Abstract
This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.
Highlights
A generally acknowledged fact is that backward stochastic differential equations (BSDEs for short) were introduced in their linear version by Bismut [20, 21] in 1973, as an adjoint equation in the Pontryagin stochastic maximum principle
Around the same time, and most probably a bit before1, Davis and Varaiya [51] studied what can be considered as a prototype of a linear BSDE for characterizing the value function and the optimal controls of stochastic control problems with drift control only
The reader may already keep in mind that m will denote the dimension of the state space of an Itointegrator, n will denote the dimension of the state space of a process associated to an integer–valued random measure and d will denote the dimension of the state space of a stochastic integral
Summary
A generally acknowledged fact is that backward stochastic differential equations (BSDEs for short) were introduced in their linear version by Bismut [20, 21] in 1973, as an adjoint equation in the Pontryagin stochastic maximum principle. As far as we know, the first articles that went beyond this assumption were developed in a very nice series of papers by Cohen and Elliott [46] and Cohen, Elliott and Pearce [48], where the only assumption on the filtration is that the associated L2 space is separable, so that a very general martingale representation result due to Davis and Varaiya [52], involving countably many orthogonal martingales, holds In these works, the martingales driving the BSDE are imposed by the filtration, and not chosen a priori, and the non–decreasing process C is not necessarily related to them, but has to be deterministic and can have jumps in general, though they have to be small for existence to hold (see [46, Theorem 5.1]). The reader may already keep in mind that m will denote the dimension of the state space of an Itointegrator, n will denote the dimension of the state space of a process associated to an integer–valued random measure and d will denote the dimension of the state space of a stochastic integral
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