Theoretical Study On FBMs And Its Application To Quasar Optical Variability

In this thesis, we introduce the related theory of Fractal Brownian Motions (FBMs) and its simple application in modeling the light curves of quasars which have different indexes of Power Spectral Density (PSD), we hope to build a theoret-ical model which can simulate the light curves of quasars with different indexes of PSD and some characteristic timescales. First, according to the auto-correlation function of FBMs, we can derive the simple expressions of the error of the re-scaled range <F(t+τ)-F(t)>, but for the re-scaled range <|F(t+τ)-F(t)|>, auto-correlation function is not applicable, based on the re-scaled range analysis method, we carry out extensive numerical simulations and give the simple ex-pressions of the error of<|F(t+τ)-F(t)|> based on these numerical simulation results. Also, we find that the error of the re-scaled range not only depends on data sampling but also varies with Hurst exponent H, where H is a variable which represents correlation between the previous result and the next result in a fractal time series; Second, with the standard deviation for the logarithm of the re-scaled range <|F(t+τ)-F(t)|> of simulated Fractal Brownian Motions F(t), the method of least squares is adopted to determine the slope, S, and intercept,I, of the log((|F(t+τ)-F(t)|)) vs log(τ) plot. It is found that the reduced χ2 of the fitting decreases with the increase of the Hurst index H(the expectation value of S,0< H< 1) and the errors of the fitting parameters S and I are usually smaller than their corresponding standard deviations. Nevertheless, considering the correlation among the fitting data, they may be used to reinterpret the fit-ting results of the least square method to determine the dimension of FBMs more self-consistently and the currency exchange rate between Euro and Dollar is used as an example to demonstrate this procedure. Last, we model the light curves of quasars with different characteristic timescales by combining a first-order continu-ous autoregressive(CAR(1)) process with a fractal process for H close to zero, and obtain their corresponding characteristic timescales from each PSD of simulated light curves.