This thesis is devoted to study the optimal liquidation strategies in a limit order book for large-tick stocks. Two frameworks are proposed.

In the first framework, we formulate a stylised limit order book that admits one-tick spread and fixed market depth cap, in which order flows arrive according to point processes with stochastic intensities. We consider an agent who wants to liquidate a position in this limit order book through market orders and pegged displayed/non-displayed limit orders within a fixed time horizon,
and whose goal is to maximise the expected utility from the terminal wealth. For this optimal liquidation problem,
we derive the associated Hamilton-Jacobi-Bellman quasi-variational inequality and prove a verification theorem giving sufficient conditions for the HJBQVI solution to be the value function. The optimal strategy is a combined stochastic and impulse control, and is then solved numerically using finite different scheme.

In the second framework, we formulate a stylised level-I limit order book whose spread is constantly one tick and whose dynamics are driven by the queueing races at the best prices. Order book events occur according to independent Poisson processes, with parameters depending on the most recent price move direction. Our goal is to maximise the expected terminal wealth of an agent who needs to liquidate a position within a fixed time horizon. By assuming that the agent trades through both limit and market orders only when the price moves, we model her liquidation procedure as a semi-Markov decision process, and compute the semi-Markov kernel using Laplace method in the language of queueing theory. The optimal liquidation policy is then solved by dynamic programming, and illustrated numerically.