Càdlàg rough differential equations with reflecting barriers

Rough paths and SPDE

“…This was done via a fixed point argument. In particular, if B is a stochastic process with Hölder continuous paths of exponent strictly larger than 1/2, then the stochastic differential equation dY t = f (Y t )dB t can be defined as a collection of equations of type (2) indexed in the canonical probability space where B takes its values. For instance, this works for a fractional Brownian motion of Hurst index strictly larger than 1/2.…”

“…of a Lipschitz continuous approximation x n of an α-Hölder control x. Thanks to such enhancement of x, T. Lyons bypassed the lack of regularity of the signal x and established a universal limit theorem giving sense to Equation (2) and proving its well-posed character. Lyons' first proof of his result involved the formulation of Equation (2) as a fixed point equation in the space of rough paths over R e .…”

“…• Since 2015, some papers also deal with constrained rough differential equations and rough differential inclusions. As for ordinary differential equations, there are at least two ways to constrain the solution to Equation (2) to live in a convex-compact subset C of R e : to assume that the map f fulfills an invariance condition or to consider a Skorokhod reflection problem associated to Equation (2). On an invariance theorem for rough differential equations, see Coutin and Marie [34].…”

“…On an invariance theorem for rough differential equations, see Coutin and Marie [34]. On the existence of solutions to rough Skorokhod reflection problems with C not depending on time, see Aida [1], and with C depending on time, see Castaing, Marie and Raynaud de Fitte [25] and Allan, Liu and Prömel [2]. To establish the uniqueness of the solution to such problems is a difficult challenge and there exists at least one example of rough Skorokhod reflection problem having multiple solutions -see Gassiat [46].…”

“…In the sequel, we use an extension of the map by identifying the nodes. We will consider the nodes of h (1) and h (2) as a subset of those of h. Therefore, the roots of the trees in h (2) will be identified with some nodes in h (1) . This property is crucial in order to define a convolution operation on tree-indexed iterated integrals.…”