Abstract
We investigate rough differential equations with a time-dependent reflecting lower barrier, where both the driving (rough) path and the barrier itself may have jumps. Assuming the driving signals allow for Young integration, we provide existence, uniqueness and stability results. When the driving signal is a càdlàg p-rough path for p∈[2,3), we establish existence to general reflected rough differential equations, as well as uniqueness in the one-dimensional case.
Highlights
We investigate rough differential equations with a time-dependent reflecting lower barrier, where both the driving path and the barrier itself may have jumps
Stochastic differential equations (SDEs) with reflecting barriers or boundary conditions have a long history in probability theory going back to Skorokhod [38]
Since the early works [26,33, 38,39,41] regarding reflected diffusions in a half-space, there has been a considerable effort to deal with various generalizations, such as more intricate boundary conditions or more complex stochastic processes, like fractional Brownian motion and general semimartingales
Summary
Stochastic differential equations (SDEs) with reflecting barriers or boundary conditions have a long history in probability theory going back to Skorokhod [38]. A fresh perspective on stochastic differential equations was initiated by Lyons, providing a pathwise analysis of SDEs, first using Young integration [31], and by introducing the theory of rough paths [40], which allows one to treat various random noises, such as fractional Brownian motion and continuous semimartingales. We prove that the solution map (A, X ) ↦→ (Y, K ) is locally Lipschitz continuous with respect to both the p-variation distance and to the Skorokhod J1 p-variation distance These results provide a comprehensive pathwise analysis of reflected Young differential equations.
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