Paracontrolled distribution approach to stochastic Volterra equations

Abstract

Based on the notion of paracontrolled distributions, we provide existence and uniqueness results for rough Volterra equations of convolution type with potentially singular kernels and driven by the newly introduced class of convolutional rough paths. The existence of such rough paths above a wide class of stochastic processes including the fractional Brownian motion is shown. As applications we consider various types of rough and stochastic (partial) differential equations such as rough differential equations with delay, stochastic Volterra equations driven by Gaussian processes and moving average equations driven by Lévy processes.

Volterra Equations Driven by Rough Signals

“…This remained an open question until late 2018, when Prömel and Trabs [27] gave a para-controlled perspective on Volterra equations driven by irregular signals. Highly influenced by the theory of rough paths, the theory of para-controlled distributions developed by Gubinelli, Imkeller and Perkowski [15] gives a path wise perspective on SDEs and SPDEs through Paley-Littlewood para-controlled calculus, and Bony's paraproduct.…”

“…This higher regularity requirement comes from the fact that we need control of the Hölder regularity of the upper argument when composing a function with a controlled Volterra path, as seen in (5.23). This is in contrast to [27] where the authors only need a C 3 b diffusion coefficients. However, [27] is restricted to the case of a coefficient f such that f (0) = 0 and to Volterra equations with kernels which can be written as k(t, s) = k(t − s).…”

“…This is in contrast to [27] where the authors only need a C 3 b diffusion coefficients. However, [27] is restricted to the case of a coefficient f such that f (0) = 0 and to Volterra equations with kernels which can be written as k(t, s) = k(t − s).…”

“…Discussion. Theorem 54 tells us that for any T > 0 there exists a solution to Equation (5.25) on [0, T ] for any singular Volterra kernel satisfying (H) (in particular, as mentioned in Remark 55, we do not require a convolutional type of kernel like in [27]). Furthermore, since the extension developed here is fully based on the framework of classical rough path, one can also construct solutions to equations driven by lower regularity noise (i.e.…”