There has been widespread use of stochastic differential equations in modeling climate change, especially the stochastic models with alpha-stable L \(\acute{e}\) vy jumps. The family of L \(\acute{e}\) vy \(\alpha\) -stable distributions is used in many applications for modeling heavy-tailed data. These distributions are crucial for capturing extreme events, such as market crashes and abrupt climate changes. Although tests like p-variation, quantile-tests, and maximum likelihood methods based on characteristic functions have proven to be successful in determining parameters in stochastic and alpha-stable L \(\acute{e}\) vy dynamics, knowing the closed-form distribution of the process helps in capturing certain properties of the process. In this work, a well-known stochastic differential equation describing a climatic state that depends on ocean circulation and atmospheric forcing is considered. The unique stationary and transition probability density functions of the atmospheric forcing process, satisfying Itô and Stratonovich dynamics, and the properties of the distributions are obtained, analyzed, and used to estimate parameters in the model. We validate our claim by comparing the results obtained in this study with simulated distributions and by applying the methodology to real-world volcanic aerosol forcing data.
