Backward stochastic differential equations with rough drivers (Q439882)

The authors study backward stochastic differential equations (BSDEs) of the form NEWLINE\[NEWLINE Y_t = \xi + \int_t^T f(r,Y_r,Z_r) \, dr + \int_t^T H(X_r,Y_r) \, d \zeta(r) - \int_t^T Z_r \,d W_r, \quad t \leq T, NEWLINE\]NEWLINE where \(X\) is an \(\mathbb{R}^n\)-valued semimartingale of the form NEWLINE\[NEWLINE X_t = x + \int_0^t \sigma_r \, d W_r + \int_0^t b_r \, dr, NEWLINE\]NEWLINE by rough path methods. The driver \(\zeta\) is a general geometric rough path, and the solution \((Y,Z)\) is defined as an appropriate limit of solutions \((Y^n,Z^n)\), \(n \in \mathbb{N}\), where \((\zeta^n)\), \(n \in \mathbb{N}\), is a sequence of smooth paths with \(\zeta^n \rightarrow \zeta\) in \(p\)-variation.NEWLINENEWLINEMoreover, the authors treat the case of Markovian BSDEs with rough drivers and establish the connection to backward doubly stochastic differential equations.