Stochastic Process Research Articles

The Application of Stochastic Processes in Financial Mathematics

Stochastic processes are crucial in financial mathematics, providing a framework to model the inherent uncertainties of financial markets. This paper explores stochastic process application across various financial domains, such as option pricing, risk management, and financial engineering. Through case studies, literature review, and case analysis, the study demonstrates the practical effectiveness of stochastic processes in finance. The findings highlight the adaptability and robustness of these models in capturing market dynamics and optimizing financial strategies. This research particularly focuses on the implementation of Brownian motion and Its processes in derivative pricing, revealing significant improvements in accuracy compared to traditional methods. Additionally, we examine the integration of machine learning techniques with stochastic models to enhance predictive capabilities in risk assessment. This study also addresses the challenges and limitations of current stochastic approaches, proposing innovative solutions for practical implementation. These insights contribute to both theoretical understanding and practical applications in quantitative finance, offering valuable guidance for practitioners and researchers in the field

Tracing microbial community across endophyte-to-saprotroph continuum of Cinnamomum camphora (L.) Presl leaves considering priority effect of endophyte on litter decomposition.

Endophytes typically coexist with plants in symbiosis and transition into the saprobic system as plant tissues senesce, participating in the decomposition process of litter. However, the dynamic changes of endophytic communities during this process and their role in litter decomposition remain unclear. This study tracked the microbial composition across the transition from live leaves to litter in Cinnamomum camphora (L.) Presl (C. camphora), evaluating the contribution of endophytes to litter decomposition by examining microbial diversity, community assembly, and co-occurrence networks along the endophyte-to-saprotroph spectrum. The results revealed increasing bacterial diversity but stable fungal diversity, and the diversity of endogenous microbes is mirrored this in the saprophytic phase. Bacterial community assembly was characterized by deterministic processes during the symbiotic phase, shifted to stochastic processes during the saprophytic phase. In contrast, fungal community assembly was predominantly driven by stochastic processes throughout the continuum. Out of the 49 keystone taxa identified, only Pseudorhodoplanes sinuspersici demonstrated a significant positive correlation with community assembly. All identified bacterial keystone taxa during the saprophytic phase originated from endophytic sources, and around 80% of the fungal keystone taxa in the initial stages of decomposition were similarly endophytic in origin. Additionally, 60% of the dominant bacterial taxa and 28% of the dominant fungal taxa at the commencement of decomposition were of endophytic descent. This suggests that endogenous microbes possess the potential to evolve into both keystone and dominant taxa during the saprophytic phase. Endogenous keystone and dominant microbes both exhibited significant correlations with microbial network, indicating their substantial ecological presence in microbial community. Both endogenous keystone and dominant taxa exerted significant potential influences on litter decomposition. Overall, during the saprophytic phase, endophytes are likely to influence the assemblage of microbial communities, the network structure, and decomposition-related functions. Specifically, it appears that bacterial endophytes may possess a greater adaptability to the decomposition processes of leaf litter compared to their fungal counterparts.

A Decentralized Storage and Security Engine (DeSSE) Using Information Fusion Based on Stochastic Processes and Quantum Mechanics

In the context of data security, this work aims to present a novel solution that, rather than addressing the topic of endpoint security—which has already garnered significant attention within the international scientific community—offers a different perspective on the subject. In other words, the focus is not on device security but rather on the protection and security of the information contained within those devices. As we will see, the result is a next-generation decentralized infrastructure that simultaneously integrates two cognitive areas: data storage and its protection and security. In this context, an innovative Multiscale Relativistic Quantum (MuReQua) chain is considered to realize a novel decentralized and security solution for storing data. This engine is based on the principles of Quantum Mechanics, stochastic processes, and a new approach of decentralization for data storage focused on information security. The solution is broken down into four main components, considered four levels of security against attackers: (i) defocusing, (ii) fogging, (iii) puzzling, and (iv) crypto agility. The defocusing is realized thanks to a fragmentation of the contents and their distributions on different allocations, while the fogging is a component consisting of a solution of hybrid cyphering. Then, the puzzling is a unit of Information Fusion and Inverse Information Fusion, while the crypto agility component is a frontier component based on Quantum Computing, which gives a stochastic dynamic to the information and, in particular, to its data fragments. The data analytics show a very effective and robust solution, with executions time comparable with cloud technologies, but with a level of security that is a post quantum one. In the end, thanks to a specific application example, going beyond purely technical and technological aspects, this work introduces a new cognitive perspective regarding (i) the distinction between data and information, and (ii) the differentiation between the owner and the custodian of data.

Independently engaging protein tethers of different length enhance synaptic vesicle trafficking to the plasma membrane.

Synaptic vesicle (SV) trafficking toward the plasma membrane (PM) and subsequent SV maturation are essential for neurotransmitter release. These processes, including SV docking and priming, are co-ordinated by various proteins, such as SNAREs, Munc13 and synaptotagmin (Syt), which connect (tether) the SV to the PM. Here, we investigated how tethers of varying lengths mediate SV docking using a simplified mathematical model. The heights of the three tether types, as estimated from the structures of the SNARE complex, Munc13 and Syt, defined the SV-PM distance ranges for tether formation. Geometric considerations linked SV-PM distances to the probability and rate of tether formation. We assumed that SV tethering constrains SV motility and that multiple tethers are associated by independent interactions. The model predicted that forming multiple tethers favours shorter SV-PM distances. Although tethers acted independently in the model, their geometrical properties often caused their sequential assembly, from longer ones (Munc13/Syt), which accelerated SV movement towards the PM, to shorter ones (SNAREs), which stabilized PM-proximal SVs. Modifying tether lengths or numbers affected SV trafficking. The independent implementation of tethering proteins enabled their selective removal to mimic gene knockout (KO) situations. This showed that simulated SV-PM distance distributions qualitatively aligned with published electron microscopy studies upon removal of SNARE and Syt tethers, whereas Munc13 KO data were best approximated when assuming additional disruption of SNARE tethers. Thus, although salient features of SV docking can be accounted for by independent tethering alone, our results suggest that functional tether interactions not yet featured in our model are crucial for biological function. KEY POINTS: A mathematical model describing the role of synaptic protein tethers to localize transmitter-containing vesicles is developed based on geometrical considerations and structural information of synaptotagmin, Munc13 and SNARE proteins. Vesicle movement, along with tether association and dissociation, are modelled as stochastic processes, with tethers functioning independently of each other. Multiple tethers cooperate to recruit vesicles to the plasma membrane and keep them there: Munc13 and Syt as the longer tethers accelerate the movement towards the membrane, whereas short SNARE tethers stabilize them there. Model predictions for situations in which individual tethers are removed agree with the results from experimental studies upon gene knockout. Changing tether length or copy numbers affects vesicle trafficking and steady-state distributions.

Renewable Energy Community Sizing Based on Stochastic Optimization and Unsupervised Clustering

Renewable Energy Communities (RECs) are emerging as significant in the global paradigm shift towards a smart and sustainable energy environment. By empowering energy consumers to actively participate in local energy generation, and sharing, using renewable energy sources, energy storage, and flexible loads, REC participants can reduce costs, and also contribute to low-carbon objectives, providing the flexibility needed to address modern smart grid challenges. This article presents a mixed integer linear programming model for optimal sizing of the solar PVs and battery energy storage systems (BESS) of REC participants who engage in P2P energy exchange. The model is formulated using a two-stage stochastic optimization to address load and PV uncertainty, and unsupervised clustering to structure the data for the stochastic optimization process. The model enables sizing solar PVs for different rooftop geometries and the objective function includes comprehensively defined electricity, operational, and scaled investment costs for solar PV and BESS, where economic fairness constraints are analyzed and implemented. The model is validated on real solar and atmospheric measured data from Zagreb, Croatia, and publicly available household consumption data from Northern Germany. The article also analyzes how tariff models, and electricity prices affect PV and BESS sizes, cost reductions, and P2P energy exchange for different REC participants with varying consumption and production profiles.

Infinite system of integral equations associated with birth-and-death stochastic process: A challenge to solve

Abstract The paper is dedicated to formulate an open problem concerning the existence of solutions of an infinite system of nonlinear differential (or integral) equations which are obtained during the modelling of the so-called birth-and-death stochastic process. Namely, creating a model of the mentioned birth-and-death process and also taking into account nonlinear components which are usually deleted during standard considerations, we obtain an infinite system of nonlinear differential (or integral) equations on a bounded or unbounded interval. It turns out that the operator associated with that infinite system is unbounded in all known classical Banach or Fréchet spaces. This fact initiates an open problem which will be described and raised in the paper.

Smith’s reduction for random processes and application for stochastic differential equations

Abstract The main focus of this work is a result related to multidimensional stochastic differential equations and one-dimensional SDEs (stochastic differential equations), using the Smith’s π-projection approach. This method is a commonly used technique in mathematics to simplify complex systems by projecting them onto lower-dimensional spaces, which can facilitate analysis and resolution. In the context of multidimensional stochastic processes (i.e., systems involving multiple random variables), extending the Smith’s π-projection approach we mention would likely involve defining a multidimensional version of this projection. This could include extending the concepts of projection and dimension reduction to multiple random variables, as well as analyzing how these projections can simplify multidimensional stochastic differential equations. As an application, we will prove the existence of almost periodic solutions to n-dimensional stochastic differential equations using the Smith projection. In fact, we will generalize the result from the article [M.-A. Boudref and A. Berboucha, Existence of almost periodic solutions of stochastic differential equations with periodic coefficients, Random Oper. Stoch. Equ. 25 2017, 1, 57–70] to dimensions n > 1 {n>1} .

MODELING CUSTOMER LIFETIME VALUE WITH MARKOV CHAIN IN THE INSURANCE INDUSTRY

In the competitive insurance industry, accurately predicting Customer Lifetime Value (CLV) is vital for sustaining long-term profitability and optimizing resource allocation. Traditional static models often fail to capture the dynamic and uncertain nature of customer behavior, which is influenced by factors such as life changes, economic conditions, and evolving product offerings. To address these limitations, this paper proposes an advanced modeling approach that integrates Markov Chains with survival analysis. Markov Chains are well-suited for modeling stochastic processes, where future states depend on current conditions, while survival analysis provides insights into event timing and likelihood for estimating the insurance premium. The proposed model combines these approaches to make a more complete and accurate prediction of CLV. This helps insurers make better decisions and improves the overall performance of their business. We employ the data of customer behavior from the insurance company in Bandung, Indonesia from 1994 to 2020. We found that CLV in the insurance industry is significantly affected by customer behavior.

Anomalous diffusion in a quantum system driven by heavy-tailed stochastic processes

Anomalous diffusion in a quantum system driven by heavy-tailed stochastic processes

Replicator dynamics on heterogeneous networks.

Networked evolutionary game theory is a well-established framework for modeling the evolution of social behavior in structured populations. Most of the existing studies in this field have focused on 2-strategy games on heterogeneous networks or n-strategy games on regular networks. In this paper, we consider n-strategy games on arbitrary networks under the pairwise comparison updating rule. We show that under the limit of weak selection, the short-run behavior of the stochastic evolutionary process can be approximated by replicator equations with a transformed payoff matrix that involves both the average value and the variance of the degree distribution. In particular, strongly heterogeneous networks can facilitate the evolution of the payoff-dominant strategy. We then apply our results to analyze the evolutionarily stable strategies in an n-strategy minimum-effort game and two variants of the prisoner's dilemma game. We show that the cooperative equilibrium becomes evolutionarily stable when the average degree of the network is low and the variance of the degree distribution is high. Agent-based simulations on quasi-regular, exponential, and scale-free networks confirm that the dynamic behaviors of the stochastic evolutionary process can be well approximated by the trajectories of the replicator equations.

A Novel Mechanism-Equivalence-Based Tweedie Exponential Dispersion Process for Adaptive Degradation Modeling and Life Prediction.

Accurately predicting the remaining useful life (RUL) of critical mechanical components is a central challenge in reliability engineering. Stochastic processes, which are capable of modeling uncertainties, are widely used in RUL prediction. However, conventional stochastic process models face two major limitations: (1) the reliance on strict assumptions during model formulation, restricting their applicability to a narrow range of degradation processes, and (2) the inability to account for potential variations in the degradation mechanism during modeling and prediction. To address these issues, we propose a novel mechanism-equivalence-based Tweedie exponential dispersion process (ME-based TEDP) for adaptive degradation modeling and RUL prediction of mechanical components. The proposed model enhances the original Tweedie exponential dispersion process (TEDP) by incorporating degradation mechanism equivalence, effectively capturing the correlation between model parameters. Furthermore, it improves prediction accuracy and interpretability by employing a dynamic testing-modeling-predicting strategy. Application of the ME-based TEDP model to high-speed rail bogie systems demonstrates its effectiveness and superiority over existing approaches. This study advances the theory of degradation modeling and significantly improves the precision of RUL predictions.

Behind the scenes of cellular organization: Quantifying spatial phenotypes of puncta structures with statistical models including random fields.

The cellular interior is a spatially complex environment shaped by non-trivial stochastic and biophysical processes. Within this complexity, spatial organizational principles-also called spatial phenotypes-often emerge with functional implications. However, identifying and quantifying these phenotypes in the stochastic intracellular environment is challenging. To overcome this challenge for puncta, we discuss the use of inference of point-process models that link the density of points to other imaged structures and a random field that captures hidden processes. We apply these methods to simulated data and multiplexed immunofluorescence images of Vero E6 cells. Our analysis suggests that peroxisomes are likely to be found near the perinuclear region, overlapping with the ER, and located within a distance of 1 μm to mitochondria. Moreover, the random field captures a hidden variation of the mean density in the order of 15 μm. This length scale could provide critical information for further developing mechanistic hypotheses and models. By using spatial statistical models including random fields, we add a valuable perspective to cell biology.

Central limit theorems for martingales-II: convergence in the weak dual topology

. A convergence theorem for martingales with càdlàg trajectories (right continuous with left limits everywhere) is obtained in the sense of the weak dual topology on Hilbert space, under conditions that are much weaker than those required for any of the usual Skorohod topologies. Examples are provided to show that these conditions are also very easy to check and yield useful asymptotic results, especially when the limit is a mixture of stochastic processes with discontinuities.

Distribution patterns and community assembly processes of bacterial communities across different sediment habitats of subsidence lakes.

Subsidence lakes, formed due to extensive underground coal mining activities, present both ecological challenges and opportunities for alternative land use practices, such as photovoltaic power generation and aquaculture. However, the ecological consequences of these anthropogenic activities on bacterial communities within subsidence lakes remain largely unexplored. To address this knowledge gap, we conducted a comprehensive investigation of bacterial communities in two typical subsidence lake districts located in Huainan, Anhui Province, China. A total of 44 sediment samples were collected across four distinct habitats: photovoltaic zones (PV), aquaculture zones (AC), photovoltaic-aquaculture zones (PV_AC), and unimpacted zones (Natural). Bacterial communities were investigated using 16S rRNA gene sequencing and various statistical methods. Our results revealed that the α-diversity and network complexity of bacterial communities in PV, PV_AC, and AC habitats are significantly higher than habitats of Natural (P<0.01). Specifically, the α-diversity is the highest in PV habitats, while the network complexity is the highest in AC habitats. The network stability is the highest in PV and PV_AC habitats, while it is the lowest in AC. There were also significant differences in community structure among different habitats, Nitrospirota in PV (31.9%) was higher than that in the other three habitats, the percentage of Firmicute in Natural (21.96%) and PV_AC (13.70%) was higher than other two habitats, Actinobacteriota has the highest proportion in AC (20.93%). Furthermore, stochastic processes dominate community assembly in PV, PV_AC, and AC habitats, it has the highest proportion in PV, particularly on dispersal limitation (DL). However, deterministic processes prevail in Natural habitats, particularly on heterogeneous selection (HeS). Collectively, these findings highlight the significant differences in sediment bacterial communities across various habitats in subsidence lakes. Our study provides valuable insights into understanding the ecological implications of different habitats in subsidence lake ecosystems.

A stochastic process-based degradation modeling framework considering measurement errors: a perspective of dual non-Gaussian assumptions

The stochastic processes are a natural choice for describing the randomness in degradation processes caused by inherent randomness and environmental factors. This article proposes a new modified skew-normal distribution to capture measurement uncertainty, and then establishes a stochastic process-based degradation modeling framework, in which both degradation increments and measurement errors do not follow the Gaussian distributions. Taking the IG process as an example, the basic reliability indicators and the alarm probabilities caused by measurement errors are derived. In addition, a multi-stage parameter estimation algorithm based on moderate particle sizes and comparative tolerances is developed for this stochastic process-based degradation model under dual non-Gaussian assumptions. Finally, the effectiveness and advantages of the proposed model along with the parameter estimation algorithm are demonstrated by a simulation study and a case application, and it is particularly pointed out that the relative size of measurement errors has a significant impact on the precision of degradation reliability assessment.

Colored stochastic multiplicative processes with additive noise unveil a third-order partial differential equation, defying conventional Fokker-Planck equation and Fick-law paradigms

Colored stochastic multiplicative processes with additive noise unveil a third-order partial differential equation, defying conventional Fokker-Planck equation and Fick-law paradigms

Smart resetting: An energy-efficient strategy for stochastic search processes

Stochastic resetting, a method for accelerating target search in random processes, often incurs temporal and energetic costs. For a diffusing particle, a lower bound exists for the energetic cost of reaching the target, which is attained at low resetting rates and equals the direct linear transportation cost against fluid drag. Here, we study smart resetting, a strategy that aims to beat this lower bound. By strategically resetting the particle only when this benefits its progress toward the target, smart resetting leverages information to minimize energy consumption. We analytically calculate the energetic cost per mean first passage time and show that smart resetting consistently reduces the energetic cost compared to regular resetting. Surprisingly, smart resting achieves the minimum energy cost previously established for regular resetting, irrespective of the resetting rate. Yet, it fails to reduce this cost further. We extend our findings in two ways: first, by examining nonlinear energetic cost functions, and second, by considering smart resetting of drift-diffusion processes. Published by the American Physical Society 2025

Use Artificial Bee Colony to Determine the Optimal Stratigraphic Boundaries for a Threshold Geometric Stochastic Process

The Artificial Bee Colony (ABC) algorithm was used in this study as a representation of a developmental artificial intelligence technique to identify the best class boundaries for modeling data on the number of daily infections with the Coronavirus (Covid-19) epidemic and for the three governorates of Iraq (Baghdad, Erbil, and Basra). The number of daily infection cases during the outbreak of a particular epidemic disease, such as the emerging acute respiratory syndrome coronavirus, frequently exhibits multiple trends: a steady increase during the epidemic's growth phase or outbreak, followed by a stable phase in the number of daily infection cases (referred to as the stabilization phase by some sources), which involves controlling the epidemic to eradicate it, and a subsequent decline, decline, or disappearance phase. In order to handle this type of data, a random stochastic model known as the geometric stochastic process model with an intelligent threshold was put forth. This model uses an intelligent technique to identify the data's turning points, or inversions.

Similar biogeographic patterns of abundant and rare bacterial communities are driven by distinct assembly mechanisms in grassland soils

AbstractBackgroundThe regional species pool and local community assembly processes shape the biogeographic patterns of soil bacterial community diversity. However, how community assembly mechanisms regulate biogeographic patterns in rare and abundant bacterial communities remains unclear.MethodsSoil samples of 16 grassland habitats across the Inner Mongolian Plateau and Qinghai‐Tibet Plateau (QTP) transects were collected to investigate the variation of β‐diversity in rare taxa (RT) and abundant taxa (AT). High‐throughput sequencing analysis of 16S rRNA gene amplicons was implemented on an Illumina MiSeq platform.ResultsSignificant distance‐decay relationships of β‐diversity in RT and AT were observed at transect and habitat scales, and the turnover rate increased from desert to meadow steppe in both taxa. For variations of β‐diversity along environmental gradient, the regional species pool had a limited effect on both taxa except RT in TP. Deterministic processes, including homogeneous selection (85.1%–97.3%) and heterogenous selection (48.1%–64.2%), dominated the assembly of RT at both the transect and habitat scales. In contrast, the assembly of AT exhibited habitat specificity and was dominated by homogeneous selection (47.2%–80.6%), heterogenous selection (42.1%–54.2%), and dispersal limitation (41.8%) in different transects and habitats. Moreover, the local assembly processes of the AT community were more stochastic than those of the RT community. Mean annual precipitation (MAP) was the dominant driver of community assembly at the transect scale, with extreme MAP (&lt;200 or &gt;400 mm) resulting in more deterministic processes and a moderate level of MAP (200–400 mm) leading to more stochastic processes. However, the effects of geographical distance and soil properties on different grassland habitats cannot be ignored.ConclusionsAlthough both bacterial taxa exhibited significant distance‐decay patterns, different assembly mechanisms shaped the β‐diversity of AT and RT communities in grassland soils. Our results suggested that MAP can mediate community assembly of soil bacteria on a large scale.

Geology and climate drive alpine plant compositional variation among peaks in the Cascade Range of Washington.

Alpine areas are host to diverse plant communities that support ecosystems through structural and floral resources and persist through specialized adaptations to harsh high-elevation conditions. An ongoing question in these plant communities is whether composition is shaped by stochastic processes (e.g., dispersal limitations) or by deterministic processes (e.g., climate, geology), and if those processes select for common phylogenetic clades across space. This study evaluates the drivers of dissimilarity in alpine vascular plant communities across 32 peaks in the Cascade Mountain Range of Washington State and examines the effects of incorporating phylogenetic relatedness to these conclusions. We documented an average of 54 species per peak and used our overall inventory of 307 taxa to construct a phylogenetic tree for the entire mountain range plant community sampled. We used multivariate techniques to quantify the phylogenetic and taxonomic differences between alpine plant communities and to relate those differences to each peak's climate, geology, and topography. Our models indicate that the age of each peak's parent material formation, precipitation, latitude, and temperature had the largest role in shaping alpine plant communities relative to the baseline effects of distance between peaks and time of sampling. A unique result was a distinct plant community in peaks with ultramafic geologic parent material formed in the Paleozoic Era, which has an extreme geochemistry that we found to form evolutionarily distinct lineages compared to all other peaks. With changing climate conditions and disturbance regimes, understanding facets of alpine plant communities like species turnover, geologic endemism, and responses to precipitation changes are vital to conserving these ecosystems.