APPLYING LIE SYMMETRY THEORY TO STOCHASTIC DIFFERENTIAL EQUATIONS INFLUENCED BY POISSON PROCESSES

Abstract

Lie symmetry theory is a well-established and powerful tool for solving deterministic differential equations, with numerous applications ranging from finding group-invariant solutions to reducing the order of higher-order differential equations and discovering conservation laws. However, its extension to stochastic differential equations (SDEs) is still in its infancy. In contrast to deterministic counterparts, Lie group theory for SDEs remains a relatively unexplored area. Gaeta and Quintero introduced the first steps towards extending Lie symmetries to stochastic ordinary differential equations (SODEs). They considered a limited class of transformations, known as fiberpreserving transformations. These transformations involve mapping the SODEs from one fiber to another in the manifold, represented as: dX = A(X)dt + B(X)dW, where X is the state variable, A(X) represents the drift term, B(X) is the diffusion term, and dW is the Wiener process. However, it's important to note that this approach is constrained to a specific subset of transformations, capable of preserving the fiber structure. The scope of these transformations is limited compared to the broader universe of possible transformations. In this manuscript, we aim to further explore the extension of Lie symmetries to SDEs, moving beyond the restrictions of fiber-preserving transformations. We strive to enhance our understanding of the relationship between symmetries in stochastic systems and their corresponding Fokker-Planck equations. This work contributes to the ongoing development of Lie symmetry theory in the realm of stochastic differential equations, offering new avenues for addressing complex stochastic systems.