Estimating the finite-time ruin probability of a surplus with a long memory via Malliavin calculus

TL;DR

This paper develops a method to estimate the probability of ruin in a surplus process modeled by a long-memory fractional Brownian motion, using Malliavin calculus to handle unknown volatility and construct confidence intervals.

Contribution

It introduces a novel approach combining Malliavin calculus and the delta method to estimate ruin probabilities in long-memory surplus models with unknown volatility.

Findings

Derived the derivative of ruin probability w.r.t. volatility using Malliavin calculus.

Established the asymptotic distribution of the estimated ruin probability.

Provided a framework for confidence interval construction for ruin probabilities.

Abstract

We consider a surplus process of drifted fractional Brownian motion with the Hurst index $H>1/2$, which appears as a functional limit of drifted compound Poisson risk models with correlated claims, and this is a kind of representation of a surplus with a long memory. Our interest is to construct confidence intervals of the ruin probability of the surplus when the volatility parameter is unknown. We will obtain the derivative of the ruin probability w.r.t. the volatility parameter via Malliavin calculus, and apply the delta method to identify the asymptotic distribution of an estimated ruin probability.