The Martingale Property in Volterra Volatility Models

Abstract

We study martingality and strict local martingality in a broad class of stochastic and path-dependent Volterra volatility models and explain the underlying mechanisms. Our martingality criteria allow rough kernels and rely on non-accessibility of an explosive direction or one-sided growth bounds on the Girsanov drift. For smooth kernels, we derive sufficient conditions for strict local martingality from an explosion criterion allowing multiplicative noise, coupled factors and general initial input curves. Our results recover classical criteria and yield new results for Volterra and path-dependent models.