Backward stochastic differential equations with jumps are stable
A backward stochastic differential equation is a stochastic differential equation whose terminal value is known, in contrast to a (forward) stochastic differential equation whose initial value is known, and whose solution has to be adapted to a given filtration. The main aim of this thesis is to provide the suitable framework for the stability of stochastic differential equations with jumps, hereinafter BSDEs or BSDE when we refer to a single object. With the term stability we understand the continuity of the operator that maps the standard data of a BSDE, a set which among others includes the terminal value of the BSDE and the filtration with respect to which the solution has to be adapted, to its solution. In other words, the stability property allows to obtain an approximation of the solution of the BSDE under interest, once we determine an approximation of the standard data of the BSDE under interest. In this thesis we provide a general wellposedness result of multidimensional BSDEs with stochastic Lipschitz generator and which is driven by a possibly stochastically discontinuous square-integrable martingale. The time horizon can be infinite and as already implicitly has been stated, the right-continuous filtration is allowed to be stochastically discontinuous. Moreover, we provide a framework under which the stability property of BSDEs is verified. This framework allows for both continuous-time and discrete-time L2 −type approximations, which can turn out to be particularly useful for the well-posedness of numerical schemes for BSDEs. These results are presented in the second and the fourth chapter of this thesis. In the third chapter the stability of martingale representations is obtained, a result which lies at the core of the stability property of BSDEs. The property of the stability of martingale representations is not only a useful tool for our current needs, but it is also an interesting result on its own. Roughly speaking, it amounts to the convergence of the spaces generated by a convergent sequence of stochastic integrators as well as of their corresponding orthogonal spaces. Apart from these main results, a series of other results have been obtained, which either improve or complement classical ones. The most interesting of them is of purely analytic nature. It provides a characterisation of the weak-convergence of finite measures on the positive real-line by means of relatively compact sets of the Skorokhod space endowed with the J1 −topology. We remain in the Skorokhod space, where we refine a classical result on convergence of the jump-times of a J1 −convergent sequence. More precisely, we deal with the case of a multidimensional J1−convergent sequence and we prove that the times that the heights of the jumps lie in a suitable fixed set form a convergent sequence in the extended positive real-line. We proceed with the theory of Young functions, where the contribution amounts to the following result. We prove that the conjugate Young function of the composition of a moderate Young function with R+ 3 x 7→ 2 1 x2 ∈ R+ is also a moderate Young function with further nice properties. Finally, a new inequality regarding generalised inverses complements a classical one.
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