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Mathematics
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Advances in Probability Theory and Stochastic Analysis
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Advances in Probability Theory and Stochastic Analysis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "D1: Probability and Statistics".

Deadline for manuscript submissions: 20 March 2025 | Viewed by 1742

Special Issue Editor

Dr. Paramahansa Pramanik

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688, USA
Interests: probability and stochastic processes; stochastic differential games; copula theory; data science

Special Issue Information

Dear Colleagues,

Recent advances in probability theory and stochastic analysis cover a wide range of theoretical and applied developments. These include the creation of stochastic differential equations (SDEs) for modeling complex systems in various fields such as physics, finance, biology, sports, and medicine. There have been significant strides in understanding and analyzing both Markovian and non-Markovian processes, developing new algorithms for stochastic control problems in finance, economics, and infectious disease modeling, and advancing the theory of mean-field games and multi-agent systems. Additionally, improved methods for parameter estimation and hypothesis testing in complex stochastic models, including Copula analysis, have been developed.  

Nearly thirty years ago, this field was an isolated branch of mathematics. However, significant discoveries over the last decade have transformed it into a crucial domain in mathematics, now closely related to analysis, number theory, and mathematical biology.

This Special Issue gathers various articles that aim to reveal theoretical and applied advances in probability theory and stochastic analysis across other disciplines, including, but not limited to, economics, finance, game theory, sports analysis, statistics, epidemiology, machine learning, and medicine.

Dr. Paramahansa Pramanik
Guest Editor

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

  • stochastic control
  • stochastic calculus and stochastic differential equations
  • statistical inference and stochastic processes
  • mathematical biology
  • stochastic analysis in sports

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  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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Research

20 pages, 327 KiB  
Article
Exponential Bounds for the Density of the Law of the Solution of an SDE with Locally Lipschitz Coefficients
by Cristina Anton
Mathematics 2025, 13(5), 798; https://doi.org/10.3390/math13050798 - 27 Feb 2025
Viewed by 150
Abstract
Under the uniform Hörmander hypothesis, we study the smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfies a monotonicity condition. We extend the approach used for SDEs [...] Read more.
Under the uniform Hörmander hypothesis, we study the smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfies a monotonicity condition. We extend the approach used for SDEs with globally Lipschitz coefficients and obtain estimates for the Malliavin covariance matrix and its inverse. Based on these estimates and using the Malliavin differentiability of any order of the solution of the SDE, we prove exponential bounds of the solution’s density law. These results can be used to study the convergence of implicit numerical schemes for SDEs. Full article
(This article belongs to the Special Issue Advances in Probability Theory and Stochastic Analysis)
19 pages, 319 KiB  
Article
σ-Martingales: Foundations, Properties, and a New Proof of the Ansel–Stricker Lemma
by Moritz Sohns
Mathematics 2025, 13(4), 682; https://doi.org/10.3390/math13040682 - 19 Feb 2025
Viewed by 280
Abstract
σ-martingales generalize local martingales through localizing sequences of predictable sets, which are essential in stochastic analysis and financial mathematics, particularly for arbitrage-free markets and portfolio theory. In this work, we present a new approach to σ-martingales that avoids using semimartingale characteristics. [...] Read more.
σ-martingales generalize local martingales through localizing sequences of predictable sets, which are essential in stochastic analysis and financial mathematics, particularly for arbitrage-free markets and portfolio theory. In this work, we present a new approach to σ-martingales that avoids using semimartingale characteristics. We develop all fundamental properties, provide illustrative examples, and establish the core structure of σ-martingales in a new, straightforward manner. This approach culminates in a new proof of the Ansel–Stricker lemma, which states that one-sided bounded σ-martingales are local martingales. This result, referenced in nearly every publication on mathematical finance, traditionally relies on the original French-language proof. We use this result to prove a generalization, which is essential for defining the general semimartingale model in mathematical finance. Full article
(This article belongs to the Special Issue Advances in Probability Theory and Stochastic Analysis)
30 pages, 482 KiB  
Article
Motivation to Run in One-Day Cricket
by Paramahansa Pramanik and Alan M. Polansky
Mathematics 2024, 12(17), 2739; https://doi.org/10.3390/math12172739 - 2 Sep 2024
Cited by 2 | Viewed by 815
Abstract
This paper presents a novel approach to identify an optimal coefficient for evaluating a player’s batting average, strike rate, and bowling average, aimed at achieving an optimal team score through dynamic modeling using a path integral method. Additionally, it introduces a new model [...] Read more.
This paper presents a novel approach to identify an optimal coefficient for evaluating a player’s batting average, strike rate, and bowling average, aimed at achieving an optimal team score through dynamic modeling using a path integral method. Additionally, it introduces a new model for run dynamics, represented as a stochastic differential equation, which factors in the average weather conditions at the cricket ground, the specific weather conditions on the match day (including abrupt changes that may halt the game), total attendance, and home field advantage. An analysis of real data is been performed to validate the theoretical results. Full article
(This article belongs to the Special Issue Advances in Probability Theory and Stochastic Analysis)
►▼ Show Figures

Figure 1

Figure 1
<p>Runs approximation of one-day matches with <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold-italic">μ</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.0009843</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold-italic">μ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.8563</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">σ</mi> <mo>=</mo> <mn>1.288</mn> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>The actual run dynamics of the last six one-day internationals played by India.</p> Full article ">Figure 3
<p>Simulation of last one-day match with <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold-italic">μ</mi> <mn>1</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0.000674</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold-italic">μ</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>0.8567</mn> </mrow> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">σ</mi> <mo>=</mo> <mn>1.22636</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Relationship between coefficient of control and total number of balls delivered.</p> Full article ">
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