  We study an optimal execution problem in illiquid markets with both
instantaneous and persistent price impact and stochastic resilience when only
absolutely continuous trading strategies are admissible. In our model the value
function can be described by a three-dimensional system of backward stochastic
differential equations (BSDE) with a singular terminal condition in one
component. We prove existence and uniqueness of a solution to the BSDE system
and characterize both the value function and the optimal strategy in terms of
the unique solution to the BSDE system. Our existence proof is based on an
asymptotic expansion of the BSDE system at the terminal time that allows us to
express the system in terms of a equivalent system with finite terminal value
but singular driver.
