This paper mainly discusses the pricing of credit default swap (CDS) in the fractional dimension environment. We assume that the default intensity of a firm depends on the default states of counterparty firms and the term structure of interest rates, but the contagious impact of the counterparty firm is decreasing over time, until disappears. The interest rate risk is reflected by the fractional Vasicek interest rate model. We model the firm's default intensity in the looping default framework and derive the pricing formulas of risky bonds and credit default swap.

The equivalent filter characteristics of variational mode decomposition (VMD) are fully evaluated when applied to the fractional Gaussian noise (fGn) and the application in separating closely spaced modes of vibration system is performed in this paper. VMD is a newly proposed signal decomposition technique, which nonrecursively decomposes a signal into a given number of subsignals (modes), and each mode is mostly compact around a center pulsation. The filter performance of VMD is largely dependent on the constraint parameter and the initialization of center frequencies. In order to extract the desired modes, criteria for the determination of decomposition parameters are established. The initial center frequencies could be simply determined by prior estimated modal frequencies of the analyzed signal, while the constraint parameter is optimized utilizing a genetic algorithm (GA). A two-degree-of-freedom parametric system is considered to evaluate the capability of VMD in the separation of closely spaced modes. Compared with the noise-assisted versions of empirical mode decomposition (EMD) and wavelet packet transform (WPT), the parameter-optimized VMD can successfully separate the closely spaced modes while recovering the most modal information simultaneously. When introduced to the ground vibration test (GVT) of a horizontal tail, the proposed method successfully extracted the first five oscillation modes and identified the modal parameters accurately.

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