Compact representations of multi-period stochastic program using simulated path model

Simulated path model was proposed by Hibiki in order to solve a multi-period optimization problem. We can use sample paths associated with asset returns using Monte Carlo simulation, and get a better accuracy of uncertainties. To meet non-anticipativity conditions, we cannot make conditional decision as in the scenario tree model, and therefore we must decide an optimal portfolio with a fixed-unit strategy. The original formulation of simulated path model does not use the decision rule with the fixed-unit strategy effectively. Though decision variables of risky asset depend on only the number of periods, respectively, the number of the decision variables of cash must depend on the set of paths and periods, and then leads to large-scale problems. In this paper, we propose a more compact formulation than the previous formulation by eliminating the decision variables of cash. This does not mean that cash is excluded from the asset allocation decision. We expect that computation time decreases by reducing the problem size. We call this formulation compact representation, and show two kinds of formulations. We examine several numerical examples in order to compare computation time of the compact form with that of original form.