Research Projects

The deterministic Taylor expansion is a fundamental result in mathematical analysis, particularly used in the numerical analysis of ordinary differential equations (ODEs). Its stochastic extension, the Ito-Taylor expansion, is a central result in stochastic analysis and is widely used to develop and analyze numerical schemes for various types of SDEs. The Ito-Taylor expansion is obtained by applying Ito's formula for semimartingales, which leads to the classical Ito-Taylor expansion for SDEs and the Ito-Taylor expansion for SDE driven by Levy noise. The main objective of this project is to generalize the stochastic Taylor expansion, where the function is no longer a deterministic function but instead depends on a stochastic process satisfying certain regularity assumptions. One possible application of the proposed generalized stochastic Taylor expansion is the Ito-Taylor expansion for regime switching SDEs, where the stochastic function is a Markov chain dependent function and the semimartingale is the solution of the SDE. The expansion can be used to derive and analyze numerical schemes of arbitrary order for regime switching SDEs. Another generalization of the stochastic Taylor expansion is to derive the Ito-Taylor expansion for stochastic delay differential equations (SDDEs) and apply it for the numerical analysis of such equations. The generalized stochastic Taylor expansion can be considered a fundamental contribution to stochastic analysis and fills a significant gap in the literature. By using these expansions in numerical analysis of SDEs, SDDEs, and interacting particle systems driven by Levy noise, the project aims to make a fundamental contribution to higher order approximations of such equations.