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22 pages, 3564 KiB

Open AccessArticle

An Efficient Numerical Scheme for a Time-Fractional Black–Scholes Partial Differential Equation Derived from the Fractal Market Hypothesis

by Samuel M. Nuugulu, Frednard Gideon and Kailash C. Patidar

Fractal Fract. 2024, 8(8), 461; https://doi.org/10.3390/fractalfract8080461 - 6 Aug 2024

Cited by 2 | Viewed by 1272

Abstract

Since the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in real-time market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems. Full article

(This article belongs to the Section Numerical and Computational Methods)

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19 pages, 761 KiB

Open AccessEditor’s ChoiceArticle

A Stochastic Model of Anomalously Fast Transport of Heat Energy in Crystalline Bodies

by Łukasz Stępień and Zbigniew A. Łagodowski

Energies 2023, 16(20), 7117; https://doi.org/10.3390/en16207117 - 17 Oct 2023

Viewed by 1088

Abstract

In this work, a new method for constructing the infinite-dimensional Ornstein–Uhlenbeck stochastic process is introduced. The constructed process is used to perturb the harmonic system in order to model anomalously fast heat transport in one-dimensional nanomaterials. The introduced method made it possible to obtain a transition probability function that allows for a different approach to the analysis of equations with such a disturbance. This creates the opportunity to relax assumptions about temporal correlations for such a process, which may lead to a qualitatively different model of energy transport through vibrations of the crystal lattice and, as a result, to obtain the superdiffusion equation on a macroscopic scale with an order of the fractional Laplacian different from the value of 3/4 obtained so far in stochastic models. Simulations confirming these predictions are presented and discussed. Full article

(This article belongs to the Topic Thermal Energy Transfer and Storage)

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13 pages, 306 KiB

Open AccessArticle

Spatial Moduli of Non-Differentiability for Time-Fractional SPIDEs and Their Gradient

by Wensheng Wang

Symmetry 2021, 13(3), 380; https://doi.org/10.3390/sym13030380 - 26 Feb 2021

Cited by 2 | Viewed by 1516

Abstract

High order and fractional PDEs have become prominent in theory and in modeling many phenomena. In this paper, we study spatial moduli of non-differentiability for the fourth order time fractional stochastic partial integro-differential equations (SPIDEs) and their gradient, driven by space-time white noise. We use the underlying explicit kernels and spectral/harmonic analysis, yielding spatial moduli of non-differentiability for time fractional SPIDEs and their gradient. On one hand, this work builds on the recent works on delicate analysis of regularities of general Gaussian processes and stochastic heat equation driven by space-time white noise. On the other hand, it builds on and complements Allouba and Xiao’s earlier works on spatial uniform and local moduli of continuity of time fractional SPIDEs and their gradient. Full article

15 pages, 1020 KiB

Open AccessArticle

A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise

by Afshin Babaei, Hossein Jafari and S. Banihashemi

Symmetry 2020, 12(6), 904; https://doi.org/10.3390/sym12060904 - 1 Jun 2020

Cited by 29 | Viewed by 3251

Abstract

A spectral collocation approach is constructed to solve a class of time-fractional stochastic heat equations (TFSHEs) driven by Brownian motion. Stochastic differential equations with additive noise have an important role in explaining some symmetry phenomena such as symmetry breaking in molecular vibrations. Finding the exact solution of such equations is difficult in many cases. Thus, a collocation method based on sixth-kind Chebyshev polynomials (SKCPs) is introduced to assess their numerical solutions. This collocation approach reduces the considered problem to a system of linear algebraic equations. The convergence and error analysis of the suggested scheme are investigated. In the end, numerical results and the order of convergence are evaluated for some numerical test problems to illustrate the efficiency and robustness of the presented method. Full article

(This article belongs to the Special Issue New developments in Functional and Fractional Differential Equations and in Lie Symmetry)

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22 pages, 339 KiB

Open AccessArticle

Fractional Diffusion in Gaussian Noisy Environment

by Guannan Hu and Yaozhong Hu

Mathematics 2015, 3(2), 131-152; https://doi.org/10.3390/math3020131 - 31 Mar 2015

Cited by 8 | Viewed by 5176

Abstract

We study the fractional diffusion in a Gaussian noisy environment as described by the fractional order stochastic heat equations of the following form: \(D_t^{(\alpha)} u(t, x)=\textit{B}u+u\cdot \dot W^H\), where \(D_t^{(\alpha)}\) is the Caputo fractional derivative of order \(\alpha\in (0,1)\) with respect to the time variable \(t\), \(\textit{B}\) is a second order elliptic operator with respect to the space variable \(x\in\mathbb{R}^d\) and \(\dot W^H\) a time homogeneous fractional Gaussian noise of Hurst parameter \(H=(H_1, \cdots, H_d)\). We obtain conditions satisfied by \(\alpha\) and \(H\), so that the square integrable solution \(u\) exists uniquely. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus and Its Applications)

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