Eulalia Nualart (the researcher, hereafter) broadly works in the field of Stochastic Calculus of Variations (Malliavin Calculus), and its applications to stochastic differential equations (SDEs) and stochastic partial differential equations (SPDEs). In the last years, she has become interested in two new applications of the Malliavin calculus, which are Statistical Inference for SDEs and applications to Mathematical Finance.The aim of this proposal is to consider two different types of SDES: (i) SDEs driven by the sum of a Brownian motion and a Poisson random measure; (ii) SDEs driven by fractional Brownian motion. The coefficients of both SDEs are assumed to depend on some parameter that needs to be estimated in the following cases: (a) when the trajectory of the SDE is observed continuously during a fixed time interval; (b) when the trajectory is observed discretely at n fixed times. Both cases will be studied, but the proposal concentrates in case (b), as is a more challenging and realistic problem.The proposal is divided into three research projects whose goals are: (1) Obtain upper and lower bounds for the density of the solution to the two types of SPDEs (i) and (ii), by means of the Malliavin Calculus; (2) Use these bounds in order to prove the local asymptotic normality (LAN) for the models (i) and (ii), and then apply Hajek-Lecam's theorem to obtain asymptotically efficient estimators for the parameter of the equations; (3) Study Monte Carlo methods and exact simulation of the SDE model with jumps (i), and apply these computational methods to the following financial problems: jump volatility models and numerical computations of greeks.In addition to the three research projects, the proposal aims to develop a research network on Stochastic Analysis and applications, by organizing weekly seminars and two international conferences at the University Pompeu Fabra, where the researcher has been offered a permanent associate professor position.