Simulation-based inference for the finite-time ruin probability of a surplus with a long-memory

We are interested in statistical inference for the finite-time ruin probability of an insurance surplus whose claim process has a long-range dependence. As an approximated model, we consider a surplus driven by a fractional Brownian motion with the Hurst parameter H > 1/2. We can compute the ruin probability via the Monte Carlo simulations if some unknown parameters in the model are decided. From discrete samples, we estimate those unknowns, by which an asymptotically normal estimator of the ruin probability is computed. An expression of the asymptotic variance is given via the Malliavin Calculus in the estimable form. As a result, we can construct a confidence interval of the finite-time ruin probability. Since the ruin is usually rare event, an importance sampling technique is sometimes usuful in computation in practice.