"A multitude of problems that fall under the scope of the field of Financial Mathematics, such as optimal investment and/or consumption, hedging of complex financial instruments, risk management and equilibrium theory, use advanced tools from the theories of Probability (including Stochastic Processes, its dynamic counterpart) and Convex Analysis. This proposal aims to develop further the methods and tools from these areas in order to address in more depth unresolved questions of theoretical and practical importance. Main focus will be given in exploring three areas: (1) Financial equilibria with heterogeneous agents in incomplete markets; (2) Viability of financial models with investment constraints and infinite number of traded assets; and (3) Hedging under model uncertainty.Versions of all three problems have been the subject of past and current scrutinised study, stemming from a desire to improve the quality of financial modelling, to allow for imperfections that appear in real markets and seek to comprehend them, as well as to address the risk involved with complicated financial positions by exploiting the structure of simple traded assets. Especially the last point is of immediate practical importance, since the field of Financial Mathematics has been criticised exactly for having failed to correctly appreciate the risks associated to introducing financial instruments of vast complexity, the incorrect valuation of which is a major factor that resulted in the recent economic crisis."