Robustness of the optimal trading strategy.

Summary The main objective of this thesis is to provide new theoretical results concerning the performance of investments based on stochastic models. To do so, we consider the optimal investment strategy in the framework of a risky asset model with constant volatility and a hidden Ornstein Uhlenbeck process. In the first chapter, we present the context and the objectives of this study. We present, also, the different methods used, as well as the main results obtained. In the second chapter, we focus on the feasibility of trend calibration. We answer this question with analytical results and numerical simulations. We close this chapter by also quantifing the impact of a calibration error on the trend estimate and exploit the results to detect its sign. In the third chapter, we assume that the agent is able to calibrate the trend well and we study the impact that the non-observability of the trend has on the performance of the optimal strategy. To do so, we consider the case of a logarithmic utility and an observed or unobserved trend. In each of the two cases, we explain the asymptotic limit of the expectation and the variance of the logarithmic return as a function of the signal-to-noise ratio and the speed of reversion to the mean of the trend. We conclude this study by showing that the asymptotic Sharpe ratio of the optimal strategy with partial observations cannot exceed 2/(3^1.5)∗100% of the asymptotic Sharpe ratio of the optimal strategy with complete information. The fourth chapter studies the robustness of the optimal strategy with calibration error and compares its performance to a technical analysis strategy. To do so, we characterize, analytically, the asymptotic expectation of the logarithmic return of each of these two strategies. We show, through our theoretical results and numerical simulations, that a technical analysis strategy is more robust than the poorly calibrated optimal strategy.