  In this study, we introduce an explicit trading-volume process into the
Almgren-Chriss model, which is a standard model for optimal execution. We
propose a penalization method for deriving a verification theorem for an
adaptive optimization problem. We also discuss the optimality of the
volume-weighted average-price strategy of a risk-neutral trader. Moreover, we
derive a second-order asymptotic expansion of the optimal strategy and verify
its accuracy numerically.
