Study shows how insurance processes converge to a specific model for better ruin probability calculations.
problem Calculating ruin probabilities in insurance processes.
method Proved convergence of insurance processes to a generalized Ornstein-Uhlenbeck process, derived approximations for ruin probabilities.
result Discrete-time insurance surplus processes converge weakly to a generalized Ornstein-Uhlenbeck process, providing insights for ruin theory.
Study investigates ruin probability with random premiums and risky investments.
problem Ruin probability with random premiums and risky investments.
method Laplace transform applied to a model with geometric Brownian motion.
result Asymptotic behavior of ruin probability for large initial capital values.
This paper studies Parisian ruin in insurance risk processes below a fixed level from the last record maximum.
problem Parisian ruin in insurance risk processes below a fixed level from the last record maximum.
method Using recent developments on fluctuation theory of drawdown of spectrally negative Levy process, the paper presents identities for the law of ruin-time and the position at ruin.
result Identities for the law of ruin-time and the position at ruin are given in terms of their joint Laplace transforms.
The paper analyzes insurance risk with Parisian ruin and capital injection.
problem Analyzing insurance risk with Parisian ruin and capital injection.
method Using fluctuation and excursion theory of spectrally negative Levy processes.
result Distributional identities and ruin probabilities are derived.
Study AR(1) series for risk model, estimating ruin probability.
problem Estimating ruin probability in risk models with dependent claim numbers.
method Established AR(1) risk model, used Newton iteration method to find adjustment coefficient, estimated ruin probability.
result Developed new method to estimate exponential upper bound of ruin probability.
Study examines ruin probability in insurance with stochastic premium and claim arrivals.
problem Analyzing ruin probability in insurance with dependent premium and claim arrivals.
method Adapted Cramer-Lundberg model for mixed Poisson processes with stochastic dependence.
result Explicit expression for ruin probability derived for exponential claim and premium sizes.
Study on Parisian ruin for a refracted Lévy process with adaptive premium rate.
problem Investigating the probability of Parisian ruin for a Lévy insurance risk process with a refracted premium rate.
method Generalization of Loeffen et al. (2013) for a refracted Lévy process with a deterministic implementation delay.
result Compact and similar structure result for the probability of Parisian ruin.
We study the ruin problem over a risk process described by a discrete-time Markov model. In contrast to previous studies that focused on the asymptotic behaviour of ruin probabilities for large values of the initial capital, we provide a new technique to compute the quantity of interest for any initial value, and with …
We reprove a result concerning certain ruin in the classical problem of the probability of ruin with risky investments and several of it's generalisations. We also provide the combined transition density of the risk and investment processes in the diffusion case.
Study on ruin probability with investment in risky assets modeled as semimartingales.
problem Analyzing ruin probability in a business process with investment in risky assets.
method Investigates ruin probability with investment in a Lévy process and semimartingale return, deriving upper bounds and conditions for ruin.
result Upper bounds on ruin probabilities decrease as a power function with increasing initial capital, and these bounds are asymptotically optimal.
Study ruin probabilities in risk processes on stochastic networks.
problem Ruin probabilities in risk processes on stochastic networks.
method Classification of agents by types, Poisson process for loss propagation, explicit ruin probabilities for infinite network size.
result Explicit ruin probabilities for agents of any type in infinite network size.
Study a dual risk model with innovation delays, focusing on ruin probability and time.
problem Analyzing risk models with innovation delays affecting ruin probability and time.
method Delayed dual risk model with innovation delays, focusing on ruin probability and time.
result Closed-form formulas for ruin probability and time in some special cases.
This note explores the mathematical theory to solve modern gamblers ruin problems. We establish a ruin framework and solve for the probability of bankruptcy. We also show how this relates to the expected time to bankruptcy and review the risk neutral probabilities associated an adjustment to asymmetrical views.
Study on Parisian ruin with hybrid observation times.
problem Calculating Parisian ruin under a hybrid observation scheme.
method Hybrid observation scheme with Poisson arrivals and continuous monitoring below zero.
result Improved results and fluctuation identities in terms of second-generation scale functions.
The paper finds optimal threshold strategies for insurance companies with a positive terminal value at creeping ruin.
problem Optimizing dividend payments in an insurance company's surplus process with a positive terminal value at creeping ruin.
method Using fluctuation theory, the paper derives explicit formulas for the objective function and shows the optimality of threshold strategies.
result Threshold strategies are optimal for the dividend optimization problem under certain conditions.
Study on risk model with claims, dividends, and random probabilities.
problem Analyzing a risk model with claims, delayed claims, and randomized dividends.
method Discrete time Compound Beta-Binomial Risk Model with recursive expressions for Gerber-Shiu function.
result Recursive relations for ruin-related quantities obtained.
In this paper a quantitative analysis of the ruin probability in finite time of discrete risk process with proportional reinsurance and investment of finance surplus is focused on. It is assumed that the total loss on a unit interval has a light-tailed distribution -- exponential distribution and a heavy-tailed distrib…
We analyze the probability of ruin in scaled Cramér-Lundberg risk process and its diffusion approximation.
problem Analyzing the probability of ruin in a scaled Cramér-Lundberg risk process and its diffusion approximation.
method Comparison method to prove convergence of ruin probability and derive the rate of convergence.
result The probability of ruin for the scaled CL process converges to the probability of ruin for the limiting diffusion process with a rate of convergence of order \( \mO\big(n^{-1/2}\big) \) and uniform with respect to surplus.
Study compares ruin probabilities under independence vs. dependence assumptions.
problem Underestimation of ruin probability when claims are dependent.
method Copulas for claim dependence analysis, sensitivity analysis.
result Dependent claims lead to underestimation of ruin probability.
Study on ruin probabilities for Lévy processes with light-tailed jumps.
problem Determining bounds on ruin probabilities for Lévy processes.
method Analyzing the Laplace exponent of the Lévy process to find bounds on ruin probabilities.
result Identification of a new case not previously considered in the literature.
The study analyzes how bonus-malus systems and delayed claims settlement affect insurance companies' financial stability.
problem Analyzing the impact of bonus-malus systems and delayed claims settlement on insurance companies' financial stability.
method Examined a discrete-time risk model with time-varying premiums, evaluating two types of claims and settlement delays.
result Delayed settlement of by-claims leads to lower ruin probabilities under specific assumptions.
This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integro-differential equations, exact and numerical solutions, asymptotic results, bounds on the ruin probability and also th…
In this paper we study the asymptotic decay of finite time ruin probabilities for an insurance company that faces heavy-tailed claims, uses predictable investment strategies and makes investments in risky assets whose prices evolve according to quite general semimartingales. We show that the ruin problem corresponds to…
Optimal insurance minimizes ruin probability with non-decreasing functions.
problem Minimizing ruin probability with insurance premiums and non-decreasing functions.
method Reformulated problem with inverse survival function as control variable.
result Deductible insurance with maximum limit is optimal.
Study shows subordinated Cramér-Lundberg model increases ruin probability.
problem Analyzing the impact of subordinated time-changed claims on insurance ruin probability.
method Examined a compound Poisson process modified by a Lévy subordinator.
result Probability of ruin decreases slowly with initial capital, despite unchanged total claim amount.
Optimizes insurance pricing to minimize ruin probability under various claim dependencies.
problem Determining optimal insurance premiums in the presence of dependencies between claim occurrences.
method Analyzes both independent and dependent claim processes, considering single and multiple risks.
result Optimal insurance premiums depend on initial reserve and claim dependencies.
Fair reinsurance premiums calculated for a perturbed risk model with capital injections.
problem Determining fair reinsurance premiums in a perturbed risk model with capital injections.
method Using a subordinator and Brownian perturbation, an explicit formula for reinsurance premiums is derived.
result An explicit formula for fair reinsurance premiums exists in a specific risk model setting.
Formula found for ruin probabilities in divided insurance companies.
problem Ruin probabilities in divided insurance companies with Lévy processes.
method Formula for supremum distribution of Lévy processes with broken drift.
result Formulas for ruin probabilities in specified proportions.
Paper introduces a new method for allocating capital based on risk measures from ruin theory.
problem Allocating capital to manage risk measures derived from ruin theory.
method Introduces a novel allocation method for dynamic value-at-risk (VaR) measures.
result Demonstrates desirable properties and compares with existing methods.
Value-at-Risk is a flawed substitute for non-ruin capital, leading to misleading financial standards.
problem Misuse of Value-at-Risk as a risk measure, replacing non-ruin capital, leads to flawed financial standards.
method Mathematical analysis of risk measures and their implications on financial standards.
result Non-ruin capital is a more accurate risk measure than Value-at-Risk, necessitating its adoption over the former.
We study solvency of insurers in a comprehensive model where various economic factors affect the capital developments of the companies. The main interest is in the impact of real growth to ruin probabilities. The volume of the business is allowed to increase or decrease. In the latter case, the study is focused on run-…
This paper approximates the Gerber-Shiu function using phase-type Levy processes.
problem Measuring the risk of insurance companies through the Gerber-Shiu function.
method Approximate the Gerber-Shiu function by fitting the underlying process with phase-type Levy processes.
result A closed-form approximation of the Gerber-Shiu function is derived.
The paper calculates ruin probabilities for insurers with phase-type distributed claims.
problem Calculating ruin probabilities for insurers with specific claim distributions.
method Change-of-measure technique applied to phase-type distributed claim amounts.
result The mixture of Erlangs best fits real-world loss data, improving risk assessment.
Paper tackles lifetime ruin with hedge funds and high-watermark fees, considering drift uncertainty.
problem Lifetime ruin problem with hedge funds and high-watermark fees under drift uncertainty.
method Employed the stochastic Perron's method to characterize the value function as the unique viscosity solution to the HJB equation.
result Characterized the value function as the unique viscosity solution without resorting to the proof of dynamic programming principle.
In this paper we analyze so-called Parisian ruin probability that happens when surplus process stays below zero longer than fixed amount of time ζ>0. We focus on general spectrally negative Lévy insurance risk process. For this class of processes we identify expression for ruin probability in terms of some other quan…
Optimizes reinsurance and investment strategies to minimize ruin probability.
problem Optimizing reinsurance and investment strategies to minimize ruin probability.
method Stochastic projected gradient method based on Malliavin calculus.
result Effectiveness of the proposed method demonstrated through numerical experiments.
We start by showing that the finite-time absolute ruin probability in the classical risk model with constant interest force can be expressed in terms of the transition probability of a positive Ornstein-Uhlenbeck type process, say X. Our methodology applies to the case when the dynamics of the aggregate claims process …
Retirees who exhaust their savings while still alive are said to experience financial ruin. These savings are typically grown during the accumulation phase then spent during the retirement decumulation phase. Extensive research into invest-and-harvest decumulation strategies has been conducted, but recommendations diff…
We assume that an agent's rate of consumption is {\it ratcheted}; that is, it forms a non-decreasing process. Given the rate of consumption, we act as financial advisers and find the optimal investment strategy for the agent who wishes to minimize his probability of ruin.
In this paper, we study a risk process modeled by a Brownian motion with drift (the diffusion approximation model). The insurance entity can purchase reinsurance to lower its risk and receive cash injections at discrete times to avoid ruin. Proportional reinsurance and excess-of-loss reinsurance are considered. The obj…
Study optimal dividend strategy with time of ruin constraint for financial firms.
problem Optimal dividend strategy with time of ruin constraint for spectrally one-sided Lévy risk models.
method Introduced a longevity feature to the classical optimal dividend problem, extended results to one-sided Lévy risk models, and characterized the solution using dual problems.
result Characterized the solution to the constrained optimal dividend problem for spectrally one-sided Lévy processes.
In this note, we explicitly solve the problem of maximizing utility of consumption (until the minimum of bankruptcy and the time of death) with a constraint on the probability of lifetime ruin, which can be interpreted as a risk measure on the whole path of the wealth process.
Two insurance companies collaborate to maximize the probability of none going bankrupt.
problem Maximizing the probability of no company bankruptcy in a correlated Brownian motion model.
method Analyzing optimal strategies and deriving explicit formulas for minimal ruin probability.
result Maximizing collaboration benefits when Brownian motions are positively correlated.
This paper analyzes the profitability of selfish mining on blockchain, considering the risk of ruin.
problem The profitability of selfish mining on blockchain, considering the risk of ruin.
method Formulated a stochastic model and used tools from applied probability and analysis to determine expected profit.
result Explicit expressions for expected profit under different scenarios were derived, identifying conditions for selfish mining as a strategic advantage.
Paper aims to minimize ruin probability in insurance companies using Sparre Andersen model.
problem Minimizing ruin probability in insurance companies with Sparre Andersen surplus process.
method Markovization of the surplus process, investigation of value function's regularity, dynamic programming principle, and comparison of viscosity solutions.
result The value function is the unique constrained viscosity solution to the Hamilton-Jacobi-Bellman equation.
We study a risk sensitive control version of the lifetime ruin probability problem. We consider a sequence of investments problems in Black-Scholes market that includes a risky asset and a riskless asset. We present a differential game that governs the limit behavior. We solve it explicitly and use it in order to find …
We show that the mutual fund theorems of Merton (1971) extend to the problem of optimal investment to minimize the probability of lifetime ruin. We obtain two such theorems by considering a financial market both with and without a riskless asset for random consumption. The striking result is that we obtain two-fund the…
In this note we give, for a spectrally negative Levy process, a compact formula for the Parisian ruin probability, which is defined by the probability that the process exhibits an excursion below zero, with a length that exceeds a certain fixed period r. The formula involves only the scale function of the spectrally ne…