Correlations and bounds for stochastic volatility models

Abstract

We investigate here, systematically and rigorously, various stochastic volatility models used in Mathematical Finance. Mathematically, such models involve coupled stochastic differential equations with coefficients that do not obey the natural and classical conditions required to make these models “well-posed”. And we obtain necessary and sufficient conditions on the parameters, such as correlation, of these models in order to have integrable or $L^p$ solutions (for $1<p<\infty$).