Volterra equations driven by rough signals

This article is devoted to the extension of theory rough paths in context Volterra equations with possibly singular kernels. We begin describe a class two parameter functions defined on simplex called paths. These are used construct so-called Volterra-signature, analogously signature Lyon’s provide detailed algebraic and analytic description this object. Interestingly, does not have multiplicative property similar classical signature, we introduce an integral product behaving like convolution extending tensor product. show that well for large paths, analogue theorem from (which guarantees particular existence signature). Moreover concept essential construction controlled which natural processes be integrated respect driving noise our situation. leads given as functional combined through The then unique solutions driven by Hölder noises An example concerning Brownian kernel treated.