Option Pricing under the Subordinated Market Models

This paper aims to study option pricing problem under the subordinated Brownian motion. Firstly, we prove that the subordinated Brownian motion controlled by the fractional diffusion equation has many financial properties, such as self-similarity, leptokurtic, and long memory, which indicate that the fractional calculus can describe the financial data well. Then, we investigate the option pricing under the assumption that the stock price is driven by the subordinated Brownian motion. The closed-form pricing formula for European options is derived. In the comparison with the classic Black–Sholes model, we find the option prices become higher, and the “volatility smiles” phenomenon happens in the proposed model. Finally, an empirical analysis is performed to show the validity of these results.

Analytical Solutions of a Class of Fluids Models with the Caputo Fractional Derivative

This paper studies the analytical solutions of the fractional fluid models described by the Caputo derivative. We combine the Fourier sine and the Laplace transforms. We analyze the influence of the order of the Caputo derivative the Prandtl number, the Grashof numbers, and the Casson parameter on the dynamics of the fractional diffusion equation with reaction term and the fractional heat equation. In this paper, we notice that the order of the Caputo fractional derivative plays the retardation effect or the acceleration. The physical interpretations of the influence of the parameters of the model have been proposed. The graphical representations illustrate the main findings of the present paper. This paper contributes to answering the open problem of finding analytical solutions to the fluid models described by the fractional operators.

Boundary stabilization and disturbance rejection for an unstable time fractional diffusion-wave equation

In this paper, we study boundary stabilization and disturbance rejection problem for an unstable time fractional diffusion-wave equation with Caputo time fractional derivative. For the case of no boundary external disturbance, both state feedback control and output feedback control via Neumann boundary actuation are proposed by the classical backstepping method. It is proved that the state feedback makes the closed-loop system Mittag-Leffler stable and the output feedback makes the closed-loop system asymptotically stable. When there is boundary external disturbance, we propose a disturbance estimator constructed by two infinite dimensional auxiliary systems to recover the external disturbance. A novel control law is then designed to compensate for the external disturbance in real time, and rigorous mathematical proofs are presented to show that the resulting closed-loop system is Mittag-Leffler stable and the states of all subsystems involved are uniformly bounded. As a result, we completely resolve, from a theoretical perspective, two long-standing unsolved mathematical control problems raised in [Nonlinear Dynam., 38(2004), 339-354] where all results were verified by simulations only.