Contrôle stochastique appliqué à la finance

This PhD dissertation presents three independent research topics in the field of stochastic target and optimal control problems with applications to financial mathematics. In a first part, we provide a PDE characterization of the super hedging price of an American option of barrier types in a Markovian model of financial market. This extends to the American case a recent works of Bouchard and Bentahar (2006), who considered European barrier options, and Karatzas and Wang (2000), who discussed the case of perpetual American barrier options in a Black and Scholes type model. Contrary to their result, we do not use the usual dual formulation, which allows to reduce to a standard control problem, but instead prove and appeal to an American version of the geometric dynamic programming principle for stochastic targets of Soner and Touzi (2002). This allows us to avoid the non-degeneracy assumption on the volatility coefficients, and therefore extends their results to possibly degenerate cases which typically appear when the market is not complete. As a by-product, we provide an extension to the case of American type targets, which is of own interest. In the second part, within a Brownian diffusion Markovian framework, we provide a direct PDE characterization of the minimal initial endowment required so that the terminal wealth of a financial agent (possibly diminished by the pay off of a random claim) can match a set of constraints in probability. Such constraints should be interpreted as a rough description of a targeted profit and loss (P&L) distribution. This allows to give a price to options under a P&L constraint, or to provide a description of the discrete P&L profiles that can be achieved given an initial capital. This approach provides an alternative to the standard utility indifference (or marginal) pricing rules which is better adapted to market practices. From the mathematical point of view, this is an extension of the stochastic target problem under controlled loss, studied in Bouchard, Elie and Touzi (2008), to the case of multiple constraints. Although the associated Hamilton-Jacobi-Bellman operator is fully discontinuous, and the terminal condition is irregular, we are able to construct a numerical scheme that converges at any continuity points of the pricing function. The last part of this thesis is concerned with the extension of the optimal control of direction of reflection problem introduced in Bouchard (2007) to the jump diffusion case. In a Brownian diffusion framework with jumps, the controlled process is defined as the solution of a stochastic differential equation reflected at the boundary of a domain along oblique directions of reflection which are controlled by a predictable process which may have jumps. We also provide a version of the weak dynamic programming principle of Bouchard and Touzi (2009) adapted to our context and which is sufficient to provide a viscosity characterization of the associated value function without requiring the usual heavy measurable selection arguments nor the a-priori continuity of the value function.