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# Special Issue "Stochastic Modelling in Financial Mathematics"

A special issue of Risks (ISSN 2227-9091).

Deadline for manuscript submissions: 31 July 2020.

## Special Issue Editor

Prof. Dr. Anatoliy Swishchuk
Website
Guest Editor
Department of Mathematics and Statistics, University of Calgary, Calgary, AB T2N 1N4, Canada
Interests: mathematical finance; energy finance; stochastic modelling; risk theory; random evolutions and their applications; modeling high-frequency and algorithmic trading; deep and machine learning in quantitative finance

## Special Issue Information

Dear Colleagues,

Financial mathematics (also known as mathematical finance and quantitative finance) is a field of applied mathematics, concerned with mathematical and stochastic modelling of financial markets.

French mathematician Louis Bachelier is considered the author of the first scholarly work on mathematical finance, published in 1900. As a discipline, financial mathematics emerged in the 1970s, following the work of Fischer Black, Myron Scholes, and Robert Merton on option pricing theory.

In financial mathematics, modelling entails the development of sophisticated mathematical and stochastic models, and one may take, for example, the share price as a given and attempt to use stochastic calculus to obtain the corresponding value of derivatives of the stock. Thus, many problems, such as derivative pricing, portfolio optimization, risk modelling, etc., are generally stochastic in nature, and hence, such models require complex stochastic analyses.

One contemporary example of such a problem is big data. Big data have now become a driver of model building and analysis in a number of areas, including finance, insurance, and energy markets, to name a few. For example, more than half of the markets in today’s highly competitive financial world now use a limit order book (LOB) mechanism to facilitate trade.

This current Special Issue is exactly devoted to modern trends in financial mathematics associated with stochastic modelling, including modelling of big data.

Topics from many areas, such as high-frequency and algorithmic trading (limit order books), energy finance, regime-switching, and stochastic volatility modelling, among others, are shown to have deep applicable values which are useful for both academics and practitioners.

Prof. Dr. Anatoliy Swishchuk
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.ynsqex.icu by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All papers will be peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Risks is an international peer-reviewed open access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1000 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

## Keywords

• Stochastic modelling
• Mathematical finance
• Regime-switching models in finance
• Energy finance
• Limit order books
• Stochastic volatility modelling

## Published Papers (2 papers)

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# Research

Open AccessArticle
General Compound Hawkes Processes in Limit Order Books
Risks 2020, 8(1), 28; https://doi.org/10.3390/risks8010028 - 14 Mar 2020
Abstract
In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regard to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional [...] Read more.
In this paper, we study various new Hawkes processes. Specifically, we construct general compound Hawkes processes and investigate their properties in limit order books. With regard to these general compound Hawkes processes, we prove a Law of Large Numbers (LLN) and a Functional Central Limit Theorems (FCLT) for several specific variations. We apply several of these FCLTs to limit order books to study the link between price volatility and order flow, where the volatility in mid-price changes is expressed in terms of parameters describing the arrival rates and mid-price process. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics)
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Open AccessFeature PaperArticle
General Conditions of Weak Convergence of Discrete-Time Multiplicative Scheme to Asset Price with Memory
Risks 2020, 8(1), 11; https://doi.org/10.3390/risks8010011 - 30 Jan 2020
Abstract
We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space $D [ 0 , T ]$ . Then we investigate the convergence of the related multiplicative scheme to a process that [...] Read more.
We present general conditions for the weak convergence of a discrete-time additive scheme to a stochastic process with memory in the space $D [ 0 , T ]$ . Then we investigate the convergence of the related multiplicative scheme to a process that can be interpreted as an asset price with memory. As an example, we study an additive scheme that converges to fractional Brownian motion, which is based on the Cholesky decomposition of its covariance matrix. The second example is a scheme converging to the Riemann–Liouville fractional Brownian motion. The multiplicative counterparts for these two schemes are also considered. As an auxiliary result of independent interest, we obtain sufficient conditions for monotonicity along diagonals in the Cholesky decomposition of the covariance matrix of a stationary Gaussian process. Full article
(This article belongs to the Special Issue Stochastic Modelling in Financial Mathematics)