Gaussian-type density estimates for mixed SDEs driven by correlated fractional Brownian motions

Abstract

In this work, we investigate the existence and properties of Gaussian-like densities for weak solutions of multidimensional stochastic differential equations driven by a mixture of completely correlated fractional Brownian motions. We consider both the short-range and long-range dependent regimes, imposing a singular drift in the short-range dependent case and a H\"older continuous drift in the long-range dependent setting. Our approach avoids the use of Malliavin calculus and stochastic dynamical systems, relying instead on the Girsanov theorem and the framework of exponential Orlicz spaces. By considering a conditionally Gaussian process, we establish the existence of a density with respect to the Lebesgue measure. Furthermore, we derive Gaussian-type upper and lower bounds for this density, illustrating the optimality of our results in the short-range dependent case.