Affine Volterra processes

Affine Rough Models

“…The two crucial assumptions are of course that (3.5) has a solution, and that the local martingale M is really a true martingale. The following theorem gives a sufficient condition that guarantees this; the proof can be found in Abi Jaber et al (2017). We now present the proof of Theorem 3.4 in the special case where v = w = 0.…”

“…Still, it is possible to construct stochastic volatility models with these features, and with an "affine structure" that produces formulas similar to (1.2). This has recently been done by Guennoun et al (2018); El Euch and Rosenbaum (2016); Abi Jaber et al (2017); Gatheral and Keller-Ressel (2018), and related ideas appear already in Comte et al (2012). The goal of this chapter is to explain and elucidate this "affine structure", and show how it leads to exponential-affine transform formulas.…”

“…The latter condition raises delicate questions of existence of solutions to (2.1) when a = 0, which we do not address here in detail. Let us however state the following result, whose proof can be found in Abi Jaber et al (2017). Part (ii) of the theorem requires the following assumption on the kernel:…”

“…The right-hand side is finite, so M is actually a square-integrable martingale. Details are given by Abi Jaber et al (2017).…”

Paracontrolled distribution approach to stochastic Volterra equations

“…Recently, Volterra equations attracted additional attention from the mathematical finance community because stochastic Volterra equations with singular kernels ϕ constitute very suitable models for the unpredictable and rough behaviour of volatility in financial markets, cf. [1,30,18].…”