Fractional smoothness and applications in finance

TL;DR

This paper reviews the concept of fractional smoothness of random variables in diffusion processes, exploring its theoretical foundations and applications in analyzing hedging errors in stochastic finance.

Contribution

It provides a comprehensive overview of fractional smoothness, connecting it to interpolation theory and highlighting its applications in finance, especially in hedging error analysis.

Findings

Connection between fractional smoothness and interpolation theory

Examples illustrating fractional smoothness in diffusion processes

Application to discrete-time hedging error analysis in finance

Abstract

This overview article concerns the notion of fractional smoothness of random variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments.