Course description
Introduction to stochastic differential equations
Syllabus
- Introduction and motivations
- Brownian motion
- Stochastic integrals, Ito formula
- Stochastic differential equations
- Applications
Example
Probabilistic approximation of the solution of the Dirichlet problem \[\begin{equation} \left \{ \begin{array}{lll} \Delta u & =0 & \textrm{in $E$}\\ u & = g & \textrm{for $\partial E$}\\ \end{array} \right . \label{Dirichlet} \end{equation}\]
\[(\forall x\in E)\quad u(x)=\mathbb{E}(g(X_{\tau_x}))\] where \(X_t\) is a brownian motion starting at \(x\in E\) and \(\tau_x\) is the time when \(X_t\) reaches \(\partial E\).
