Distribution function of the blow up time of the solution of an anticipating random fatigue equation

TL;DR

This paper investigates the distribution of the explosion time in a stochastic fatigue crack model with anticipating initial conditions, using advanced stochastic calculus techniques to derive the law of blow-up time.

Contribution

It introduces a novel approach by considering anticipating initial conditions and forward stochastic integrals in analyzing the blow-up time of the model.

Findings

Derived the distribution function of the blow-up time.

Applied Russo and Vallois's substitution formula for local solutions.

Established results on barrier crossing probabilities of Brownian bridges.

Abstract

In this paper, we study the distribution function of the time of explosion of a stochastic differential equation modeling the length of the dominant crack due to fatigue. The main novelty is that initial condition is regarded as an anticipating random variable and the stochastic integral is in the forward sense. Under suitable conditions, we use the substitution formula from Russo and Vallois to find the local solution of this equation. Then, we find the law of blow up time by proving some results on barrier crossing probabilities of Brownian bridge.