Right-continuity of solutions to Brownian–Poissonian BSDEs

Determine sufficient structural and integrability conditions on the driver g, terminal condition X, and Lévy measure ν that guarantee the existence of right-continuous (in time) solution processes for backward stochastic differential equations with jumps in a Brownian–Poissonian filtration, i.e., for solutions (Y,Z,U) to Y_t = X + ∫_t^T g(s,Y_s,Z_s,U_s) ds − ∫_t^T Z_s · dW_s − ∫_{(t,T]×ℝ^d_*} U_s(x) dÑ(s,x). Establishing such conditions would enable the right-limit representation of g(s,Y_s,Z_s,U_s) at stopping times and extend the bouncing-drift (resilience rate) formulas beyond the purely Brownian setting.

Background

In Section 3 the authors derive representation formulas for the resilience rate (bouncing drift) both at deterministic times and, under stronger regularity assumptions, at stopping times. In the Brownian case, continuity of the generator along the solution can be verified using known Malliavin-calculus-based regularity results.

To extend these stopping-time representations to the Brownian–Poissonian setting (BSDEs with jumps), additional path regularity is required—specifically, right-continuity of the solution processes so that the limit of g(s,ρ_s,Z_s,U_s) as s approaches the stopping time from the right exists. The authors note that while such continuity assumptions can be verified for Brownian BSDEs, identifying sufficient conditions in the Brownian–Poissonian framework remains unresolved.