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We analyze an optimal trade execution problem in a financial market withstochastic liquidity. To this end we set up a limit order book model incontinuous time. Both order book depth and resilience are allowed to evolverandomly in time. We allow for trading in both directions and for c\`adl\`agsemimartingales as execution strategies. We derive a quadratic BSDE that underappropriate assumptions characterizes minimal execution costs and identifyconditions under which an optimal execution strategy exists. We alsoinvestigate qualitative aspects of optimal strategies such as, e.g., appearanceof strategies with infinite variation or existence of block trades and discussconnections with the discrete-time formulation of the problem. Our findings areillustrated in several examples.

C\`adl\`ag semimartingale strategies for optimal trade execution in stochastic order book models