The Analysis of Currencies for Different Countries via Complexity and Brownian Motion

Abstract

The foreign exchange (Forex) market is a pillar of global economic development, and the United States Dollar (USD) influences nations such as India, Britain, Mexico, South Africa, China, Brazil, and Australia. Daily exchange-rate movements against the Dollar over 2024–2025 for these seven currencies are examined here as cyclical signals, each carrying its own underlying rhythm rather than a simple straight-line trend. A periodic Laplace transform is applied to extract the periodic component from the raw time series, while Brownian motion is used to model the random, continuous-time fluctuations that characterise currency movement. Two further tools probe the shape of the data itself. The HarFA fractal (box-counting) analysis measures self-similarity and multi-scale irregularity in the exchange-rate curves, and Pixel boundary analysis tracks the distance between chosen points on the graphs to gauge how sharply each currency's path bends and swings. All computation, simulation, and plotting were carried out in Scilab. The fractal box dimension and pixel boundary distances place China, the United Kingdom, and Mexico among the most volatile and structurally complex of the seven series, while India's exchange rate follows a comparatively smooth, steadily rising path with the lowest fractal dimension and shortest pixel distance. The periodic Laplace transforms, summarised for a family of representative square-wave functions, confirm that the exchange-rate series behave as periodic signals superimposed on a broader stochastic trend. Taken together, the four methods show that exchange-rate behaviour is stochastic, fractal, and nonlinear all at once, and that it tracks the broader economic health of a nation closely. This study illustrates the power of computational tools in finance.