| Abstract [eng] |
This thesis undertakes a comprehensive exploration of stochastic differential equations (SDEs), spanning diverse problem domains and addressing a variety of challenges. A key focus is on fractional stochastic differential equations (fSDEs), where conventional assumptions are relaxed, allowing for coefficients that do not adhere to linear growth conditions and non-Lipschitz diffusion coefficients. The research establishes conditions for solution positivity, demonstrates convergence rates for the backward Euler approximation scheme, and devises high-quality estimators for the Hurst index. Concrete examples of classical models adhering to theorem conditions are provided for practical applicability. Additionally, the thesis investigates one-dimensional SDEs driven by stochastic processes with Holder continuous paths when its order is between 0.5 and 1. We construct and analyze approximation schemes with the aim of surpassing existing convergence rates and identifying classical models compatible with the SDEs under consideration. Furthermore, the research explores integrated fractional Brownian motion and introduces a novel concept: fractional SDEs with soft walls, featuring repulsion instead of reflection. It establishes a mathematically rigorous model for this soft wall model, defining conditions for unique solutions and developing an implicit Euler approximation scheme with high convergence rates. Mathematical modelling tools are employed to illuminate the distinct behaviour of this model. In summary, this thesis covers a wide range of SDE topics, offering innovative solutions, efficient approximation methods, and valuable insights for both theoretical advancements and practical applications. |
