Short-Time Asymptotic Methods In Financial Mathematics

In this talk, we are concerned with the average values of certain functionals of the path of a stochastic process during a given time period. High-order asymptotic characterizations of such values when the time period shrinks to \(0\) have a wide range of applications. In statistics, they are crucial in obtaining the infill asymptotic properties of high-frequency based statistical methods of stochastic processes. In finance, they have been used as model selection and calibration tools based on near expiration option prices. In some Engineering problems, they also show up as a method to solve a problem in continuous time by looking at the analogous problem in discrete time and shrinking the time step. These short-time asymptotic methods are especially useful to study complex models with jumps and stochastic volatility due to the lack of tractable formulas and efficient numerical procedures. I will discuss some recent advances in the area and illustrate their broad relevance in several contexts.