SDEs with constraints driven by semimartingales and processes with bounded p-variation

Penalisation techniques for one-dimensional reflected rough differential equations

“…In the case β > 1 2 , the uniqueness of the reflected solution is due to Falkowski and Słomiński [13]. In the case β ≤ 1 2 , the uniqueness of the reflected RDE has been proven recently by Deya et al [8].…”

“…Hence, the inequality (42) implies that Z k converges uniformly to Z. Since Y k has the representation (13), then so has Y .…”

“…Similarly, observe that (L, 0) is solution of SP(0, 0, L). We call Skorokhod mapping the function that takes any continuous paths (z, l) and maps it to (y, k), where y = z + k is a path reflected on l. The Skorokhod mapping is Lipschitz continuous in the uniform topology (call C S the Lipschitz constant), see, for instance, Equations (2.1)-(2.2) in [13]. Thus, one gets that…”

“…Now to prove that (Y t ) t∈[0,τ ] and (U X;Z t t←0 ) t∈[0,τ ] coincide, one can follow the scheme of proof of Proposition 3 in [15]: let (X k ) k∈N a sequence of geometric rough paths such that (X k ) k∈N is a sequence of Lipschitz paths with uniform β-Hölder bound, which converges pointwise to X. Denote by Y k the solution to (8) where X k replaces X. Then (Y k , Z k ) is easily seen to solve (13) with X replaced by X k . As stated in [15], it suffices to prove the uniform convergence of Z k to Z to get the result.…”

“…In the last few years, this problem has attracted a lot of attention when the driver X is a β-Hölder continuous path: in the "regular" case β ∈ ( 1 2 , 1), existence of a solution has been established in a multidimensional setting by Ferrante and Rovira [14] and uniqueness was then obtained by Falkowski and Słomiński [13]. In that case, the integral can be constructed by a Riemann sum approximation and is known as a Young integral [33].…”