Stochastic Invariance and Stochastic Volterra Equations.

Summary This thesis deals with the theory of stochastic equations in finite dimension. In the first part, we derive necessary and sufficient geometric conditions on the coefficients of a stochastic differential equation for the existence of a solution constrained to remain in a closed domain, under weak regularity conditions on the coefficients.In the second part, we address existence and uniqueness problems of stochastic Volterra equations of convolutional type. These equations are in general non-Markovian. We establish their correspondence with infinite-dimensional equations which allows us to approximate them by finite-dimensional Markovian stochastic differential equations. Finally, we illustrate our results by an application in mathematical finance, namely the modeling of rough volatility. In particular, we propose a stochastic volatility model that provides a good compromise between flexibility and tractability.