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This paper analyzes the pricing of collateralized derivatives, i.e. contracts where counterparties are not only subject to financial derivatives cash flows but also to collateral cash flows arising from a collateral agreement. We do this along the lines of the brilliant approach of the first part of Moreni and Pallavicini (2017): in particular we extend their framework where underlyings are continuous processes driven by a Brownian vector, to a more general setup where underlyings are semimartingales (and hence jump processes). First of all, we briefly derive from scratch the theoretical foundations of the main subsequent achievements, i.e. the extension of the classical No-Arbitrage theory to dividend paying semimartingale assets, where by dividend we mean any cash flow earned/paid from holding the asset. In this part we merge, in the same treatment and under the same notation, the principal known results with some original ones. Then we extend the approach of Moreni and Pallavicini (2017) in different directions and we derive not only the pricing formulae but also the dynamics and forward prices of collateralized derivatives (extending the achievements of the first part of Gabrielli et al. (2019)). Finally, we study some important applications (Repurchase Agreements, Securities Lending and Futures contracts) of previously established theoretical frameworks, obtaining some results that are commonly used in practitioners literature, but often not well understood.