# Penalizing fractional Brownian motion for being negative

## TL;DR

This paper investigates a modified fractional Brownian motion that is penalized for leaving the positive half-axis, deriving its weak limit and comparing it to the classical Brownian meander through explicit SDE representations.

## Contribution

It introduces a novel penalization approach for fractional Brownian motion and characterizes its weak limit, extending understanding beyond the classical Brownian case.

## Key findings

- Derived the weak limit of penalized fractional Brownian motion.
- Provided explicit SDE representations for the limiting processes.
- Compared fractional and classical Brownian meander behaviors.

## Abstract

We study a modification of the fractional analogue of the Brownian meander, which is Brownian motion conditioned to be positive on the time interval ${[0,1]}$. More precisely, we determine the weak limit of a fractional Brownian motion which is penalized -- instead of being killed -- when leaving the positive half-axis. In the Brownian case, we give a representation of the limiting process in terms of an explicit SDE and compare it to the SDE fulfilled by the Brownian meander.