Blog-Series

Stochastic Differential Equations

Welcome to the Blog-Series

Integrals

Stochastic Processes

Application & Tools

Content of the Series

Literature

Welcome to the Blog-Series

In our digitally and mathematically dominated world Time-Series-Modelling is an indispensable subject in many areas (e.g. in Quantitative Financial Analysis, Financial Mathematics, Blockchain & Cryptocurrencies as well as applied Risk Management). Stochastic Differential Equations (SDE) are used more frequently in order to analyze time-series in the most appropriate manner . (Oksendal, 2013).
The term extends the mathematical generalization of Ordinary Differential Equations (ODE) to Stochastic Processes (Oksendal, 2013).
To provide an introduction for interested readers like you, it is necessary to briefly describe the terms Differential Equation and Stochastic Process without mathematical formalism.

A Stochastic Process (also known as Random Process ) is a mathematical object defined as a collection of random variables (in chronological order and driven by random occurences) (Oksendal, 2013).
A Differential Equation can be described as a mathematical equation relating parameters, variables and some function with its derivatives. Therefore, it is qualified for mathematical modelling (Arens, et al., 2018).

In short Stochastic Differential Equations (SDE) enable the analysis of time-series which follow the laws of a Stochastic Process (the goals of the analysis are the same as with deterministic functions) (Oksendal, 2013).

In order to make such a fascinating but simultaneously complex subject area more accessible to the reader, I have decided to write a whole series of articles about the discussed topic (these articles will be found on the blog using the abbreviation "SDE" and the specified subject).
The aim of this abstract is to provide you with a comprehensive overview (with no claim for completeness) of SDEs. Especially, the historical background, theoretical-mathematical implications as well as practical applications ensure comprehension of a global view of the subject.

lnitially, I will provide you with an overview of the universe of Integrals & Stochastic Processes and conclude with the discussed topics in this series. These will contain a selection of the most relevant and most widely used models and insights.

All relevant terms are presented in a non-mathematical, but formal way, using Mindmaps . You will find (without explanations & mathematical formalisms) a overview of different Integrals, Processes & Applications.

As you proceed, you may refer to these overviews to see the "whole picture" and which part of it can be linked to a certain article. Mathematical formalism and explanations can be found in the upcoming posts.

Articles that refer to Practical Applications and Programming (especially Source Code in Python or R ) are only accessible for our Network Partners and selected, veryfied & acquainted Readers . Contact us to get exclusive access to our collected knowledge.
If you want to become part of our network, we are looking forward to a personal meeting, once you signed a  Confidentiality Agreement .

Integrals

Integrals are presented because an ODE can also be formulated as an Integral Equation (Arens, et al., 2018).

Integrals

Stochastic Processes

Stochastic processes

Application & Tools

Applications & Tools

Content of the Series

Hereinafter, you will find a Selective Choice which faciliates the understanding of the subject area. Furthermore, my aim is to provide you with a deeper insight and to allow you to assess other questions of the area on your own.

The didactical concept of the articles is outlined below:

  • At first, the concept of the commonly known Random Walk as well as its underlying Theorems and Properties are introduced
  • Subsequently, the concept of Brownian Motion followed by Wiener Processes are explained
  • Finally, Itô ´s integrative and generalized formulations conclude discussed models
  • The Approximations of the discussed models in relation to one and another will also be deduced
  • Having illustrated the models, their properties & theorems as well as their approximations, I will demonstrate Practical Applications & Programming using Financial Markets & Operative Risk Management as an example.
Please note, that I will refrain from presenting long lasting mathematical proofs and derivations. Instead, I will supply interested readers with proper references for further study. The intention of the articles is to focus on the understanding of key concepts rather than providing detailed proofs of formalisms. 
Stochastische Differentialgleichung

With this in mind, I am looking forward to your opinions, messages, ideas and future projects.

Yours Respectfully,

Markus Vogl

Literature

Arens, T. et al., 2018. Mathematik. 4. Hrsg. s.l.:Springer.

Oksendal, B., 2013. Stochastic Differential Equations. Norway: Springer.

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