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Mathematics
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First SDE: New Advances in Stochastic Differential Equations
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First SDE: New Advances in Stochastic Differential Equations

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 4804

Special Issue Editors

Dr. Nuno M. Brites

E-Mail Website
Guest Editor
ISEG—School of Economics and Management, REM—Research in Economics and Mathematics, CEMAPRE, Universidade de Lisboa, Rua do Quelhas, 6, Lisboa, Portugal
Interests: stochastic differential equations; stochastic processes; stochastic optimal control; statistics
Dr. João M. Guerra

E-Mail Website
Guest Editor
REM—Research in Economics and Mathematics, CEMAPRE, Rua do Quelhas 6, 1200-781 Lisboa, Portugal
Interests: stochastic differential equations; Lévy processes; mathematical finance; fractional brownian motion
Dr. Paula Milheiro-Oliveira

E-Mail Website
Guest Editor
Faculty of Engineering, University of Porto, 4200-465 Porto, Portugal
Interests: probability and statistics; applications in engineering; stochastic systems

Special Issue Information

Dear Colleagues,

We cordially invite you to submit your articles to the Special Issue of Mathematics entitled “First SDE: New Advances in Stochastic Differential Equations”. The title of the Special Issue not only reflects the topicality of the Special Issue itself but also provides a direct link to the First SDE – Stochastic Days Encounters: Stochastic Differential Equations and Statistics (https://cemapre.iseg.ulisboa.pt/sde2023/), held in Lisbon, Portugal, from May 25 to 26, 2023.

The primary goal of this event is to bring together researchers in some of the most active and promising areas of research on stochastic differential equations and statistics in order to exchange ideas and foster future collaborations. Another important goal is to expose academics, young researchers, and post-graduate students to the most recent developments in the above active areas.

The meeting will cover a broad range of topics, including theoretical and applied contributions to the following:

  • Stochastic differential equations;
  • Stochastic models;
  • Statistics;
  • Stochastic optimal control;
  • Numerical methods;
  • Applications.

This Special Issue aims to collect the most recent developments in the theory and applications of stochastic differential equations. You are cordially invited to contribute an original research article or comprehensive review to this Special Issue on "First SDE: New Advances in Stochastic Differential Equations". The most recent developments in the theory of stochastic differential equations and their applications in terms of concepts as well as techniques are emphasized, and the journal will focus on a wide range of mathematical, scientific, and engineering disciplines. Applications to mathematical statistical physics, ergodic theory, mathematical biology, mathematical statistics, telecommunications modeling, reliability, mathematical finance, operations research, and theoretical computer science are of interest, in addition to the main topic of stochastic differential equations theory. This Special Issue is one of such typical post-conference Special Issues; however, it is also absolutely open to submissions from authors who are interested in the topic even if they do not participate in the First SDE event at all.

Dr. Nuno M. Brites
Dr. João M. Guerra
Dr. Paula Milheiro-Oliveira
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com 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 submissions that pass pre-check are 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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • boundary-value problems
  • filtering problems
  • first passage times
  • stochastic control
  • stochastic differential equations
  • stochastic partial differential equations
  • stochastic models
  • stochastic processes
  • mathematical finance
  • optimal stopping
  • energy
  • structural mechanics

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Further information on MDPI's Special Issue polices can be found here.

Published Papers (4 papers)

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Research

17 pages, 315 KiB  
Article
Functional Solutions of Stochastic Differential Equations
by Imme van den Berg
Mathematics 2024, 12(8), 1258; https://doi.org/10.3390/math12081258 - 21 Apr 2024
Viewed by 787
Abstract
We present an integration condition ensuring that a stochastic differential equation dXt=μ(t,Xt)dt+σ(t,Xt)dBt, where μ and σ are sufficiently regular, [...] Read more.
We present an integration condition ensuring that a stochastic differential equation dXt=μ(t,Xt)dt+σ(t,Xt)dBt, where μ and σ are sufficiently regular, has a solution of the form Xt=Z(t,Bt). By generalizing the integration condition we obtain a class of stochastic differential equations that again have a functional solution, now of the form Xt=Z(t,Yt), with Yt an Ito process. These integration conditions, which seem to be new, provide an a priori test for the existence of functional solutions. Then path-independence holds for the trajectories of the process. By Green’s Theorem, it holds also when integrating along any piece-wise differentiable path in the plane. To determine Z at any point (t,x), we may start at the initial condition and follow a path that is first horizontal and then vertical. Then the value of Z can be determined by successively solving two ordinary differential equations. Due to a Lipschitz condition, this value is unique. The differential equations relate to an earlier path-dependent approach by H. Doss, which enables the expression of a stochastic integral in terms of a differential process. Full article
(This article belongs to the Special Issue First SDE: New Advances in Stochastic Differential Equations)
19 pages, 319 KiB  
Article
Controlled Reflected McKean–Vlasov SDEs and Neumann Problem for Backward SPDEs
by Li Ma, Fangfang Sun and Xinfang Han
Mathematics 2024, 12(7), 1050; https://doi.org/10.3390/math12071050 - 31 Mar 2024
Viewed by 768
Abstract
This paper is concerned with the stochastic optimal control problem of a 1-dimensional McKean–Vlasov stochastic differential equation (SDE) with reflection, of which the drift coefficient and diffusion coefficient can be both dependent on the state of the solution process along with its law [...] Read more.
This paper is concerned with the stochastic optimal control problem of a 1-dimensional McKean–Vlasov stochastic differential equation (SDE) with reflection, of which the drift coefficient and diffusion coefficient can be both dependent on the state of the solution process along with its law and control. One backward stochastic partial differential equation (BSPDE) with the Neumann boundary condition can represent the value function of this control problem. Existence and uniqueness of the solution to the above equation are obtained. Finally, the optimal feedback control can be constructed by the BSPDE. Full article
(This article belongs to the Special Issue First SDE: New Advances in Stochastic Differential Equations)
20 pages, 800 KiB  
Article
On the Bias of the Unbiased Expectation Theory
by Renato França and Raquel M. Gaspar
Mathematics 2024, 12(1), 105; https://doi.org/10.3390/math12010105 - 28 Dec 2023
Viewed by 1421
Abstract
The unbiased expectation theory stipulates that long-term interest rates are determined by the market’s expectations of future short-term interest rates. According to this hypothesis, if investors have unbiased expectations about future interest rate movements, the forward interest rates should be good predictors of [...] Read more.
The unbiased expectation theory stipulates that long-term interest rates are determined by the market’s expectations of future short-term interest rates. According to this hypothesis, if investors have unbiased expectations about future interest rate movements, the forward interest rates should be good predictors of future spot interest rates. This hypothesis of the term structure of interest rates has long been a subject of debate due to empirical and theoretical challenges. Despite extensive research, a satisfactory explanation for the observed systematic difference between future spot interest rates and forward interest rates has not yet been identified. In this study, we approach this issue from an arbitrage theory perspective, leveraging on the connection between the expectation hypothesis and changes in probability measures. We propose that the observed bias can be explained by two adjustments: a risk premia adjustment, previously considered in the literature, and a stochastic adjustment that has been overlooked until now resulting from two measure changes. We further demonstrate that for specific instances of the Vasicek and Cox, as well as the Ingersoll and Ross, stochastic interest rate models, quantifying these adjustments reveals that the stochastic adjustment plays a significant role in explaining the bias, and ignoring it may lead to an overestimation of the required risk premia/aversion adjustment. Our findings extend beyond the realm of financial economic theory to have tangible implications for interest rate modelling. The capacity to quantify and distinguish between risk and stochastic adjustments empowers modellers to make more informed decisions, leading to a more accurate understanding of interest rate dynamics over time. Full article
(This article belongs to the Special Issue First SDE: New Advances in Stochastic Differential Equations)
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20 pages, 722 KiB  
Article
Modelling French and Portuguese Mortality Rates with Stochastic Differential Equation Models: A Comparative Study
by Daniel dos Santos Baptista and Nuno M. Brites
Mathematics 2023, 11(22), 4648; https://doi.org/10.3390/math11224648 - 15 Nov 2023
Viewed by 1094
Abstract
In recent times, there has been a notable global phenomenon characterized by a double predicament arising from the concomitant rise in worldwide life expectancy and a significant decrease in birth rates. The emergence of this phenomenon has posed a significant challenge for governments [...] Read more.
In recent times, there has been a notable global phenomenon characterized by a double predicament arising from the concomitant rise in worldwide life expectancy and a significant decrease in birth rates. The emergence of this phenomenon has posed a significant challenge for governments worldwide. It not only poses a threat to the continued viability of state-funded welfare programs, such as social security, but also indicates a potential decline in the future workforce and tax revenue, including contributions to social benefits. Given the anticipated escalation of these issues in the forthcoming decades, it is crucial to comprehensively examine the extension of the human lifespan to evaluate the magnitude of this matter. Recent research has focused on utilizing stochastic differential equations as a helpful means of describing the dynamic nature of mortality rates, in order to tackle this intricate issue. The usage of these models proves to be superior to deterministic ones due to their capacity to incorporate stochastic variations within the environment. This enables individuals to gain a more comprehensive understanding of the inherent uncertainty associated with future forecasts. The most important aims of this study are to fit and compare stochastic differential equation models for mortality (the geometric Brownian motion and the stochastic Gompertz model), conducting separate analyses for each age group and sex, in order to generate forecasts of the central mortality rates in France up until the year 2030. Additionally, this study aims to compare the outcomes obtained from fitting these models to the central mortality rates in Portugal. The results obtained from this work are quite promising since both stochastic differential equation models manage to replicate the decreasing central mortality rate phenomenon and provide plausible forecasts for future time and for both populations. Moreover, we also deduce that the performances of the models differ when analyzing both populations under study due to the significant contrast between the mortality dynamics of the countries under study, a consequence of both external factors (such as the effect of historical events on Portuguese and French mortality) and internal factors (behavioral effect). Full article
(This article belongs to the Special Issue First SDE: New Advances in Stochastic Differential Equations)
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