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In this paper, to cope with the shortage of sufficient theoretical supportresulted from the fast-growing quantitative financial modeling, we investigatetwo classes of generalized stochastic volatility models, establish theirwell-posedness of strong solutions, and conduct the stability analysis withrespect to small perturbations. In the first class, a multidimensionalpath-dependent process is driven by another multidimensional path-dependentprocess. The second class is a generalized one-dimensional stochasticvolatility model with H\"older continuous coefficients. What greatlydifferentiates those two classes of models is that both the process and itscorrelated driving process have their own subdifferential operators, whose onespecial case is the general reflection operators for multi-sided barriers.Hence, the models investigated fully cover various newly explored variants ofstochastic volatility models whose well-posedness is unknown, and naturallyserve as the rigorous mathematical foundation for new stochastic volatilitymodel development in terms of multi-dimension, path-dependence, and multi-sidedbarrier reflection.

Well-posedness and Stability Analysis of Two Classes of Generalized Stochastic Volatility Models