Level dependent Levy risk processes

Teoria prawdopodobieństwa i modelowanie stochastyczne
Osoba referująca: 
Tomasz Rolski (Uniwersytet Wrocławski)
Data spotkania seminaryjnego: 
czwartek, 18. Maj 2017 - 12:15
We consider level dependent Lévy risk process whose dynamics is given by sde $$\mathrm{d}U(t) = \mathrm{d}X(t) - \phi(U(t))dt,$$ where $X$ is a spectrally negative Lévy process. A special case is when $$\phi(x)=\left(\delta_1 \mathbf{1}_{\{U_k(t) > b_1\}}+\delta_2 \mathbf{1}_{\{U_k(t) > b_2\}}+...+\delta_k \mathbf{1}_{\{U_k(t) > b_k\}} \right)\mathrm{d}t , \quad t \geq 0 ,$$ which gives $k$-multi-refracted Levy process. We define a scale function $w^{(q)}$ (and also $z^{(q)}$) as the solutions of some Volterra equations and show that formulas for one and two sided exit problems written in their terms. The major part of the seminar will be devoted to solutions of Volterra equations.