Structure condition under initial enlargement of filtration

We address the question of how the structure condition is affected when one possesses some additional information at the very beginning of the investment period. The structure condition represents essentially an alternative to non-arbitrage conditions for the Markowitz’s portfolio optimization framework, and is crucial for the existence of the optimal portfolio in quadratic utility settings. Herein, we provide practical assumption on the initial market model and the additional information to preserve the structure condition. The stochastic tools that drive this result are a generalization of the Lazaro-Yor representation by Lazaro and Yor (1978) and optional stochastic integral.