Stability And Numerical Analysis For Jump Diffusions With Markovian Switching

The classical theory of difusions process has been widely applied to variousfields, such as finance and engineering. However, in the realistic world, many systemsnot only have substantially instantaneous changes due to some unexpected events, butalso may be subjected to frequent unpredictable structural changes in regime. The clas-sical difusions theory could not deal with these unpredictable situations. Recently, inorder to make up for the deficiency of the existing studies, many scholars have triedto introduce various kinds of stochastic systems such as jump difusions systems withMarkovian switching. The jump difusions with Markovian switching is a combinationof difusions with both random jumping and Markovian switching, which is an ef-cient model to solve the problems above. However, since the appearance of Markovianswitching and jump in the system, some new challenges will inevitably be appearedin the model analyzing, which make the model have important theoretical significanceand realistic value.The dissertation will focus on two main problems of jump difusions with Marko-vian switching. Firstly, we will discuss theψγstability and exponential stability indetail. Secondly, we will construct two numerical schemes to solve this process andanalyze the convergence and stability properties for both schemes in this dissertation.In the following, we will give the main innovations and novelties of this disserta-tion briefly.(1) We extend firstly the concept ofψγstability in classical difusions process tojump difusions with Markovian switching. We also propose and prove sometheorems to determine theψγstability for such process, which widen the researchidea for such problems.(2) For linearized jump difusions with Markovian switching, the corresponding ex-ponential stability has been investigated deeply. In one-dimensional linear case,some necessary and sufcient conditions for determining the p-th moment and almost sure exponential stability are derived, and also the corresponding suf-cient conditions in multi-dimensional linearized case are obtained.(3) The compensated stochasticθmethods used to numerically solve the classicstochastic diferential equations with Poisson jumps has been extended to thecase of jump difusions with Markovian switching. For the new numerical meth-ods for jump difusions with Markovian switching, we prove that the order of thestrong convergence is1/2. We also give the necessary and sufcient conditionsto determine the mean square stability for linear test systems.(4) We propose firstly the predictor-correctorθmethods based on the compensatedstochasticθmethods. In this dissertation, we prove the strong1/2order conver-gence and also discuss the mean square stability for this new scheme.