On optimal retirement

We pose an optimal control problem arising in a perhaps new model for retirement investing. Given a control function f and our current net worth X ( t ) for any t , we invest an amount f ( X ( t )) in the market. We need a fortune of M ‘superdollars’ to retire and want to retire as early as possible. We model our change in net worth over each infinitesimal time interval by the Ito process d X ( t ) = (1 + f ( X ( t )))d t + f ( X ( t ))d W ( t ). We show how to choose the optimal f = f 0 and show that the choice of f 0 is optimal among all nonanticipative investment strategies, not just among Markovian ones.