Systems of rough stochastic differential equations I: weak existence and Yamada--Watanabe

Abstract

For systems of rough stochastic differential equations in the sense of Friz--Hocquet--Lê, with a fixed deterministic rough driver, we prove two results. The first is weak existence with unbounded coefficients. The Itô data is assumed continuous and of linear growth, with no Hölder or Lipschitz continuity and no ellipticity. The rough vector field is a controlled pair of linear growth with bounded derivatives, subject either to a norm-curvature decay or to diagonality with coordinatewise curvature decay; in the diagonal case a coordinatewise upper bound on the diffusion matrix is added. The existing results ask the Itô data and the rough field to be bounded. We remove that boundedness, and ask in exchange for a curvature hypothesis on the rough field, bounds on its derivatives, and a narrower index configuration. The conclusion is weaker on two counts: the bounds that define a solution are imposed in the mean rather than uniformly on the probability space, and the solution's moments stop at the order the initial law carries. The second is the Yamada--Watanabe implication for rough equations: weak existence, together with pathwise uniqueness, yields a strong solution and joint uniqueness in law. A solution here is not defined by a pathwise identity but by two bounds on a remainder, one of them conditional, and an arbitrary enlargement of the filtration need not preserve that one. The enlargements the Yamada--Watanabe construction produces are immersions, however, and along an immersion both bounds transfer unchanged.