Filtration enlargement and applications to finance

Abstract : This thesis consists of four independent parts. The topic in common is the filtration enlargement. In the first part, we present classical results for filtration enlargement in discrete time. We study some examples in initial enlargement of filtration. For the progressive enlargement of filtration, we give conditions for immersion martingale property. We also provide various characterizations of pseudo-stopping times and properties for honest times. In the second part, we are interested in determining the indifference price for variable annuities products. For this we consider two models, in both models we suppose that the market is incomplete and we adopt the approach of indifference price. In the first model we assume that the insured performs random withdrawals. Following indifference pricing theory, we define indifference fee rate for the insurer as a solution of an equation involving two stochastic control problems. Relating these problems to backward stochastic differential equations with a jump, we provide a verification theorem and give the optimal strategies associated to our control problems. From these, we derive a computation method to get indifference fee rates. We conclude this part with numerical illustrations of indifference fees sensibilities with respect to parameters. In the second model we propose the same approach as in the first model but we assume that the insured makes withdrawals that match the worst case for the insurer. In the third part, we study the relation of the solutions of BSDEs in two filtrations. As an application, one of our goals is to find the indifference price of information, i.e. the price at which an agent would have the same expected utility level using extra information as by not doing so. In the fourth part, we investigate advanced backward stochastic differential equations (ABSDE) with a jump. We study the existence and uniqueness of the solution to these ABSDEs. For this we relate the solution of the ABSDEs wth jumps to Brownian ABSDEs associated to the original ABSDE before and after the time jump.
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Submitted on : Friday, July 29, 2016 - 12:03:16 AM
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Ricardo Romo Romero. Filtration enlargement and applications to finance. Probability [math.PR]. Université Paris-Saclay; Universite d'Evry Val d'Essonne, 2016. English. ⟨NNT : 2016SACLE012⟩. ⟨tel-01349831⟩



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