have a minimum guarantee for the final annuity, we consider a safety level for the wealth process which consequently yields a Hamilton-Jacobi-Bellman (HJB) equation on a bounded domain.