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Let be a d-dimensional bifractional Brownian motion with Hurst parameters and . Assuming , we prove that the renormalized self-intersection local time exists in if and only if , where δ denotes the Dirac delta function. Our work generalizes the result of the renormalized self-intersection local time for fractional Brownian motion.

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In this study, a wavelet denoising method is first used to eliminate the influence of noise. Then, an overlapping smooth window technique is introduced into the asymmetric multifractal detrended cross-correlation analysis method, which was combined with the multiscale multifractal analysis method, resulting in the proposed asymmetric multiscale multifractal detrended cross-correlation analysis method. This method not only remedies the pseudo-fluctuation defect of the traditional method, but also explores the asymmetric multifractal cross-correlation between China’s rebar futures and spot markets at different scales. The results show the existence of an asymmetric multifractal cross-correlation between rebar futures and spot markets with upward and downward trends at different scales. This cross-correlation is highly complex at the small-scale, and more pronounced when the futures market is in an uptrend.