Fractional Stochastic Differential Equations: Unlocking the Secrets of Stochastic Processes

Stochastic processes are a crucial component of modern mathematics and find applications in various fields such as physics, finance, and biology. Understanding the behavior of such processes is essential in modeling and predicting real-world phenomena. One particular area of interest is Fractional Stochastic Differential Equations (FSDEs) - a powerful tool that provides insights into complex dynamics.

What are Stochastic Processes?

In simple terms, a stochastic process is a mathematical model that describes the random evolution of a system over time. It involves the study of random variables and their behavior in various scenarios. Unlike deterministic models, which yield precise outcomes, stochastic processes account for uncertainty and randomness in the system.

Stochastic processes are widely used to model and analyze phenomena with inherent randomness, such as stock prices, weather patterns, or the spread of diseases. They provide a framework for understanding how a system evolves over time based on random factors.

Fractional Stochastic Differential Equations: Applications to Covid-19 Modeling (Industrial and Applied Mathematics)

by Abdon Atangana(Kindle Edition)

4.7 out of 5

Language	:	English
File size	:	14960 KB
Screen Reader	:	Supported
Print length	:	20 pages
X-Ray for textbooks	:	Enabled

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The Role of Fractional Stochastic Differential Equations (FSDEs)

Building upon the foundation of standard stochastic differential equations (SDEs),FSDEs introduce a fractional order derivative operator to capture the memory effects and long-range dependencies observed in many real-world processes. While traditional SDEs describe systems with short-range memory, FSDEs offer a more comprehensive approach that accounts for the complex dynamics associated with long-range memory.

This long-range dependence is an inherent feature of various natural processes, such as earthquakes, DNA sequences, and financial markets. FSDEs allow us to model and analyze such phenomena more accurately, leading to a deeper understanding of their underlying dynamics.

Understanding Fractional Calculus

Fractional calculus, the mathematical framework behind FSDEs, extends the concept of differentiation and integration to non-integer orders. It provides a mathematical tool to describe the behavior of systems with memory and long-range dependence.

Traditional calculus deals with integer-order derivatives and integrals, which correspond to instantaneous changes or accumulations. Fractional calculus, on the other hand, generalizes these operations to fractional orders, capturing the history and memory of the system.

For instance, a fractional order derivative of a function represents the rate of change of that function over time, considering a fractional memory window. This enables the modeling and analysis of complex processes with memory effects.

Applications of FSDEs

FSDEs have found applications in various fields, including finance, physics, telecommunications, geophysics, and environmental sciences. By incorporating long-range dependence and memory effects, FSDEs provide more accurate models for processes with inherent randomness.

In finance, FSDEs offer a valuable tool for pricing financial derivatives, risk management, and portfolio optimization. They capture the complex dynamics of financial markets, which are characterized by long-range dependence and volatility clustering.

In physics, FSDEs have been applied to model diffusion processes in complex media, such as porous materials and biological tissues. By considering the memory effects and spatial heterogeneity, FSDEs improve the accuracy of predicting particle transport and dispersion.

In telecommunications, FSDEs help address issues related to signal propagation and noise in wireless communication networks. Their ability to model long-range dependencies enhances the understanding of signal degradation and interference effects.

Challenges and Future Directions

Despite their numerous applications and benefits, FSDEs also pose challenges in terms of computational complexity and numerical simulation. Due to the non-Markovian nature of FSDEs, traditional numerical methods may not be directly applicable, necessitating the development of specialized techniques.

Researchers are constantly working towards developing efficient algorithms and numerical schemes to solve FSDEs. This includes Monte Carlo methods, finite difference methods, and numerical approximation techniques tailored specifically for FSDEs.

Moreover, further investigation is needed to explore the theoretical properties of FSDEs, such as existence and uniqueness of solutions, stability, and convergence properties. These advancements will pave the way for more reliable and accurate modeling of complex dynamic systems.

Fractional Stochastic Differential Equations (FSDEs) emerge as a powerful tool for understanding and quantifying complex systems' dynamics under uncertainty. By incorporating long-range dependence and memory effects, FSDEs bridge the gap between traditional stochastic methods and real-world phenomena.

With their applications spanning various fields, FSDEs provide valuable insights into financial markets, physics phenomena, telecommunications systems, and more. Though challenging, ongoing research in efficient numerical methods and theoretical properties will drive the future advancements in this intriguing branch of mathematics.

Fractional Stochastic Differential Equations: Applications to Covid-19 Modeling (Industrial and Applied Mathematics)

by Abdon Atangana(Kindle Edition)

4.7 out of 5

Language	:	English
File size	:	14960 KB
Screen Reader	:	Supported
Print length	:	20 pages
X-Ray for textbooks	:	Enabled

FREE

This book provides a thorough conversation on the underpinnings of Covid-19 spread modelling by using stochastics nonlocal differential and integral operators with singular and non-singular kernels. The book presents the dynamic of Covid-19 spread behaviour worldwide. It is noticed that the spread dynamic followed process with nonlocal behaviours which resemble power law, fading memory, crossover and stochastic behaviours. Fractional stochastic differential equations are therefore used to model spread behaviours in different parts of the worlds. The content coverage includes brief history of Covid-19 spread worldwide from December 2019 to September 2021, followed by statistical analysis of collected data for infected, death and recovery classes.