Estimating the time value of ruin in a Lévy risk model under low-frequency observation

The Gerber-Shiu discounted penalty function: A review from practical perspectives

“…Indeed, the advent of technology has increased the means of such high-frequency sampling. On the other hand, if the frequency is much lower (for instance, only weakly or monthly data are available), then it is more sensible to impose the low-frequency condition (for instance, [163,197]) with a fixed step size h n ≡ h(> 0), no matter how many observations are available. From a technical point of view, it is often impossible to construct a (statistically) efficient estimator for unknown parameters in low-frequency scenarios.…”

“…The Fourier-cosine series expansion is first employed [183] in estimating the discounted density of the deficit at ruin f δ (u, x) = ∞ 0 e −δt f (u, x,t)dt on the Cramér-Lundberg model (Section 2.3.1). The estimator of the Gerber-Shiu function on a risk model with compound Poisson random income is represented by Fourier-cosine series expansion [165,166] (without constant premium in the latter), and on a Lévy risk model [152,163] (Section 2.3.2). We note that the work [163] is focused on low-frequency scenarios (h n ≡ h(> 0)), under which theoretical results, such as the rate of convergence, differ from ones under high-frequency scenarios that we present in what follows.…”