Young Differential Equations with Power Type Nonlinearities

Abstract

In this note we give several methods to construct nontrivial solutions to the equation $d{y}_t=\sigma (y_t)d{x}_t$, where $x$ is a $\gamma$-Hölder $R^d$-valued signal with $\gamma \in (1/2,1)$ and $\sigma$ is a function behaving like a power function $|\xi {|}^{\kappa }$, with $\kappa \in (0,1)$. In this situation, classical Young integration techniques allow to get existence and uniqueness results whenever $\gamma (\kappa +1)>1$, while we focus on cases where $\gamma (\kappa +1)\le 1$. Our analysis then relies on some extensions of Young's integral allowing to cover the situation at hand.