Journey to the heart of second-order EDSRs and other contemporary problems in financial mathematics.

Summary This thesis presents two main independent research topics, the last one being declined as two distinct problems. In the first part of the thesis, we focus on the notion of second order stochastic backward differential equations (in the following 2EDSR), first introduced by Cheridito, Soner, Touzi and Victoir [25] and recently reformulated by Soner, Touzi and Zhang [107]. We first prove an extension of their existence and uniqueness results when the considered generator is only continuous and linearly growing. Then, we continue our study by a new extension to the case of a quadratic generator. These theoretical results then allow us to solve a utility maximization problem for an investor in an incomplete market, both because constraints are imposed on his investment strategies, and also because the market volatility is assumed to be unknown. We prove in our framework the existence of optimal strategies, characterize the value function of the problem thanks to a second-order RLS and solve explicitly some examples that allow us to highlight the modifications induced by the addition of volatility uncertainty compared to the usual framework. We end this first part by introducing the notion of second order EDSR with reflection on an obstacle. We prove the existence and uniqueness of the solutions of such equations, and provide a possible application to the problem of shorting American options in a market with uncertain volatility. The first chapter of the second part of this thesis deals with an option pricing problem in a model where market liquidity is taken into account. We provide asymptotic developments of these prices in the vicinity of infinite liquidity and highlight a phase transition phenomenon depending on the regularity of the payoff of the considered options. Some numerical results are also proposed. Finally, we conclude this thesis by studying a Principal/Agent problem in a moral hazard framework. A bank (playing the role of the agent) owning a number of loans, wishes to exchange their interest for capital flows. The bank can influence the default probabilities of these loans by performing or not performing costly monitoring activity. These choices of the bank are known only to the bank. Investors (who play the role of principal) wish to set up contracts that maximize their utility while implicitly incentivizing the bank to perform constant monitoring activity. We solve this optimal control problem explicitly, describe the associated optimal contract and its economic implications, and provide some numerical simulations.