# Re-weighted estimation of the transition probability density for second-order diffusion processes

## TL;DR

This paper introduces a new method to estimate probability densities for second-order diffusion processes, improving accuracy and reducing bias.

## Contribution

A re-weighted estimator is proposed that reduces boundary bias and maintains nonnegativity in density estimation.

## Key findings

- The new estimator outperforms existing methods in Monte Carlo simulations.
- The estimator achieves better theoretical properties under standard conditions.
- The method preserves nonnegativity while reducing boundary bias.

## Abstract

The transition probability density of second-order diffusion processes plays a fundamental role in statistical inference and practical applications such as financial derivatives pricing. This paper combines nonparametric Nadaraya-Watson kernel smoothing and local linear smoothing techniques to devise a re-weighted estimator for the transition probability density of second-order diffusion processes. The proposed estimator effectively addresses the persistent boundary bias inherent in Nadaraya-Watson estimation while preserving the nonnegativity constraint essential for probability densities. Under standard regularity conditions, we establish the asymptotic properties of the proposed estimator, demonstrating its theoretical superiority over existing approaches. Furthermore, Monte Carlo simulations show that the new estimator has better performance than Nadaraya-Watson estimator and local linear estimator.