Differential equations driven by rough signals

Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation

“…We comment briefly on the relation of our results to those of [7]. When p ≥ 2 the notion of solution developed in [7] contains more than the solution path y(t) which we obtain.…”

“…When p ≥ 2 the notion of solution developed in [7] contains more than the solution path y(t) which we obtain. Just as the driving path x(t) has to be accompanied by the associated 'iterated integrals', the approach taken in [7] is that the solution y(t) should also be accompanied by its iterated integrals, and the solution obtained there incorporates these as well as y(t). This leads to a more complete theory at the expense of greater complexity.…”

“…If p ≥ 3 then it is necessary to assume higher-order iterated integrals of x(t) are given. In this setting Lyons in [7] proves existence and uniqueness of solutions of (1) provided f ∈ C γ where γ > p.…”

“…The approach to (1) in [7] mirrors the standard approach to ODE's by writing them as integral equations and using Picard iteration or a contraction mapping argument. So one writes y i (t) = y i 0 + t 0 f i j (y(s))dx j (s) and the problem then is to interpret the integral on the right.…”

“…In [7] x(t) is assumed to have finite p-variation for some positive p. The study of equations driven by such rough paths is motivated by the case of stochastic differential equations driven by Brownian motion, which has finite p-variation only if p < 1 2 . Rough path theory gives an approach to such stochastic equations by viewing them as deterministic equations, for a fixed choice of driving path, which is in contrast to the more classical stochastic approach.…”