A Highly Sensitive Mean-Reverting Process With Markovian Switching And Euler-Maruyama Approximations

In recent decades, stochastic differential equation theory in physics, chemistry, mechanics, biology, economic and financial, control theory, aerospace engineering, and other fields, plays an important role. Mature stochastic differential equation theory tools to solve various kinds of economic activity in the capital market plays an important role. For example, the interest rate model, can be used for the analysis and research of the economic and financial pricing, etc. Interest rates in the financial markets play a role of the most fundamental and most important, the short-term interest rate is the hot issues in the economic and financial research. Along with the deepening of the research on interest rates, up to now there has been a lot of stochastic differential equation model for its volatility related simulation. Interest rate forecast more accurate, this will greatly reduce the risk of all kinds of financial activities, is advantageous to the economic and social stability.Past research has shown that successful consecutive time short-term interest rate model is to allow interest fluctuation change is sensitive to high interest rates. From mathematical perspective, however, is sensitive to high interest rates mean differential equation with coefficient does not meet the linear growth condition, so can't use the traditional mathematical methods to examine the nature of it. On the other hand, exploring the characteristics of short-term interest rate of China's financial markets, we find that short-term interest rates in China has a large fluctuation, characteristics of time-varying, and structure change, is influenced by many random factors. Therefore, this article introduced Markov switching process, review Markov switching process under short-term riskless interest rate model. On the basis of this model, this paper probes into Markov switching process under the interest rate model under the condition of nonlinear growth of math problems. Including whether the equation has a nonnegative solution and the existence and uniqueness, the solution of the stochastic boundedness and asymptotic behavior of numerical solution, and investigate its properties and numerical solution convergence in probability. The convergence result can be used to calculate the expected return some financial products. Using E- M method to calculate the financial product pricing problems.for example, we can apply the results to calculate bond rates of prices under highly sensitive mean reverting process, etc.