  We consider the problem of optimization of contributions of a financial
planner such as a working individual towards a financial goal such as
retirement. The objective of the planner is to find an optimal and feasible
schedule of periodic installments to an investment portfolio set up towards the
goal. Because portfolio returns are random, the practical version of the
problem amounts to finding an optimal contribution scheme such that the goal is
satisfied at a given confidence level. This paper suggests a semi-analytical
approach to a continuous-time version of this problem based on a controlled
backward Kolmogorov equation (BKE) which describes the tail probability of the
terminal wealth given a contribution policy. The controlled BKE is solved
semi-analytically by reducing it to a controlled Schrodinger equation and
solving the latter using an algebraic method. Numerically, our approach amounts
to finding semi-analytical solutions simultaneously for all values of control
parameters on a small grid, and then using the standard two-dimensional spline
interpolation to simultaneously represent all satisficing solutions of the
original plan optimization problem. Rather than being a point in the space of
control variables, satisficing solutions form continuous contour lines
(efficient frontiers) in this space.
