Before revision:  We model return dynamics as exogenous state variables and model portfolio weights, price dynamics and portfolio value as endogenous state variables . This separation allows for incorporation of any formation of transaction cost, liquidity cost and market impact. We first perform a forward simulation for both exogenous and endogenous state variables, then use a least-squares regression to approximate the backward recursive dynamic programs on a discrete grid of controls. Finally, we use a local interpolation and an adaptive refinement grid to enhance the optimal allocation estimates.
After revision:  We model return dynamics as exogenous state variables and model portfolio weights, price dynamics and portfolio value as endogenous state variables , namely the portfolio value and the asset prices subject to permanent market impacts. To do so, we improve the accuracy of the control randomization approach in the case of discrete controls, and propose a global iteration procedure to further improve the allocation estimates.