Decompositions of g-Frames and Duals and Pseudoduals of g-Frames in Hilbert Spaces

Firstly, we study the representation of g-frames in terms of linear combinations of simpler ones such as g-orthonormal bases, g-Riesz bases, and normalized tight g-frames. Then, we study the dual and pseudodual of g-frames, which are critical components in reconstructions. In particular, we characterize the dual g-frames in a constructive way; that is, the formulae for dual g-frames are given. We also give some g-frame like representations for pseudodual g-frame pairs. The operator parameterizations of g-frames and decompositions of bounded operators are the key tools to prove our main results

g-Bases in Hilbert Spaces

The concept of g-basis in Hilbert spaces is introduced, which generalizes Schauder basis in Hilbert spaces. Some results about g-bases are proved. In particular, we characterize the g-bases and g-orthonormal bases. And the dual g-bases are also discussed. We also consider the equivalent relations of g-bases and g-orthonormal bases. And the property of g-minimal of g-bases is studied as well. Our results show that, in some cases, g-bases share many useful properties of Schauder bases in Hilbert spaces

Some New Characterizations and g-Minimality and Stability of g-Bases in the Hilbert Spaces

The concept of g-basis in the Hilbert spaces is introduced by Guo (2012) who generalizes the Schauder basis in the Hilbert spaces. g-basis plays the similar role in g-frame theory to that the Schauder basis plays in frame theory. In this paper, we establish some important properties of g-bases in the Hilbert spaces. In particular, we obtain a simple condition under which some important properties established in Guo (2012) are still true. With these conditions, we also establish some new interesting properties of g-bases which are related to g-minimality. Finally, we obtain a perturbation result about g-bases

Operator Characterizations and Some Properties of g-Frames on Hilbert Spaces

Given the g-orthonormal basis for Hilbert space H, we characterize the g-frames, normalized tight g-frames, and g-Riesz bases in terms of the g-preframe operators. Then we consider the transformations of g-frames, normalized tight g-frames, and g-Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums and g-dual frames of g-frames by applying the results of characterizations

A Shannon Wavelet Method for Pricing American Options under Two-Factor Stochastic Volatilities and Stochastic Interest Rate

In the paper, the pricing of the American put options under the double Heston model with Cox–Ingersoll–Ross (CIR) interest rate process is studied. The characteristic function of the log asset price is derived, and thereby Bermuda options are well evaluated by means of a state-of-the-art Shannon wavelet inverse Fourier technique (SWIFT), which is a robust and highly efficient pricing method. Based on the SWIFT method, the price of American option can be approximated by using Richardson extrapolation schemes on a series of Bermudan options. Numerical experiments show that the proposed pricing method is efficient, especially for short-term American put options

A Fourier-Cosine Method for Pricing Discretely Monitored Barrier Options under Stochastic Volatility and Double Exponential Jump

In this paper, the valuation of the discrete barrier options on the condition that the underlying asset price process follows the GARCH volatility and double exponential jump is studied. We derived an analytical approximation of the characteristic function for the underlying log-asset price. Then, a quasianalytical approximate formula of the price of the discrete barrier option is obtained based the on Fourier-cosine method. Numerical examples show that the Fourier-cosine method is fast and efficient for pricing discrete barrier options compared with the Monte Carlo simulation method. Finally, the influences of some important parameters on the prices of discrete barrier options are studied to further illustrate the rationality of the model