  We establish a super-replication duality in a continuous-time financial model
where an investor's trades adversely affect bid- and ask-prices for a risky
asset and where market resilience drives the resulting spread back towards zero
at an exponential rate. Similar to the literature on models with a constant
spread, our dual description of super-replication prices involves the
construction of suitable absolutely continuous measures with martingales close
to the unaffected reference price. A novel feature in our duality is a
liquidity weighted $L^2$-norm that enters as a measurement of this closeness
and that accounts for strategy dependent spreads. As applications, we establish
optimality of buy-and-hold strategies for the super-replication of call options
and we prove a verification theorem for utility maximizing investment
strategies.
