As is the case for deterministic dynamic systems or stochastic differential equations, closed-form solutions for switching diffusions are often dificult to obtain, and numerical approximation is frequently a viable or possibly the only alternative. Being extremely important, numerical methods have drawn much attention. To date, a number of works (e.g., [120, 121, 122, 171]) have focused on numerical approximations where the switching process is independent of the continuous component and is modeled by a continuous-time Markov chain. In addition to the numerical methods, approximation to invariant measures and non-Lipschitz data were dealt with. Nevertheless, it is necessary to be able to handle the coupling and dependence of the continuous states and discrete events. This chapter is devoted to numerical approximation methods for switching diffusions whose switching component is x-dependent. Section 5.2 presents the setup of the problem. Section 5.3 suggests numerical algorithms. Section 5.4 establishes the convergence of the numerical algorithms. Section 5.5 proceeds with a couple of examples. Section 5.6 gives a few remarks concerning the rates of convergence of the algorithms and the study on decreasing stepsize algorithms. Finally Section 5.7 concludes the chapter.
KeywordsNumerical Approximation Weak Convergence Erential Equation Sample Path Switching Process
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