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Deterministic and stochastic Volterra integral equations with path-dependency

Oberseminar Finanz- und Versicherungsmathematik

Deterministic and stochastic Volterra integral equations with path-dependency

Abstract

Three intrinsic features describe Volterra processes, arising as solutions to stochastic Volterra integral equations with irregular coefficients.
They fail to be semimartingales, are not Markovian and admit paths that are almost surely of lower Hölder regularity than the driving Brownian motion. In particular, the last effect, called roughness, is used in mathematical finance to model the volatility of the price process of a risky asset. New fields on rough volatility modelling and
pricing therein evolved with a focus on affine Volterra processes.
Convinced by the range of applications, we seek to provide unique nonextendible solutions to deterministic and stochastic Volterra integral equations that are allowed to be path-dependent, provide growth and error estimates for Picard iterations and establish regularity of solutions relative to the initial data.