By: Igor Halperin

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 g... more

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.
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By: Shady E Ahmed, Omer San, Sivaramakrishnan Lakshmivarahan, John M Lewis

The four-dimensional variational data assimilation methodology for
assimilating noisy observations into a deterministic model has been the
workhorse of forecasting centers for over three decades. While this method
provides a computationally efficient framework for dynamic data assimilation,
it is largely silent on the important question concerning the minimum number
and placement of observations. To answer this question, we demonstrate the ... more

The four-dimensional variational data assimilation methodology for
assimilating noisy observations into a deterministic model has been the
workhorse of forecasting centers for over three decades. While this method
provides a computationally efficient framework for dynamic data assimilation,
it is largely silent on the important question concerning the minimum number
and placement of observations. To answer this question, we demonstrate the dual
advantage of placing the observations where the square of the sensitivity of
the model solution with respect to the unknown control variables, called
forward sensitivities, attains its maximum. Therefore, we can force the
observability Gramian to be of full rank, which in turn guarantees efficient
recovery of the optimal values of the control variables, which is the first of
the two advantages of this strategy. We further show that the proposed strategy
of placing observations has another inherent optimality: the square of the
sensitivity of the optimal estimates of the control with respect to the
observations (used to obtain these estimates) attains its minimum value, a
second advantage that is a direct consequence of the above strategy for placing
observations. Our analytical framework and numerical experiments on linear and
nonlinear systems confirm the effectiveness of our proposed strategy.
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