Non-standard backward stochastic differential equations and multiple optimal stopping problems with applications to securities pricing

This thesis elaborates on the wealth maximization problem of a small investor who invests in a financial market. Key tools for our studies come across in the form of several classes of BSDEs with particular non-linearities, casting them outside the standard class of Lipschitz continuous BSDEs. We first give a characterization of a small investor''s optimal wealth and its associated optimal strategy by means of a systems of coupled equations, a forward-backward stochastic differential equation (FBSDE) with non-Lipschitz coefficients, where the backward component is of quadratic growth. We then examine how specifying concrete utility functions give rise to another class of non-standard BSDEs. In this context, we also investigate the relationship to a modeling approach based on random fields techniques, known by now as the backward stochastic partial differential equations (BSPDEs) approach. We continue with the presentation of a numerical method for a special type of quadratic BSDEs. This method is based on a stochastic analogue to the Cole-Hopf transformation from PDE theory. We discuss its applicability to numerically solve indifference pricing problems for contingent claims in an incomplete market. We then proceed to BSDEs whose drifts explicitly incorporate path dependence. Several analytical properties for this type of non-standard BSDEs are derived. Finally, we devote our attention to the problem of a small investor who is equipped with several exercise rights that allow her to collect pre-specified cashflows. We solve this problem by casting it into the language of multiple optimal stopping and develop a martingale dual approach for characterizing the optimal possible outcome. Moreover, we develop regression based Monte Carlo algorithms which simulate efficiently lower and upper price bounds.