The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two mai...

This communication deals with first introducing the exponential type hybrid B − (b, ρ, θ, p̃, r̃)invexities and then establishing a class of the ε− efficiency conditions applying to multiobjective fractional programming problems. The exponential type hybrid B − (b, ρ, θ, p̃, r̃)-invexities encompass most of the existing generalized higher order invexities as well as the exponential type generalized...

Hadamard transform is an important tool in discrete signal processing. In this paper, we define the discrete fractional Hadamard transform which is a generalized one. The development of discrete fractional Hadamard is based upon the same spirit of discrete fractional Fourier transform.

In this paper, we develop some methods to save the bandwidth required in the fractional domain. The fractional domain is the transformed domain of the fractional Fourier transform (FRFT). It is the intermediate of the time domain and the frequency domain. We find that, with the aid of the fractional Hilbert transform and other techniques, we can save 112 or 314 of the bandwidth in the fractiona...

For time-invariant degradation models and stationary signals and noise, the classical Fourier domain Wiener filter, which can be implemented in O(N logN) time, gives the minimum mean-square-error estimate of the original undistorted signal. For time-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N2) time for implementation. We consider filterin...

First, a class of comprehensive higher order exponential type generalized B-(b, ρ, η, ω, θ, p̃, r̃, s̃)-invexities is introduced, which encompasses most of the existing generalized invexity concepts in the literature, including the Antczak type first order B-(b, η, p̃, r̃)-invexities as well as the Zalmai type (α, β, γ, η, ρ, θ)-invexities, and then a wide range of parametrically sufficient optimali...

In this article we have applied a numerical finite difference method to solve the Black-Scholes European and American option pricing both presented by fractional differential equations in time and asset.

The inverse tempered stable subordinator is a stochastic process that models power law waiting times between particle movements, with an exponential tempering that allows all moments to exist. This paper shows that the probability density function of an inverse tempered stable subordinator solves a tempered time-fractional diffusion equation, and its ‘‘folded’’ density solves a tempered time-fr...