Narrow Results

In this paper we provide numerical evidence of the richer behavior of the connectivity degrees in heterogeneous preferential attachment networks in comparison to their homogeneous counterparts. We analyze the degree distribution in the threshold model, a preferential attachment model where the affinity between node states biases the attachment probabilities of links. We show that the degree densities exhibit a power-law multiscaling which points to a signature of heterogeneity in preferential attachment networks. This translates into a power-law scaling in the degree distribution, whose exponent depends on the specific form of heterogeneity in the attachment mechanism.

This study reveals the two-scale characteristics in prime number distribution. It is observed a sub-diffusion process of power law decay at the small scale of the natural number $1\le N\le 10^{35}$, but is found to obey the classical Brownian motion of an exponential decay at the large scale $N\ge 10^{220}$. Such two-scale mechanism gives rise to the multi-fractal scaling from the power law to the exponential law distributions in a transition region of the natural number $10^{35}\le N\le 10^{220}$. In the small range, the sub-diffusion of prime number distribution is well depicted by the fractional derivative equation model, and in the large scale, exponential decay distribution can accurately be described by a classical diffusion equation model. The Riemann diffusion equation proposed recently by the present authors can accurately model the prime distribution from small to moderate to large scales and is reduced to the fractional derivative sub-diffusion equation at small scale and the classical Brownian motion diffusion equation at large scale, respectively.

It has been recently discovered that some random processes may satisfy limit theorems even though they exhibit intermittency, namely an unusual growth of moments. In this paper, we provide a deeper understanding of these intricate limiting phenomena. We show that intermittent processes may exhibit a multiscale behavior involving growth at different rates. To these rates correspond different scales. In addition to a dominant scale, intermittent processes may exhibit secondary scales. The probability of these scales decreases to zero as a power function of time. For the analysis, we consider large deviations of the rate of growth of the processes. Our approach is quite general and covers different possible scenarios with special focus on the so-called supOU processes.

A generalized Black–Scholes–Merton economy is introduced. The economy is driven by Brownian motion in random time that is taken to be continuous and independent of Brownian motion. European options are priced by the no-arbitrage principle as conditional averages of their classical values over the random time to maturity. The prices are path dependent in general unless the time derivative of the random time is Markovian. An explicit self-financing hedging strategy is shown to replicate all European options by dynamically trading in stock, money market, and digital calls on realized variance. The notion of the average price is introduced, and the average price of the call option is shown to be greater than the corresponding Black–Scholes price for all deep in- and out-of-the-money options under appropriate sufficient conditions. The model is implemented in limit lognormal random time. The significance of its multiscaling law is explained theoretically and verified numerically to be a determining factor of the term structure of implied volatility.

An arbitrage-free CEV economy driven by Brownian motion in independent, continuous random time is introduced. European options are priced by the no-arbitrage principle as conditional averages of their classical CEV values over the CEV-modified random time to maturity. A novel representation of the classical CEV price is used to investigate the asymptotics of the average implied volatility. It is shown that the average implied volatility of the at-the-money call option is lower and of deep out-of-the-money call options, under appropriate sufficient conditions, greater than the implied CEV volatilities. Unlike in the classical CEV model, the shape of the out-of-the-money tail can be both downward and upward sloping depending on the tails of random time. The model is implemented in limit lognormal time. Its multiscaling law is shown to imply a term structure of implied volatility that is qualitatively more sensitive to changes in the time to maturity than is the classical CEV model.

China has taken important steps to reform its economy and capital markets in the past 20 years. Despite these efforts there is a lack of quantitative evidence on how these measures have impacted price returns in the stock exchanges. The purpose of this research was to determine the randomness of Chinese equity returns and to measure the scaling property of volatility over time. The main assumption of the Efficient Market Hypothesis is that security returns follow the path of a random walk and that volatility scales with the square root of time. This notion was tested by analyzing daily stock returns of the Shanghai- and Shenzhen Composite Indexes between 1990 and 2013. The Kolmogorov–Smirnov (KS) test rejected the log-normal distribution and random walk hypothesis. The measured Hurst exponents revealed a multiscaling property of fractal Brownian motion and indicated the presence of long-range dependence. The findings also showed that the degree of persistence and cycle length has reduced over time.

Frame multiresolution analysis (FMRA) in $L^2(ℝ)$ is an important topic in frame theory and its applications. In this paper, we consider the so-called multiscaling FMRA in $L^2({ℝ}^n)$, which has matrix dilations and a finite number of scaling functions. This framework is a generalization of the theories both on monoscaling FMRA and on the classical MRA of multiplicity $d$. We characterize wavelet frames and Parseval wavelet frames for $L^2({ℝ}^n)$ under the circumstances that they can be associated with a multiscaling FMRA. We give two necessary and sufficient conditions for given functions ${\{{\psi }_k\}}_{k=1}^K$ in $V_1$ to be multiframe generators of $W_0=V_1\ominus V_0$. Especially, the second condition depends on the multiscaling FMRA and ${\{{\psi }_k\}}_{k=1}^K$ only, does not require the existence of other functions, and is relatively easier to verify. Moreover, for any finitely-generated frame of integer translates, we give explicitly the Fourier transforms of the generators of its canonical dual frame. We illustrate the implementation and an application of the theory with an example.