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This system extends Geometric Brownian Motion (GBM) to two correlated assets with constant initial prices captures co-movement via correlation. Results show that common shocks create co-movement, while bank-specific components and ρ determine the degree of divergence and systemic risk. Existence and uniqueness of a strong solution is proved using multidimensional SDE theory, yielding explicit solutions for. The system is used to quantify diversification benefits and joint tail risk within the banking sector. This section models the joint future share price dynamics of Access Bank and First Bank using a 2-asset coupled GBM framework. Finally, we present graphical results which represent the behavior of the economic investments and discuss the effect of the relevant parameters.

This system models the stock price of a bank using Geometric Brownian Motion(GBM) with a deterministic seasonal initial condition. The Stochastic Differential Equation (SDE) is formulated to capture annual dividend, earnings, and macroeconomic cycles in the Nigerian banking sector. Three sample paths are simulated examines the future share price dynamics of First Bank under a seasonal Stochastic Differential Equation (SDE) framework using three figures. The analysis considers stochastic Brownian paths, deterministic seasonal cycles, and sensitivity to amplitude parameter A.  Results show that while seasonality drives recurring patterns, random shocks and amplitude changes determine short-term dispersion and volatility. Under Lipschitz and linear growth conditions, existence and uniqueness of a strong solution is proved via Itô’s Lemma, yielding. To this end, the model provides a framework for timing entry and exit around quarterly results.

Many institution-level credential-verification systems generate a random-looking reference code and treat it as secure by virtue of being unique, without examining whether that code is actually resistant to guessing. This paper re-examines a deployed reference-number scheme built for the Academic Document Verification System at Northwest University (NWU), Kano, and shows that its uniqueness suffix - four hexadecimal characters drawn from a UUID4 - carries only 16 bits of entropy (65,536 possible values). We show analytically and by Monte Carlo simulation that, while the system's retry-on-collision loop guarantees no reference is ever issued twice, the same 16-bit space can be exhaustively enumerated against the public verification endpoint in under 11 minutes at a modest 100 requests per second, since no rate limiting is applied. We then design, specify, and analyse a drop-in replacement: an HMAC-SHA256-derived tag computed over the year, document-type, department, and a monotonic serial number, keyed by a server-side secret. At a 48-bit truncation this raises brute-force exhaustion time to an estimated 89,000 years under the same attack rate, at a measured computational cost of 3.2 microseconds per generation or verification - statistically negligible against the system's sub-second query latency. The two schemes are compared against manual, QR-code-based, blockchain-anchored, and zero-knowledge credential-verification approaches reported in the literature on a five-dimension feature matrix, supplemented by a quantitative verification-time comparison. The system was validated through 34 functional test cases (100% pass rate) and an exploratory, formative user-acceptance evaluation (n=8; overall satisfaction 4.5-4.8/5.0, not powered for statistical inference), in which security confidence was the lowest-rated criterion - a finding consistent with, though not statistically proof of, the entropy weakness this paper identifies and addresses. The contribution of this paper is therefore not the verification system as an artefact, but a transferable method for quantifying and closing the guessability gap in identifier-based credential-verification schemes generally.

Miniature hearing aids operate under severe constraints in volume, mass, battery capacity, acoustic performance, moisture resistance, and user comfort. This paper investigates whether otherwise dissipated mechanical motion associated with the ear canal and temporomandibular joint can be converted into supplementary electrical energy by piezoelectric transduction. The study uses a physics-based contemporary review and conceptual engineering analysis. Direct piezoelectric constitutive relations, transducer capacitance, open-circuit voltage, time-varying power, mechanical strain energy, resonance, acoustic intensity, storage energy, and end-to-end efficiency are used to establish a quantitative framework. Published in-ear studies demonstrate that jaw-related ear-canal deformation is a measurable mechanical energy source and that flexible PVDF structures can generate electrical output in the microwatt range under defined conditions. A free-field acoustic calculation also shows why ordinary sound alone is a weak source for a hearing-aid-sized capture area. Literature from 2024-2026 emphasizes flexible PVDF-based harvesters, mechanically amplified structures, MEMS integration, and ultra-low-power power-management circuits. The proposed architecture combines a compliant piezoelectric transducer, low-loss rectification, energy buffering, an ultra-low-power power-management unit, and micro-storage or an auxiliary load. The analysis does not support continuous full self-powering of the hearing aid; the more defensible near-term application is battery support, gradual energy accumulation, or intermittent operation of low-power auxiliary electronics. The governing physics establishes plausibility, while experimental validation remains necessary to determine the net energy benefit in a real device.

This system presents the classical Geometric Brownian Motion (GBM) benchmark with constant initial price. The Stochastic Differential Equation (SDE) assumes no deterministic trend, seasonality or regime effects. The simulation results for Fidelity Bank share prices were presented using a constant GBM model with varying sample paths. Four graphical solutions were analyzed to examine path behavior, expected growth, and sensitivity to drift and volatility. The results highlight the interaction between deterministic trend and stochastic noise in shaping Fidelity’s price dynamics. The model serves as a benchmark for market efficiency with no deterministic trend. Also, the existence and uniqueness of the strong solution is proved under standard Lipschitz conditions. This system serves as the null model for comparing the other stochastic systems.