Power Variation Theory Of Fractal Integral Processes With Jumps And Its Application To High Frequency Financial Data

High-frequency data generally refers to data that is collected at a very rapid rate. Financial high-frequency data often refers to intraday data which often has jumps and long memory. In this dissertation, we study some continuous time processes that has the two properties, i.e., some fractional integral processes with jumps, and discuss the asymptotic properties of power variations for the processes. Using these properties, we build some statistics for testing whether the process can describe the dynamic of financial asset price. In detail, the main performances and results of this study are described as follows.(1)?A class of processes that has jumps and long memory was advanced. The processes are the processes that is a sum of a fractional Brownian motion and a non-Gaussian Levy process, we also give a general class that is a integral process of a fractional Brownian motion plus a semimartingale which has not a continuous martingale part, or a integral process of a stationary Gaussian process plus a semimartingale which has not a continuous martingale part.(2)?The asymptotic properties of power variations for these processes were discussed. The large number law was analyzed and the associated large number law was obtained. We study the central limit theorems and get some central limit theorems. The case of central limit theorems is complicated, and the asymptotic distributions are different for different cases.(3)?The asymptotic properties of multipower variations, threshold power variations and threshold multipower variations for these processes were investi-gated. In detail, we analyze the asymptotic behavior of realized bipower variations, realized threshold power variations and realized threshold bipower variations and obtain some large number law and central limit theorems.(4)?The asymptotic properties of power variations for a special process that is a sum of a fractional Brownian motion and an?-stable process were discussed. We obtain some conclusions about the large number law in this special case, and also get some central limit theorems for many cases. In the proof of these re-sults, we produce an inequality and the inequality can be applied in other similar situations.(5)?Some appropriate statistic were advanced for testing the model. Using the theory of power variations, multipower variations, threshold multipower vari-ations, threshold power variations, we build three statistics to test whether the underlying process has long memory for the case the process has jumps, and we also discuss the performance of the tests in finite sample situation.(6)?The empirical application was conducted to the financial high-frequency data. We analyze the actual financial high-frequency data by these test statistics. The results show the financial data has long memory, and the proposed process would prefer to describe the data in comparison with the semimartingales. The conclusions provide an alterative model for study the financial market and the microstructure of financial market.The innovations of the achievements in this study are described as follows. Firstly, we advance a continuous-time process that has jumps and long memory, which provide an alterative model for describing the dynamics of financial asset price or for other applications. Secondly, we study the asymptotic properties of the realized power variations for the processes. We obtain the large number law and central limit theorems for realized power variations, realized bipower variations, threshold power variations and threshold bipower variations. Thirdly, we provide an inequality in the proceed of the proofs, and the inequality will has other uses. At last, we show that the model is accepted by the test statistics for the real financial high-frequency data, which will provide an alterative model for studying the financial market and other applications.The innovations of the methodologies in this study are described as follows. Firstly, by noticing that the process is neither a semimartingale nor a Gaussian process, the existed approach can not directly employ here when we study the asymptotic properties of the realized power variations. Hence, we use a new de-composition approach to settle the problems, and the approach can be applied other similar situations. Secondly, we build a new inequality in the procedure of the proof for the special process, and the inequality settle some problems in establishing some central limit theorems. Thirdly, in order to resolve the tests, we advance three test statistics and the approach of building the statistics will provide a new method and idea to analyze some similar problems.The conclusions and methodologies have developed the theory of stochastic processes, generalize from Levy processes, semimartingales, fractional integral pro-cesses to the fractional integral processes with jumps. The results are helpful to apply stochastic processes to other fields and provide an alternative model. For the theory of power variations for the processes, it is benefit to gasp the asymp-totic properties for stochastic processes and employ the power variations to study the stochastic processes. Moreover, those are benefit to the more applications of the processes and to treat some similar problems. Undoubtedly, the theory power variations will help to study the characteristic of high-frequency data, furthermore, and analyze the theory of financial market by integrating the realized volatility and other methods. In conclusion, the related studies of processes in this paper can not only perfect the statistical inference of the processes, but also enrich the effective methods to deal with the real data.