This work establishes existence and uniqueness of strong solutions to stochastic Volterra differential equations (SVDEs) whose drift is spatially singular (non-Lipschitz, possibly discontinuous) and expressed as a convergent power series in the time-lag, driven by an additive Wiener noise. Its novel contribution is showing, via Malliavin-calculus techniques rather than classical Lipschitz/Yamada-Watanabe arguments, that for drifts in the admissible class the SVDE has a unique strong solution that is Malliavin differentiable and locally Sobolev differentiable with respect to the initial value, and deriving the associated weak convergence and unique weak solution. This pushes singular-drift regularization-by-noise results from the Markovian SDE case into the non-Markovian Volterra memory setting used in rough-volatility modeling.
This paper constructs unique strong solutions to a class of stochastic Volterra differential equations (SVDEs) driven by a spatially singular (non-Lipschitz, possibly discontinuous) drift vector fiel…