The Bismut-Elworthy-Li formula for semi-linear distribution-dependent SDEs driven by fractional Brownian motion

TL;DR

This paper extends the Bismut-Elworthy-Li formula to semi-linear mean-field SDEs driven by fractional Brownian motion, providing new tools for sensitivity analysis in financial derivatives.

Contribution

It introduces an extended Bismut-Elworthy-Li formula for fractional Brownian motion-driven mean-field SDEs, with applications in finance.

Findings

Proves existence, uniqueness, and weak differentiability of solutions.

Extends Bismut-Elworthy-Li formula to fractional Brownian motion.

Applies results to sensitivity analysis of financial derivatives.

Abstract

In this work, we will show the existence, uniqueness, and weak differentiability of the solution to semi-linear mean-field stochastic differential equations driven by fractional Brownian motion. We prove an extension of the Bismut-Elworthy-Li formula and show some applications in the sensitivity analysis of variance swaps and also the price of derivatives with respect to the initial point.