Stochastic control applied in the theory of decision in a discrete time non- dominated multiple-priors framework

Abstract : This dissertation evolves around the following thematics: uncertainty, utility functions and no-arbitrage. In the first chapter, there is no uncertainty and we establish the existence of an optimal portfolio for an investor trading in a multi-period and discrete-time financial market and maximising its terminal wealth expected utility. We consider general non-concave and non-smooth random utility function defined on the half real-line. The proof is based on dynamic programming and measure theory tools. In the next chapters, we introduce the concept of Knightian uncertainty and adopt the non-dominated multi-priors framework introduced in [25] in discrete time. We first study in the second chapter the notion of quasi-sure no- arbitrage introduced in [25] and propose two equivalent definitions: a quantitative and geometric characterisation. We also introduce a stronger no-arbitrage condition that simplifies some of the measurability difficulties. In the third chapter, we build on these results to study the maximisation of non-dominated multiple-priors worst- case expected utility for investors trading in a multi-period and discrete-time financial for general concave utility functions defined on the half-real line and unbounded from above. The proof uses again a dynamic programming framework together with measurable selection tools. Finally the last chapter formulates a utility indifference pricing model for investor trading in a multi-period and discrete-time financial market. We prove that under suitable conditions the utility indifference prices of a contingent claim converge to its superreplication price.
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Romain Blanchard. Stochastic control applied in the theory of decision in a discrete time non- dominated multiple-priors framework. Quantitative Finance [q-fin]. Université de Reims Champagne-Ardenne, 2017. English. ⟨tel-01883439⟩

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