Smoothing Stochastic Bang-Bang Problems with Application to the Optimal Execution Problem D I S S E R TAT I O N zur Erlangung des akademischen Grades Dr. rerum naturalium

Motivated by the problem of how to optimally execute a large stock position, this thesis considers a stochastic control problem with two special properties. First, the control problem has an exponential delay in the control variable, and so the present value of the state process depends on the moving average of past control decisions. Second, the coefficients are assumed to be linear in the control variable. It is shown that a control problem with these properties generates a mathematically challenging problem. Specifically, it becomes a stochastic control problem whose solution (if one exists) has a bang-bang naturei.e. , the optimal solution switches discontinuously between a given upper and lower bound. This discontinuity creates difficulties in proving the existence of an optimal solution and in solving the problem with numerical methods. A sequence of stochastic control problems with state processes is constructed, whose diffusion matrices are invertible and approximate the original degenerate diffusion matrix. The cost functionals of the sequence of control problems are convex approximations of the original linear cost functional. To prove the convergence of the solutions, the control problems are written in the form of forward-backward stochastic differential equations (FBSDEs). It is then shown that the solutions of the FBSDEs corresponding to the constructed sequence of control problems converge in law, at least along a subsequence. By assuming convexity of the coefficients, it is then possible to construct from this limit an admissible control process which, for an appropriate reference stochastic system, is optimal for our original stochastic control problem. In addition to proving the existence of an optimal (bang-bang) solution, another useful result is obtained: The optimal solutions of the sequence of approximating control problems exhibit a smooth approximation of the discontinuous optimal bang-bang solution and can be used for the numerical solution of the problem. These results are then applied to the optimal execution problem to find an optimal strategy for executing a given stock position such that the expected execution price is maximized. The trading activity is assumed to have an impact on the stock priceeither because the stock is very illiquid or because the size of the trade is large enough to affect the market price. This problem is formulated as a stochastic control problem where the control variable denotes the trading speed. The impact on the stock price is assumed to have a transient component which decays over time. This means that the choice of trading speed influences not only the current but also the future stock price. Arguments are presented concerning the assumption that the impact of the trading speed on the stock price is linear and the decay of the transient price impact is exponential. With these assumptions the solution of the mathematically challenging problem which was previously treated theoretically is obtained. An approximating control problem for the numerical solution of the kind derived earlier is chosen and hence a smooth approximation of the optimal bang-bang problem is achieved. Numerical results for the case of a fast decay of the transient impact are compared with the case of a slow decay for both constant and time-varying liquidity of the stock.