FST Graduate Research Topic

Research Topic: A Generalized Framework for Stochastic Differential Equations with State-Dependent Nonlinearities
Suggested By: Maiga Chang
Last Update: December 19th, 2025

Note: This research topic can be taken by both Thesis and Project route students Note: This topic is co-supervised with Dr. Maiga Chang by Dr. Mustafa Avci (https://www.athabascau.ca/science-and-technology/our-people/mustafa-avci.html) who is the leading supervisor. A detailed version PDF can be accessed at: https://maiga.athabascau.ca/share/Dr.Mustafa_Avci.pdf Research Background and Motivation This research topic involves the development, implementation, and validation of a Generalized Stochastic Model (GSM) with state-dependent nonlinearities. The project outcome will be evaluated through rigorous computational testing, numerical validation, and application to real-world datasets (e.g., financial time series, physical systems, biological dynamics). Stochastic Differential Equations (SDEs) are foundational in modeling systems influenced by randomness, with applications in finance, physics, biology, and beyond. However, classical SDEs often assume constant or linear coefficients, limiting their ability to capture complex, state-dependent behaviors observed in real systems. This research proposes a novel generalized SDE framework (GSM) with state-dependent variable exponents in the drift and diffusion terms. This model allows nonlinear, adaptive responses to the system state, offering a more expressive tool for modeling real-world stochastic dynamics. Prospective MSc students will focus primarily on the computational implementation and validation of this model, including numerical simulation, parameter calibration, and performance benchmarking. Required Skills Proficiency in Python and scientific libraries (NumPy, SciPy, Pandas, Matplotlib/Seaborn) Experience with stochastic simulation (Monte Carlo methods, random number generation, variance reduction) Knowledge of numerical methods for SDEs (Euler–Maruyama, Milstein, convergence analysis) Familiarity with data fitting, parameter estimation, and model calibration Technical writing in LaTeX for thesis and documentation Research Objectives and Methodology The research is structured into three phases: Theoretical Foundation: Understanding the well-posedness and properties of the GSM. Numerical Development: Designing and implementing efficient and stable numerical solvers for the GSM.  Validation and Application: Testing the model against known benchmarks and real-world data. Important: MSc students will contribute only to Phases 2 and 3, with opportunities to engage with theoretical aspects (however, this requires strong mathematical foundation in: Probability, Stochastic Processes, and Advanced Calculus). Key Tasks for MSc Students: Implement numerical schemes (Euler–Maruyama, Milstein) for the GSM. Conduct convergence and stability analyses. Calibrate model parameters using historical data (e.g., financial time series). Benchmark the GSM against classical models (e.g., Geometric Brownian Motion). Perform error analysis and statistical validation. Visualize results and prepare technical reports. Students are expected to produce reproducible, well-documented code and contribute to research papers and presentations. Expected Outcomes A robust Python-based simulation framework for state-dependent SDEs. Validation studies demonstrating improved modeling capability over classical SDEs. Potential for co-authorship in conference or journal publications. A completed MSc thesis or project report with full implementation and analysis. Student Responsibilities by Route For Thesis Route Students: Develop and implement custom numerical solvers for the GSM. Design and perform comprehensive validation studies using synthetic and real-world data. Compare the GSM with existing SDE models in terms of accuracy, stability, and computational efficiency. Write a thesis documenting the methodology, experiments, results, and contributions. Present findings at research group meetings and potentially at conferences. For Project Route Students: Implement and optimize the numerical solvers for the GSM based on provided specifications. Conduct performance evaluations and benchmark tests. Prepare a detailed project report including code documentation, test cases, and validation results. Demonstrate the functionality of the implemented system through clear examples and visualizations. Contact for inquiries about this research topic Dr. Mustafa Avci Applied Mathematics, FST Email: mavci@athabascau.ca https://www.athabascau.ca/science-and-technology/our-people/mustafa-avci.html https://orcid.org/0000-0002-6001-627X