In this thesis, we first present in detail the mathematical foundations of the Kalman filter, including the derivation of the prediction and update equations, the properties of the optimal estimator, and the influence of initialization. We then demonstrate its use in two different contexts. In the first part, we analyze a simple simulated example of object motion, where the filter estimates position and velocity based on noisy measurements, which allows for an intuitive understanding of its functioning. In the second, central part of the thesis, we apply the filter in a financial context to implement a statistical arbitrage strategy. Using the Kalman filter, we estimate the time-varying parameters of the spread between cointegrated stock pairs and generate trading signals based on these estimates. We empirically test the strategy on historical data using several estimation methods (OLS, moving OLS, and the Kalman filter), comparing their returns, volatility, and Sharpe ratio. The results show that the Kalman filter achieves the highest return and the best return-to-risk ratio.
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