Differential equations driven by rough paths: An approach via discrete approximation.

*(English)*Zbl 1163.34005Author’s abstract: A theory of systems of differential equations of the form

\[ dy^i = \sum_jf^i_j(y)dx^i, \] where the driving path \(x(t)\) is nondifferentiable, has recently been developed by Lyons. I develop an alternative approach to this theory, using (modified) Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. I also give some other examples showing that the main results are reasonably sharp.

\[ dy^i = \sum_jf^i_j(y)dx^i, \] where the driving path \(x(t)\) is nondifferentiable, has recently been developed by Lyons. I develop an alternative approach to this theory, using (modified) Euler approximations, and investigate its applicability to stochastic differential equations driven by Brownian motion. I also give some other examples showing that the main results are reasonably sharp.

Reviewer: Sergiu Aizicovici (Athens/Ohio)

##### MSC:

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

34F05 | Ordinary differential equations and systems with randomness |