# First Passage Time of Nonlinear Diffusion Processes with Singular Boundary Behavior

## TL;DR

This paper develops new theorems for calculating the moments of first passage times in nonlinear stochastic processes with singular boundaries, validated through existing results and applied to ship roll dynamics.

## Contribution

It introduces analytical theorems for moments of first passage times in nonlinear processes with singular boundaries, applicable to physical systems like ship dynamics.

## Key findings

- Validated theorems using existing analytical results.
- Provided fast computational methods for moments of first passage times.
- Applied results to model dangerous ship roll behavior.

## Abstract

New theorems for the moments of the first passage time of one dimensional nonlinear stochastic processes with an entrance boundary are formulated. This important class of one dimensional stochastic processes results among others from approximations of the energy or amplitude of second order nonlinear stochastic differential equations. Since the diffusion of a stochastic process vanishes at an entrance boundary, the entrance boundary is called a singular point of the stochastic process. The theorems for the moments of the first passage times are validated based on existing analytical results. In addition, the first passage times of a nonlinear stochastic differential equation, which is important for the determination of dangerous ship roll dynamics, are calculated. The proposed analytical expressions for the moments of the first passage times can be calculated very fast using standard quadrature formulas.