Backward stochastic differential equations in a general filtration and applications

Résumé

This thesis is devoted to the study of backward stochastic differential equations in a general filtration and their applications to mathematical finance and stochastic game theory in two types of filtration enlargements, including initial enlargement with insider trading problems and progressive enlargement involving default risk. In the first part, we are interested in the problem of existence and uniqueness of the solution to a backward stochastic differential equation with two completely separated reflecting barriers, along with a stochastic Lipschitz driver, in a general filtration that supports a general RCLL square-integrable martingale. In the first application, we establish the connection with the problem of valuing American game options in a public financial market model driven by a normal martingale, between two agents sharing additional information on the price of shares in the market. The second application consists in characterizing the state process of our doubly reflected BSDE as the value function of a certain classical Dynkin game within a general setting. The third field of application is a direct generalization of the second one to the case where the classical linear expectation is replaced by its non-linear extension, so we are dealing with a general situation of generalized Dynkin games. In the second part, we provide answers to the natural extension of the question of existence and uniqueness of solutions to generalized BSDEs with jumps in the case of a general filtration that supports a Brownian motion and an independent integer-valued random measure, when the drivers are only monotonic with linear growth. This is achieved by approximating monotone drivers with uniformly Lipschitz coefficients. Furthermore, we extend this result to the case where, on one hand, the filtration supports a general square-integrable martingale, and on the other, the coefficients are stochastic monotones along with a stochastic linear growth condition. By employing a method rooted in the Yosida approximation of monotone operators, we establish both existence and uniqueness of the solution. Finally, we study doubly reflected BSDEs with a stochastic Lipschitz driver and two completely separated RCLL barriers in a defaultable setting, using two approaches. In the first approach, we assume that the (H)-Hypothesis, also called the immersion property, is satisfied. After proving the existence and uniqueness of a solution, we study the connection with the problem of valuing American game options in a financial market exposed to default risk, where the game payoff is expressed via a non-linear expectation. Following this approach, and since the (H)-Hypothesis is regarded as too strong, we relax this assumption in the second approach, where it is not necessarily satisfied. By proceeding with a change of measure to one equivalent to the historical probability, we recover the (H)-Hypothesis under this new measure. This allows us to prove the well-posedness of the problem and to study many interesting cases, including those related to a class of generalized Dynkin games with default time and late payments.

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