Rubin, Tomas. Stochastic Evolution Systems and Their Applications. Prague, 2016. Master’s thesis. Charles University,
My Master’s thesis studies a class of stochastic partial differential equations driven the fractional Brownian motion, a stochastic driving force featuring correlation of its schock. While most of the thesis presents a theoretical overview of such equations, I succeeded in extending one particular inequality that allows for solving the aforementioned class of equations without the notion of analytic semigroups, and thus enabling the theoretical analysis of the Heath-Jarrow-Morton model for yield curves driven by the infinite dimensional singular fractional Brownian motion.
My Master’s thesis is a purely theoretical contribution to the field of stochastic analysis.
In the Thesis, linear stochastic differential equations in a Hilbert space driven by a cylindrical fractional Brownian motion with the Hurst parameter in the interval H < 1/2 are considered. Under the conditions on the range of the diffusion coefficient, existence of the mild solution is proved together with measurability and continuity. Existence of a limiting distribution is shown for exponentially stable semigroups. The theory is modified for the case of analytical semigroups. In this case, the conditions for the diffusion coefficient are weakened. The scope of the theory is illustrated on the Heath-Jarrow-Morton model, the wave equation, and the heat equation.
My Master’s thesis won the following awards:
- First place in the student scientific competition “SVOČ“, 2016,
- Second best Master’s thesis in probability theory and statistics at Charles University in Prague, 2016,
- Second place in the Competition for the Best Student Paper in Theoretical Economics, 2016.