Otunuga et al. (2026) Statistical analysis of stationary and transition probability densities for atmospheric forcing in a climate change model

Identification

• Journal: Stochastic Environmental Research and Risk Assessment
• Year: 2026
• Date: 2026-01-01
• Authors: Olusegun Michael Otunuga, Sher Chhetri, Hongwei Long
• DOI: 10.1007/s00477-025-03153-3

Research Groups

• Department of Mathematics, Augusta University, Augusta, GA, USA
• Division of Science, Mathematics, and Engineering, University of South Carolina Sumter, Sumter, SC, USA
• Department of Mathematics and Statistics, Florida Atlantic University, Boca Raton, FL, USA

Short Summary

This study derives and analyzes the unique stationary and transition probability density functions for an unmeasured atmospheric forcing process within a stochastic climate change model, validating these analytical solutions through numerical simulations and application to real-world volcanic aerosol forcing data.

Objective

• To obtain explicit, closed-form expressions for the unique stationary and transition probability density functions of an unmeasured atmospheric forcing process, V(t), described by a stochastic differential equation (SDE) in a climate change model, and to analyze their properties and use them for parameter estimation.

Study Configuration

• Spatial Scale: Global (application to volcanic aerosol forcing data covers global optical depth across 64 latitudinal columns).
• Temporal Scale: Model analysis is general for time t. Application to real-world data covers monthly observations from January 1890 to December 1999, with a specific simulation period of 12 months from October 1992 to September 1993. Numerical simulations use a time step of 0.2 (unitless, as V(t) is generic), with 7,000 steps and 10,000 simulations.

Methodology and Data

• Models used:
  • Stochastic Differential Equations (SDEs), specifically a Langevin equation for atmospheric forcing (Itô and Stratonovich forms).
  • Fokker-Planck equation for deriving probability density functions (PDFs).
  • Sturm-Liouville equation for solving for eigenfunctions.
  • Milstein scheme for numerical solutions of SDEs.
  • Maximum Likelihood Estimation (MLE) for parameter estimation.
• Data sources:
  • Simulated distributions generated from the Milstein scheme.
  • Real-world Volcanic Aerosol Forcing data (optical depth) from Ammann et al. (2003), comprising monthly global data from January 1890 to December 1999.

Main Results

• Explicit, closed-form expressions for the unique stationary and transition probability density functions (PDFs) of the atmospheric forcing process V(t) were derived for both Itô and Stratonovich dynamics.
• The stationary PDF, when a parameter ω > 0, was shown to follow a non-standardized Student-t distribution lst(0, 1/(2ω), 2ω).
• If ω = 0, the process V(t) is non-stationary, and its transition PDF was derived as a specific log-normal-like distribution.
• Key properties of the stationary distribution were calculated, including its k-th moments, mean, variance, and skewness, confirming its symmetry and a mean of zero for 2ω > 1.
• It was demonstrated that the transition PDF converges to the stationary PDF as time approaches infinity, with the mean approaching 0 and variance approaching 1/(2(ω-1)) for ω > 1.
• Numerical simulations using the Milstein scheme validated the derived analytical PDFs and moment functions, showing strong agreement.
• Application to real-world volcanic aerosol forcing data yielded estimated model parameters for a 12-month period (October 1992 to September 1993): mean reversion rate a ≈ 3.802 x 10^-11 s^-1 (0.0001 per month) and noise intensity σ ≈ 1.233 x 10^-7 s^-1/2 (0.0002 per month^(1/2)).
• The analysis of volcanic aerosol forcing data showed a decreasing mean and increasing variance over time, reflecting aerosol decay and growing uncertainty.

Contributions

• Provides novel explicit, closed-form analytical solutions for the stationary and transition probability density functions of an unmeasured atmospheric forcing process, which is a critical component in stochastic climate change models.
• Offers a robust and computationally efficient method for parameter estimation in stochastic differential equations where direct observation of processes or closed-form PDFs are typically unavailable, overcoming significant challenges in existing methodologies.
• Validates theoretical derivations through comprehensive numerical simulations and demonstrates practical applicability by analyzing real-world volcanic aerosol forcing data, providing insights into its statistical behavior and temporal evolution of uncertainty.
• Establishes a connection between the atmospheric forcing process and known statistical distributions (Student-t, normal approximation), which can facilitate further analysis of extreme events and climate dynamics.

Funding

• 2021 RISE (Research Initiative for Summer Engagement) grant from the Office of the Vice-President for Research, University of South Carolina Columbia (partially supported Sher Chhetri).

Citation

@article{Otunuga2026Statistical,
  author = {Otunuga, Olusegun Michael and Chhetri, Sher and Long, Hongwei},
  title = {Statistical analysis of stationary and transition probability densities for atmospheric forcing in a climate change model},
  journal = {Stochastic Environmental Research and Risk Assessment},
  year = {2026},
  doi = {10.1007/s00477-025-03153-3},
  url = {https://doi.org/10.1007/s00477-025-03153-3}
}