Enlargement of Filtration, Backward Stochastic Differential Equations and Optimal Stopping Problems - Soane, Andrew (African Institute of Financial Markets and Risk Management, 2023)
This thesis focuses on the application of the enlargement of filtration to backward stochastic differential equations (BSDEs) and optimal stopping problems. In particular, the thesis develops the theory of the progressive enlargement of filtration with multiple random times and their associated marks. Several extensions of the classical progressive enlargement of filtration are derived, including a semimartingale decomposition theorem and a martingale representation theorem. The extensions then allow for the study of BSDEs and optimal stopping problems in an enlarged filtration. BSDEs are a very useful tool in stochastic optimal control and mathematical finance, the usefulness in the latter being that the solutions provide simultaneous calculation of derivative prices and their corresponding hedging strategies. Enlargement of filtration has a very intuitive application to BSDEs in a financial context, it models the effect that additional information has on the valuation of derivatives and their hedging strategies. This thesis develops certain classical results on BSDEs in the context of enlargement of filtration. The thesis then progresses to studying the effect of additional information on the value process of an optimal stopping problem. This again has an intuitive application to finance, as the effect of valuing American contingent claims in the presence of additional information. A very useful decomposition of the Snell envelope is derived. The thesis is rounded out with several applications of certain key results to topical fields in mathematical finance such as utility optimisation, risk metrics and Snell envelopes.