SIMBa: Stochastic Differential Equations

04/04/2014 by blocmat Deixa un comentari

Dilluns, 7 d’abril, se celebrarà una nova sessió del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: David Ruiz
Universitat: Universitat d’Oslo (Department of Mathematics)

Data: dilluns, 7 d’abril de 2014
Hora: 17:45, cafè; 18:00, inici
Lloc: Aula IMUB (al terrat), Facultat de Matemàtiques de la Universitat de Barcelona.

Títol: Stochastic Differential Equations.
Resum: In this talk we want to explain what a stochastic differential equation (SDE) is. As an idea, we will recall what a (deterministic) ordinary differential equation (ODE) is and then we will add some “noise” to it. Such equations are widely used to study and model phenomena in nature such as for instance: the “random” movement of a particle in a fluid due to collisions with the molecules of the fluid, variable uncertainty, macroeconomic dynamics, etc. The solution of an SDE is therefore a stochastic process, i.e. think of it as a function whose image is not a real number, but a random variable. It is a big area of research to study the solutions of SDE’s and the densities of such solutions. It is very difficult to say something about the densities.

An SDE looks typically like

$dx(t) = f(t,x(t))dt + "noise", t_0\leq t \leq a, x(t_0)=x_0\in \mathbb{R}$

where the initial condition $x(t_0)=x_0$ is typically taken as a deterministic point $x_0 \in \mathbb{R}$ that we have observed or might also be taken as $x(t_0) = Z$ where $Z$ is a random variable (e.g. normal distributed).

Finally, we will mention some properties of SDE’s, like… For example, you know that the solution when $f$ is Lipschitz exists and it is unique for ODE’s (Picard’s theorem). In the case of SDE’s even if $f$ is very “ugly” a unique solution also exists! Meaning that, the “noise” somehow regularizes $x(t)$ .

Si voleu rebre informació dels propers seminaris us podeu subscriure a la llista de correu. Si voleu contactar amb els responsables podeu escriure un missatge a seminari(dot)simba(at)ub(dot)edu.