2025/05/26 by Becherer, Dirk, Sun, Yuchen
#37H30 #60H15 #60H20 #60J76 #60L90 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2505.20437
We study solutions to backward differential equations that are driven hybridly by a deterministic discontinuous rough path W of finite q-variation for q ∈ [1, 2) and by Brownian motion B. To distinguish between integration of jumps in a forward- or Marcus-sense, we refer to these equations as forward- respectively Marcus-type rough backward stochastic differential equations (RBSDEs). We establish global well-posedness by proving global apriori bounds for solutions and employing fixed-point arguments locally. Furthermore, we lift the RBSDE solution and the driving rough noise to the space of decorated paths endowed with a Skorokhod-type metric and show stability of solutions with respect to perturbations of the rough noise. Finally, we prove well-posedness for a new class of backward doubly stochastic differential equations (BDSDEs), which are jointly driven by a Brownian martingale B and an independent discontinuous stochastic process L of finite q-variation. We explain, how our RBSDEs can be understood as conditional solutions to such BDSDEs, conditioned on the information generated by the path of L.