Studies On Some Problems Of Self-Similar Processes

Self-similar processes are stochastic processes which have identical distribu-tions under suitable scaling of time and space. Since Lamperti gave the rigorousdefinition of general self-similar processes in the middle of last century, the theo-retical platform of self-similar processes has been established by a large number ofscholars. Nowadays, self-similar processes have been widely applied in many fieldssuch as statistical physics, mathematical finance, communication network, econo-metrics, hydrology, etc.. Brownian motion is the most outstanding representative inthe theoretical and applied studies of self-similar processes. However, some idealmodels are implausible. Thus, we need to find and study some other new self-similarprocesses. In recent years, fractional Brownian motion, sub-fractional Brownian mo-tion, Rosenblatt process et al. have been well applied in some related fields, whichhave inspired a lots of scholars to study these processes. The discussions of theseprocesses have now became the hotspot issues on self-similar processes.In this thesis, following the frontier of research, we study some theoretical prob-lems about three sorts of self-similar processes which have given rise to particularattention in recent years. We also discuss the existence and uniqueness as well asregularity property of solutions to two-parameter stochastic Volterra equation withnon-Lipschitz coefficients. At last, an application of the self-similar process is given.Firstly, fractional Brownian motion is a centered Gaussian self-similar processwith stationary increments. In Chapter3, adopting Doss-Sussmann transformation,we firstly solve the ordinary differential equations with parameters and give thegeneralized sample solutions of a class of stochastic Volterra integral equation andstochastic differential equation driven by fractional Brownian motion. Moreover, twocomparison theorems of generalized sample solution are obtained.Secondly, the auto-covariance matrix of discrete time sub-fractional Brownianmotion (dsfBm) with length m is studied. This process has many properties as sameas fractional Brownian motion except for stationary increment property. In fact, weusually deal with discrete time signals for all practical problems. Thus, the studyto the properties of dsfBm is the precondition of applications. In Chapter4, byvirtue of the perturbation theory of linear operator in matrix theory, we obtain theapproximation and structure theorems of auto-covariance matrix of dsfBm and prove the increments of dsfBm are asymptotic stationary. At last, The data simulation andanalysis are implemented by Maple.Thirdly, Rosenblatt process is the most commonly Hermite process, and it isalso self-similarity, stationary increments, long-range dependence and non-Gaussianstochastic process. In Chapter5, we get a Fubini theorem by the Wiener integral ofRosenblatt process. On the other hand, we give the Riemann-Stieltjes integral withrespect to Rosenblatt process by its Ho¨lder continuous property and Young integral.Moreover, another stochastic Fubini theorem for Riemann-Stieltjes integral with re-spect to the process is given.Fourthly, in the past three decades, two-parameter stochastic differential equa-tions driven by Brownian sheets have been widely discussed. But, the mostly articlesinvolved stochastic Volterra equation with Lipschitz coefficients. In Chapter6, withBihari’s inequality in the plane, extended Minkowski inequality and Picard iterationprocedure, we prove the existence and uniqueness as well as regularity property of so-lutions to two-parameter stochastic Volterra equation with non-Lipschitz coefficientsand driven by Brownian sheet.Finally, In Chapter7, we give an application of self-similar processes in thebursty of behavior in web traffic. the self-similarity has been demonstrated in thebursty of I/O access behavior in web server. Moreover, we point out the meanings ofthese discoveries.