Vol. 9, Issue 6, Part F (2023)

Mathematical stochastic calculus on Riemannian manifolds is a burgeoning field that merges differential geometry with probability theory to study stochastic processes on curved spaces. This abstract presents an overview of key concepts and results in this interdisciplinary area.

The foundation of stochastic calculus on Riemannian manifolds lies in extending the notions of stochastic differential equations (SDEs) and stochastic integration from Euclidean spaces to manifolds. Key tools include the theory of connections and parallel transport, which provide a framework for defining covariant derivatives and stochastic flows on manifolds.

Central to the study of stochastic processes on Riemannian manifolds is the development of stochastic parallel transport, which generalizes the concept of parallel transport to incorporate stochastic effects. This allows for the analysis of stochastic differential equations driven by Wiener processes or other types of stochastic processes on curved spaces.

Applications of mathematical stochastic calculus on Riemannian manifolds abound in various fields such as mathematical finance, physics, and biology. For instance, in finance, it enables the modeling of asset prices and portfolio optimization in markets where asset prices evolve on curved spaces. In physics, it provides a framework for describing the behavior of particles subject to random fluctuations in curved spacetimes. In biology, it aids in understanding the dynamics of biological systems evolving on complex manifolds.

Overall, mathematical stochastic calculus on Riemannian manifolds offers a rich theoretical framework for studying stochastic processes in curved spaces, with diverse applications across different scientific disciplines. Ongoing research in this area continues to deepen our understanding of the interplay between geometry and randomness, paving the way for new insights and applications in various fields.