Numerical Stochastic Dynamics

The aim of this book is to provide a well-structured and coherent overview of existing mathematical modeling approaches for biochemical reaction systems, investigating relations between both the conventional models and several types of deterministic-stochastic hybrid model recombinations.

The Abel Symposia volume at hand contains a collection of high-quality articles written by the world’s leading experts, and addressing all mathematicians interested in advances in deterministic and stochastic dynamical systems, numerical analysis, and control theory.

In financial and actuarial modeling and other areas of application, stochastic differential equations with jumps have been employed to describe the dynamics of various state variables. The numerical solution of such equations is more complex than that of those only driven by Wiener processes, described in Kloeden & Platen: Numerical Solution of Stochastic Differential Equations (1992).

This book is focused on the numerical discretization of ordinary differential equations (ODEs), under several perspectives. The attention is first conveyed to providing accurate numerical solutions of deterministic problems. Then, the presentation moves to a more modern vision of numerical approximation, oriented to reproducing qualitative properties of the continuous problem along the discretized dynamics over long times. The book finally performs some steps in the direction of stochastic differential equations (SDEs), with the intention of offering useful tools to generalize the techniques introduced for the numerical approximation of ODEs to the stochastic case, as well as of presenting numerical issues natively introduced for SDEs.

This book presents advanced methods for the electronic modeling of dynamical systems governed by ordinary differential equations.
It offers a comprehensive toolkit and practical solutions for specialists in nonlinear dynamics who seek experimental validation of their mathematical models. The aim is to empower readers without an extensive background in electronics or circuit theory to translate their theoretical concepts into real-world devices, facilitating the rapid experimental confirmation of numerical and theoretical findings. For experts in electronic engineering, the book showcases how a wide range of non-electronic systems and their unique characteristics can be effectively modeled using electronic circuits.

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in the fields of physics and mathematics, among others, as highly correlated nonlinear phenomena, evolving over a large range of time and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods.