Distributional identities for a Lévy process Xt, its quadratic variation process Vt and its maximal jump processes, are derived, and used to make "small time" (as t ↓ 0) asymptotic comparisons between them. The representations are constructed using properties of the underlying Poisson point process of the jumps of X. Apart from providing insight into the connections between X, V , and their maximal jump processes, they enable investigation of a great variety of limiting behaviours. As an application, we study "self-normalised" versions of Xt, that is, Xt after division by sup 0

For non-negative integers r, s, let (r,s) X t be the Lévy process X t with the r largest positive jumps and the s smallest negative jumps up till time t deleted, and let (r) X t be X t with the r largest jumps in modulus up till time t deleted. Let a t ∈ R and b t > 0 be non-stochastic functions in t. We show that the tightness of ( (r,s) X t − a t )/b t or ( (r) X t − a t )/b t as t ↓ 0 implies the tightness of all normed ordered jumps, hence the tightness of the untrimmed process (X t −a t )/b t at 0. We use this to deduce that the trimmed process ( (r,s) X t −a t )/b t or ( (r) X t − a t )/b t converges to N (0, 1) or to a degenerate distribution as t ↓ 0 if and only if (X t − a t )/b t converges to N (0, 1) or to the same degenerate distribution, as t ↓ 0.

Abstract:We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer's point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company's ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this.