The Løkka–Zervos Alternative for a Cramér–Lundberg Process with Exponential Jumps

In this paper, we study a stochastic control problem faced by an insurance company allowed to pay out dividends and make capital injections. As in (Løkka and Zervos (2008); Lindensjö and Lindskog (2019)), for a Brownian motion risk process, and in Zhu and Yang (2016), for diffusion processes, we will show that the so-called Løkka–Zervos alternative also holds true in the case of a Cramér–Lundberg risk process with exponential claims. More specifically, we show that: if the cost of capital injections is low, then according to a double-barrier strategy, it is optimal to pay dividends and inject capital, meaning ruin never occurs; and if the cost of capital injections is high, then according to a single-barrier strategy, it is optimal to pay dividends and never inject capital, meaning ruin occurs at the first passage below zero.

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A Review of First-Passage Theory for the Segerdahl-Tichy Risk Process and Open Problems

Risks ◽

10.3390/risks7040117 ◽

2019 ◽

Vol 7 (4) ◽

pp. 117 ◽

Cited By ~ 1

Author(s):

Florin Avram ◽

Jose-Luis Perez-Garmendia

Keyword(s):

Interest Rates ◽

Risk Model ◽

Diffusion Processes ◽

Risk Process ◽

First Passage ◽

Open Problems ◽

Theoretical Approaches ◽

Basic Functions ◽

Capital Injections ◽

And Diffusion

The Segerdahl-Tichy Process, characterized by exponential claims and state dependent drift, has drawn a considerable amount of interest, due to its economic interest (it is the simplest risk process which takes into account the effect of interest rates). It is also the simplest non-Lévy, non-diffusion example of a spectrally negative Markov risk model. Note that for both spectrally negative Lévy and diffusion processes, first passage theories which are based on identifying two “basic” monotone harmonic functions/martingales have been developed. This means that for these processes many control problems involving dividends, capital injections, etc., may be solved explicitly once the two basic functions have been obtained. Furthermore, extensions to general spectrally negative Markov processes are possible; unfortunately, methods for computing the basic functions are still lacking outside the Lévy and diffusion classes. This divergence between theoretical and numerical is strikingly illustrated by the Segerdahl process, for which there exist today six theoretical approaches, but for which almost nothing has been computed, with the exception of the ruin probability. Below, we review four of these methods, with the purpose of drawing attention to connections between them, to underline open problems, and to stimulate further work.

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The Level-Crossing Problem: First-Passage, Escape and Extremes

Fluctuation and Noise Letters ◽

10.1142/s0219477514300018 ◽

2014 ◽

Vol 13 (04) ◽

pp. 1430001 ◽

Cited By ~ 1

Author(s):

Jaume Masoliver

Keyword(s):

Brownian Motion ◽

Diffusion Processes ◽

Extreme Values ◽

Level Crossing ◽

First Passage ◽

One Dimensional ◽

Absolute Value ◽

Dimensional Diffusion

We review the level-crossing problem which includes the first-passage and escape problems as well as the theory of extreme values (the maximum, the minimum, the maximum absolute value and the range or span). We set the definitions and general results and apply them to one-dimensional diffusion processes with explicit results for the Brownian motion and the Ornstein–Uhlenbeck (OU) process.

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On Risk Model with Dividends Payments Perturbed by a Brownian Motion – An Algorithmic Approach

Astin Bulletin ◽

10.1017/s0515036100015130 ◽

2008 ◽

Vol 38 (1) ◽

pp. 183-206 ◽

Cited By ~ 1

Author(s):

Esther Frostig

Keyword(s):

Brownian Motion ◽

Risk Model ◽

Risk Process ◽

Insurance Company ◽

Algorithmic Approach ◽

Compound Poisson ◽

Expected Time ◽

Premium Rate ◽

Surplus Process ◽

Phase Type

Assume that an insurance company pays dividends to its shareholders whenever the surplus process is above a given threshold. In this paper we study the expected amount of dividends paid, and the expected time to ruin in the compound Poisson risk process perturbed by a Brownian motion. Two models are considered: In the first one the insurance company pays whatever amount exceeds a given level b as dividends to its shareholders. In the second model, the company starts to pay dividends at a given rate, smaller than the premium rate, whenever the surplus up-crosses the level b. The dividends are paid until the surplus down-crosses the level a, a < b . We assume that the claim sizes are phase-type distributed. In the analysis we apply the multidimensional Wald martingale, and the multidimensional Asmussesn and Kella martingale.

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Double-barrier first-passage times of jump-diffusion processes

Monte Carlo Methods and Applications ◽

10.1515/mcma-2013-0005 ◽

2013 ◽

Vol 19 (2) ◽

pp. 107-141

Author(s):

Lexuri Fernández ◽

Peter Hieber ◽

Matthias Scherer

Keyword(s):

Diffusion Processes ◽

First Passage Time ◽

Asset Price ◽

Jump Diffusion ◽

Mathematical Finance ◽

Passage Time ◽

First Passage ◽

Double Barrier ◽

Jump Diffusion Processes ◽

Wide Range

Abstract. Required in a wide range of applications in, e.g., finance, engineering, and physics, first-passage time problems have attracted considerable interest over the past decades. Since analytical solutions often do not exist, one strand of research focuses on fast and accurate numerical techniques. In this paper, we present an efficient and unbiased Monte-Carlo simulation to obtain double-barrier first-passage time probabilities of a jump-diffusion process with arbitrary jump size distribution; extending single-barrier results by [Journal of Derivatives 10 (2002), 43–54]. In mathematical finance, the double-barrier first-passage time is required to price exotic derivatives, for example corridor bonus certificates, (step) double barrier options, or digital first-touch options, that depend on whether or not the underlying asset price exceeds certain threshold levels. Furthermore, it is relevant in structural credit risk models if one considers two exit events, e.g., default and early repayment.

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On Risk Model with Dividends Payments Perturbed by a Brownian Motion – An Algorithmic Approach

Astin Bulletin ◽

10.2143/ast.38.1.2030410 ◽

2008 ◽

Vol 38 (01) ◽

pp. 183-206 ◽

Cited By ~ 5

Author(s):

Esther Frostig

Keyword(s):

Brownian Motion ◽

Risk Model ◽

Risk Process ◽

Insurance Company ◽

Algorithmic Approach ◽

Compound Poisson ◽

Expected Time ◽

Premium Rate ◽

Surplus Process ◽

Phase Type

Assume that an insurance company pays dividends to its shareholders whenever the surplus process is above a given threshold. In this paper we study the expected amount of dividends paid, and the expected time to ruin in the compound Poisson risk process perturbed by a Brownian motion. Two models are considered: In the first one the insurance company pays whatever amount exceeds a given levelbas dividends to its shareholders. In the second model, the company starts to pay dividends at a given rate, smaller than the premium rate, whenever the surplus up-crosses the levelb. The dividends are paid until the surplus down-crosses the levela,a<b. We assume that the claim sizes are phase-type distributed. In the analysis we apply the multidimensional Wald martingale, and the multidimensional Asmussesn and Kella martingale.

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Loss portfolio transfer treaties within Solvency II capital system: a reinsurer’s point of view

Insurance Markets and Companies ◽

10.21511/ins.11(1).2020.01 ◽

2020 ◽

Vol 11 (1) ◽

pp. 1-10

Author(s):

Nicolino Ettore D’Ortona ◽

Gabriella Marcarelli ◽

Giuseppe Melisi

Keyword(s):

Cost Of Capital ◽

Solvency Ii ◽

Capital Requirements ◽

Point Of View ◽

Insurance Company ◽

Specific Level ◽

Collective Risk ◽

Risk Appetite ◽

Reinsurance Treaty ◽

The Cost

Loss portfolio transfer (LPT) is a reinsurance treaty in which an insurer cedes the policies that have already incurred losses to a reinsurer. This operation can be carried out by an insurance company in order to reduce reserving risk and consequently reduce its capital requirement calculated, according to Solvency II. From the viewpoint of the reinsurance company, being a very complex operation, importance must be given to the methodology used to determine the price of the treaty.Following the collective risk approach, the paper examines the risk profiles and the reinsurance pricing of LPT treaties, taking into account the insurance capital requirements established by European law. For this purpose, it is essential to calculate the capital need for the risk deriving from the LPT transaction. In the case analyzed, this requirement is calculated under Solvency II legislation, considering the measure of variability determined via simulation. This quantification was also carried out for different levels of the cost of capital rate, providing a range of possible loadings to be applied to the premium. In the case of the Cost of Capital (CoC) approach, the results obtained provide a lower level of premium compared to the percentile-based method with a range between 2.69% and 1.88%. Besides, the CoC approach also provides the advantage of having an explicit parameter, the CoC rate whose specific level can be chosen by the reinsurance company based on the risk appetite.

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Optimal Dividend and Capital Injection Strategies for a Risk Model under Force of Interest

Mathematical Problems in Engineering ◽

10.1155/2013/750547 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8

Author(s):

Ying Fang ◽

Zhongfeng Qu

Keyword(s):

Risk Model ◽

Risk Process ◽

Insurance Company ◽

Threshold Strategy ◽

Dividend Payments ◽

Optimal Dividend ◽

Optimal Injection ◽

Force Of Interest ◽

Capital Injections ◽

Injection Strategies

As a generalization of the classical Cramér-Lundberg risk model, we consider a risk model including a constant force of interest in the present paper. Most optimal dividend strategies which only consider the processes modeling the surplus of a risk business are absorbed at 0. However, in many cases, negative surplus does not necessarily mean that the business has to stop. Therefore, we assume that negative surplus is not allowed and the beneficiary of the dividends is required to inject capital into the insurance company to ensure that its risk process stays nonnegative. For this risk model, we show that the optimal dividend strategy which maximizes the discounted dividend payments minus the penalized discounted capital injections is a threshold strategy for the case of the dividend payout rate which is bounded by some positive constant and the optimal injection strategy is to inject capitals immediately to make the company's assets back to zero when the surplus of the company becomes negative.

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