The Existence of Solutions for Rough Differential Equations

• Main Author: Caruana, Michael
• Published: University of Oxford 2007
• Subjects: 510
• Online Access: http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.491331

In one of the last Saint Flour lectures in 2004, T. Lyons remarked that a Peano theorem for rough differential equations had not yet been proved. The first successful attempt to prove this existence result was carried out by A.M. Davie. Using generalized Euler schemes, Davie showed that when working in finite dimensions, differential equations driven by geometric p-rough paths, for 1 ~ p < 3, have solutions whenever the vector fields have one degree less of smoothness than what is required for the existence of a unique solution in Lyons' Universal Limit theorem. P. Friz and N. Victoir have used geodesic approximations in the free nilpotent group to construct higher order Euler schemes and thus generalize Davie's results to the general case p ~ 1. Again this construction works in finite dimensions. In this thesis we present a new proof for Peano's theorem for rough differential equations, which is valid in infinite dimensions under an appropriate compactness assumption on the vector fields. Our approach, which is different from the ones mentioned above, makes full use of Lyons' Universal Limit theorem and is based on the construction of a family of rough polynomial approximations, each of which is a concatenation of rough path solutions of different equations. After giving a brief overview of the theory of rough paths, we prove several compactness results for multiplicative functionals and integrals along rough paths. \Ve then present our proof for Peano's theorem, which in the infinite dimensional case uses these compactness arguments. In the last part of the thesis, we focus on the relationship between the solutions of It6-Stratonovich stochastic differential equations and the rough path solutions of the equations driven by the Brownian rough path. In particular we address the issue concerning the different conditions imposed on the vector fields in these two approaches. \Ve prove that if the vector field of the rough differential equation driven by the Brownian rough path is Cl+c and can be expressed as a limit in V' (pi> p) of a sequence of C~+c functions which satisfy an appropriate ellipticity condition, then the rough path solution of the equation can be defined as the almost sure limit in the p-variation metric of a sequence of solutions of stochastic differential equations.