Paper: arXiv 2507.01989

Authors: Yazdan Babazadeh Maghsoodlo, Amin Safaeesirat

Abstract

Understanding the stochastic behavior of currency exchange rates is critical for assessing financial stability and anticipating market transitions. In this study, we investigate the empirical dynamics of the USD exchange rate in three economies, including Iran, Turkey, and Sri Lanka, through the lens of the Kramers-Moyal expansion and Fokker-Planck formalism. Using log-return data, we confirm the Markovian nature of the exchange rate fluctuations, enabling us to model the system with a second-order Fokker-Planck equation. The inferred Langevin coefficients reveal a stabilizing linear drift and a nonlinear, return-dependent diffusion term, reflecting both regulatory effects and underlying volatility. A rolling-window estimation of these coefficients, paired with structural breakpoint detection, uncovers regime shifts that align with major political and economic events, offering insight into the hidden dynamics of currency instability. This framework provides a robust foundation for detecting latent transitions and modeling risk in complex financial systems.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced stochastic calculus including Kramers-Moyal expansion and Fokker-Planck equations, demonstrating high mathematical density. It uses real-world currency data with rolling-window estimation and structural break detection, indicating substantial empirical implementation and backtest-ready methodology.

Research Flowchart

  flowchart TD
  A["Research Goal:\nModel Exchange Rate Instability"] --> B["Data Acquisition\nUSD Exchange Rate Data: Iran, Turkey, Sri Lanka"]
  B --> C["Methodology:\nKramers-Moyal Expansion & Fokker-Planck Formalism"]
  C --> D{"Markovian Analysis"}
  D -- Verified --> E["Computational Process:\nInfer Langevin Coefficients via Rolling-Window Estimation"]
  E --> F["Outcome 1:\nStabilizing Linear Drift & Nonlinear Diffusion"]
  E --> G["Outcome 2:\nRegime Shifts via Structural Breakpoint Detection"]
  F --> H["Final Insight:\nFramework for Latent Transition Detection & Risk Modeling"]
  G --> H

Cited References (42)

  1. Kramers–Moyal Expansion and Fokker–Planck Equation (2019) — link
  2. The ergodic theorem (1939)
  3. Approaching complexity by stochastic methods: From biological systems to turbulence (2011) — link
  4. Official exchange rate (LCU per US, period average) - Iran, Islamic Rep — link
  5. Approximation of the linear Boltzmann equation by the Fokker-Planck equation (1967)
  6. Influence of macroeconomic variables on exchange rates (2015)
  7. Forecasting the Romanian financial system stability using a stochastic simulation model (2010)
  8. Evidence of Markov properties of high frequency exchange rate data (2001)
and 34 more
  1. Dynamical stochastic processes of returns in financial markets (2007)
  2. Modelling the Wealth and Probability of Default of Unsecured Loans: The Stochastic Partial Differential Equation Approach (2025)
  3. PRICING ASIAN OPTIONS IN UNCERTAIN STOCHASTIC MARKETS WITH JUMPS (2024)
  4. Topics in rough and stochastic analysis with applications in finance (2025)
  5. New numerical method via RBF approach to the price of fixed-rate mortgages (2025)
  6. The Hull-White Model for Interest Rates: A Comprehensive Overview
  7. Stable Multilevel Deep Neural Networks for Option Pricing and xVAs Using Forward-Backward Stochastic Differential Equations (2024)
  8. A Langevin equation for the rates of currency exchange based on the Markov analysis (2007)
  9. Geopolitical risks and exchange rates (2025)
  10. Fokker–Planck equation of distributions of financial returns and power laws (2001)
  11. Medium and small-scale analysis of financial data (2007)
  12. Modelling high-frequency economic time series (2000)
  13. Stochastic modeling of fat-tailed probabilities of foreign exchange rates (2002)
  14. A theory ofnon-Gaussian option pricing (2002)
  15. Financial market dynamics (2003)
  16. Nonstationary increments, scaling distributions, and variable diffusion processes in financial markets (2007)
  17. Early warning signals for critical transitions in complex systems (2023)
  18. Critical slowing down as an early warning signal for financial crises? (2019)
  19. Early warning signals for critical transitions: a generalized modeling approach (2012)
  20. Slowing down as an early warning signal for abrupt climate change (2008)
  21. Generic indicators for loss of resilience before a tipping point leading to population collapse (2012)
  22. Deep learning for early warning signals of tipping points (2021)
  23. Early warning signals for stock market crashes: empirical and analytical insights utilizing nonlinear methods (2024)
  24. A Kramers-Moyal Approach to the Analysis of Third-order noise with applications in option valuation (2015)
  25. Disentangling the stochastic behavior of complex time series (2016)
  26. The impact of macroeconomic news on exchange rate volatility (2004)
  27. Markov analysis and Kramers-Moyal expansion of nonstationary stochastic processes with application to the fluctuations in the oil price (2007)
  28. Applications of Stochastic Markov Models to Financial Markets (2024)
  29. The Volterra Stein-Stein model with stochastic interest rates — → 2503.01716
  30. Regime shifts in the behaviour of international currency and equity markets: A Markov-switching analysis (2021)
  31. Control and system theory of discrete-time stochastic systems (2021)
  32. Fokker-planck equation (1996)
  33. Smooth approximation of stochastic differential equations (2016)
  34. Analysis of sliding window techniques for exponentiation (1995)

1 of 42 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.