This thesis project addresses hedging in financial markets in the presence of basis risk. Basis risk in financial markets essentially refers to the inability to perfectly hedge the risk on a financial product (typically an option) based on an asset (usually illiquid) using another liquid asset due to their imperfect correlation. This situation leads to what is known as market incompleteness. Although basis risk is not the only source of market incompleteness, it is nonetheless a significant one. However, hedging in the presence of basis risk has received little attention, particularly with quadratic hedging techniques. Moreover, most studies on basis risk are conducted in continuous time, whereas financial markets are characterized by the impossibility of trading continuously. It thus seems crucial to investigate the use of quadratic hedging techniques under basis risk in discrete time. This thesis makes significant methodological contributions in three main areas. First, a semi-explicit formula is proposed for local and global quadratic hedging in a multivariate model where log-returns follow stationary independent increments (SII) processes. This approach generalizes existing work by considering multiple assets in the hedging portfolio. Next, the results are extended to bivariate GARCH models, which capture the dynamics of conditional volatility observed in financial markets. In this context, a semi-closed solution is derived for the local quadratic hedging strategy under a risk-neutral measure. Simulations and empirical analyses show the robustness and effectiveness of the proposed strategy compared to traditional methods, including delta hedging. Finally, the thesis addresses the integration of stochastic interest rates into global mean-variance hedging, a dimension often overlooked in the literature. A new formula is derived for the global hedging strategy in this context, and interesting application cases are presented. The results obtained in this thesis provide efficient solutions and formulas applicable in complex financial environments, where uncertainty and volatility prevail. This work paves the way for future research on extending these methods to other market dynamics, including the integration of transaction costs.