Resumen de Adapted numerical methods for stochastic differential equations
This thesis explores three key aspects of Stochastic Differential Equations (SDEs), which are mathematical models incorporating randomness often absent in simpler deterministic equations.
First, it focuses on the numerical simulation of SDEs, particularly the mean-reverting Constant Elasticity of Variance (CEV) equation and its special case, the Cox-Ingersoll-Ross (CIR) equation, due to their wide applicability in finance, biology, and physics. The work analyzes existing numerical methods (like semi-implicit schemes) and develops new ones, studying their convergence, ability to preserve important solution properties (like positivity and long-term moments), and monotonicity.
Second, the thesis investigates the stability of systems modeled by SDEs. It does this by extending two deterministic models for malware spread in wireless sensor networks into stochastic versions. The stability of the disease-free state is then analyzed using mean-square stability techniques.
Third, it examines an alternative, operator-theoretic approach to analyzing SDEs by approximating associated operators like the Koopman and Perron-Frobenius operators using data. The thesis contributes improved error bounds for these data-driven approximations under less restrictive conditions than previously established.
Overall, the research advances the understanding and application of SDEs through contributions to their numerical solution, stability analysis, and operator approximation theory, supported by numerical experiments.
