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In this note we consider differential equations driven by a signal $x$ which is $gamma$-Holder with $gamma>1/3$, and is assumed to possess a lift as a rough path. Our main point is to obtain existence of solutions when the coefficients of the equation behave like power functions of the form $|xi|^{kappa}$ with $kappain(0,1)$. Two different methods are used in order to construct solutions: (i) In a 1-d setting, we resort to a rough version of Lampertis transform. (ii) For multidimensional situations, we quantify some improved regularity estimates when the solution approaches the origin.
We develop the rough path counterpart of It^o stochastic integration and - differential equations driven by general semimartingales. This significantly enlarges the classes of (It^o / forward) stochastic differential equations treatable with pathwise methods. A number of applications are discussed.
We investigate rough differential equations with a time-dependent reflecting lower barrier, where both the driving (rough) path and the barrier itself may have jumps. Assuming the driving signals allow for Young integration, we provide existence, uni
We study inverse problems for semilinear elliptic equations with fractional power type nonlinearities. Our arguments are based on the higher order linearization method, which helps us to solve inverse problems for certain nonlinear equations in cases
In this paper we solve real-valued rough differential equations (RDEs) reflected on an irregular boundary. The solution $Y$ is constructed as the limit of a sequence $(Y^n)_{ninmathbb{N}}$ of solutions to RDEs with unbounded drifts $(psi_n)_{ninmathb
We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,mathcal{L}(X_t))dt + F(X_t,mathcal{L}(X_t))dW_t $$ where $W$ is a random rough pa