| Abstract [eng] |
The aim of the thesis has been to construct simple, yet effective first- and second-order weak approximations for the solution of the Wright-Fisher (WF) model that use only generation of discrete random variables at each approximation step. The WF process was originally used to model gene frequencies, i.e., the proportions of genes in a population. In recent years WF and Jacobi processes have started appearing in the finance applications. These processes are restricted to a finite interval, and due to this, they seem appropriate to model dynamic bounded variables such as a regime probability or a default probability. However, closed-form solutions of the WF or Jacobi models are unknown therefore a need to approximate them arises. In the doctoral thesis, we construct first- and second-order weak approximations for the WF model using split-step, moments matching, and approximate moment matching techniques. Also, we provide a probabilistic proof of the regularity of solutions of the backward Kolmogorov equations for the WF equation and, in addition, for the CIR and general Stratonovich equations with square-root diffusion coefficient without relying on existing transition density formulas. Such a regularity is needed for rigorous proofs that potential (“candidate”) weak approximations are indeed weak approximations of the corresponding order. |
