Il teorema di Girsanov con applicazioni alle equazioni differenziali stocastiche

The main goal of this thesis is to present Girsanov's theorem and some of its applications to Stochastic Differential Equations (SDEs) and mathematical finance. After recalling the basic tools of stochastic analysis and probability theory, we study the Wiener process and It\^o stochastic integration. In particular, we discuss the reflection principle, providing a full proof, and derive Bachelier's theorem. A central part of the thesis is devoted to the construction of the It\^o integral. We provide a detailed construction following the approach of N.~V.~Krylov \cite{krylov2002}, which is more general than some classical treatments such as \cite{baldi2017} or \cite{ks}, since it does not require progressive measurability. The core of the thesis is to provide a complete proof of Girsanov's theorem. We present the elegant approach by N.~V.~Krylov \cite{krylov2002}, while including several details that are omitted in his exposition. We also clarify some points of the Krylov method. Particular attention is devoted to the exponential martingale $$ \rho_t(b)=\exp \left( \int_0^t b_s dW_s - \frac{1}{2} \int_0^t b_s^2 ds \right)$$ involved in the change of measure, to Novikov's condition, and to further sufficient criteria ensuring the martingale property of $\rho_t$. We then apply Girsanov's theorem to prove weak existence and uniqueness in law for stochastic differential equations of the form \[ dX_t=b(t,X_t)\,dt+dW_t, \] assuming that b is Borel and grows at most linearly. Finally, we discuss an application to mathematical finance and derive the Black--Scholes formula for European options.