Optimal execution under price impact in a heterogenous characteristic timescale

The classical Almgren-Chriss price impact model is generalized to incorporate contributions to the transacted price from a crowd of infinitesimal market makers, each identified by a characteristic time during which their inventory mean reverts. Upon completion of the execution, these market makers revert their capacities back to zero. The resulting price dynamics, in general, are neither Markovian nor semimartingale. Determining the optimal execution scheme for the liquidation problem thus becomes an infinite-dimensional stochastic control problem. Despite this complexity, the problem remains linear-quadratic, allowing its solution to be reduced to a system of operator Riccati equations that characterize the optimal value process and the associated optimal liquidation strategy. Remarkably, the operator Riccati differential equations can be explicitly solved in the observed relevant case where the price impact model reproduces the empirically detected power-law decay. A numerical implementation illustrates the theoretical findings.