Ruin Problem Research Articles

On the Moments of the Asymmetric n-Player Gambler’s Ruin Problem

In this article, we study the asymmetric n-player gambler’s ruin problem with ties allowed, where each player has equal capital of d units of money. In each round, the probabilities of winning from each player are fixed and the outcomes are independent from previous rounds. The game ends when at least one player run out of the money. The duration of the game is called the ruin time. For 1≤d<n, the formula of the moments of ruin time is found by Hashemiparast and Sabzevari (2011). Moreover, for d=n or n+1, the expectation and variance of ruin time were found by Hashemiparast and Sabzevari Hashemiparast and Sabzevari (2011) and Sabzevari(2017) respectively. In this work, we obtain the explicit formulas of the moments of ruin time for the cases d=n,n+1. Recurrence relation techniques are used to prove the results. Our results agree with the results from Monte carlo simulations. These novel formulas allow us to compute all the moments of the ruin time of the game, including expectation and variance.

On the longest/shortest negative excursion of a Lévy risk process and related quantities

In this paper, we analyze some distributions involving the longest and shortest negative excursions of spectrally negative Lévy processes using the binomial expansion approach. More specifically, we study the distributions of such excursions and related quantities such as the joint distribution of the shortest and longest negative excursions and their difference (also known as the range) over a random and infinite horizon time. Our results are applied to address new Parisian ruin problems, stochastic ordering and the number near-maximum distress periods showing the superiority of the binomial expansion approach for such cases.

A singularly perturbed ruin problem for a two-dimensional Brownian motion in the positive quadrant

Abstract We consider the following problem: the drift of the wealth process of two companies, modelled by a two-dimensional Brownian motion, is controllable such that the total drift adds up to a constant. The aim is to maximize the probability that both companies survive. We assume that the volatility of one company is small with respect to the other, and use methods from singular perturbation theory to construct a formal approximation of the value function. Moreover, we validate this formal result by explicitly constructing a strategy that provides a target functional, approximating the value function uniformly on the whole state space.

An excursion theoretic approach to Parisian ruin problem

Applying excursion theory, we re-express several well studied fluctuation quantities associated to Parisian ruin for Lévy risk processes in terms of integrals with respect to the corresponding excursion measure. We show that these new expressions reconcile with the previous results on the Parisian ruin problem.

Based on Symmetric Jump Risk Market: Study on the Ruin Problem of a Risk Model with Liquid Reserves and Proportional Investment

In order to deal with complex risk scenarios involving claims, uncertainty, and investments, we consider the ruin problems in a compound Poisson risk model with liquid reserves and proportional investments and study the expected discounted penalty function under threshold dividend strategies. Firstly, the integral differential equation of the expected discounted penalty function is derived. Secondly, since the closed-form solution of the equation cannot be obtained, a sinc method is used to obtain the numerical approximation solution of the equation. Finally, the feasibility and superiority of the sinc method are illustrated by error analysis. In addition, based on a symmetric jump risk market, we discuss the influence of some parameters on the ruin probability with some examples. This study can help actuaries develop more robust risk management strategies and ensure the long-term stability and profitability of insurance companies. It provides a theoretical basis for actuaries to carry out risk management.

Asymptotics for the ruin probability in a proportional reinsurance risk model with dependent insurance and financial risks

. This article studies the joint ruin problem for two insurance companies that divide claims in positive proportions (modeling an insurance and re-insurance company). The arrival times of claims are delayed by a common random time. Suppose that the two insurance companies are allowed to make risk-free and risky investments, and the price processes of the corresponding investment portfolios are exponentials of jump-diffusion processes with common jumps. Furthermore, assuming that the claim sizes and their corresponding investment return jump possess a dependence structure, this article establishes an asymptotic formula for the ruin probability.

A Random Journey Through the Math of Gambling

The laws of chance are often subtle and deceptive. This is why games of chance work. People are convinced that they obey seemingly intuitive laws, while the underlying mathematical structure reveals a different and more complex reality. This article is a brief and rigorous journey through the implications that the mathematical laws governing stochastic processes have on gambling. It addresses a specific process, the random walk, and analyze some instances of fair and unfair games by highlighting the fallacy of many of our intuitions and beliefs. The paper gradually moves from the analysis of the random walk properties to a comprehensive description of the ruin problem. The introduction of the idea of transient and persistent states concludes the discussion. Much emphasis is placed on concrete examples and on the numerical values, in particular of the involved probabilities, and the interpretation of the results is always more central than the demonstrative technical details, which are nevertheless available to the reader.

Different topological solution structures in a two-dimensional controlled ruin problem depending on the optimization criterion

We consider two insurance companies with wealth processes given by independent Brownian motions with controllable non-negative drift. The drift rates sum up to 1. The companies aim at finding a strategy for the drift rates to maximize the probability that at least one of them survives forever. We prove that the strategy, where the company with higher wealth obtains the maximal drift rate, is optimal. Our result differs considerably from the numerical result of McKean and Shepp in [H.P. McKean and L.A. Shepp, The advantage of capitalism vs. socialism depends on the criterion, J. Math. Sci. 139(3) (2006), pp. 6589–6594]. Furthermore, we numerically obtain candidates for the optimal strategy if the common aim of the two companies is to maximize a convex combination of the probability that both firms survive and the probability that exactly one firm survives forever. Our numerical results indicate that in general it is optimal to assign the maximal drift rate to one company but whether it is the company with higher or less wealth depends on the size of both wealth processes.

Stochastic ordering results on the duration of the gambler’s ruin game

AbstractIn the classical gambler’s ruin problem, the gambler plays an adversary with initial capitals z and $a-z$ , respectively, where $a&gt;0$ and $0&lt; z &lt; a$ are integers. At each round, the gambler wins or loses a dollar with probabilities p and $1-p$ . The game continues until one of the two players is ruined. For even a and $0&lt;z\leq {a}/{2}$ , the family of distributions of the duration (total number of rounds) of the game indexed by $p \in [0,{\frac{1}{2}}]$ is shown to have monotone (increasing) likelihood ratio, while for ${a}/{2} \leq z&lt;a$ , the family of distributions of the duration indexed by $p \in [{\frac{1}{2}}, 1]$ has monotone (decreasing) likelihood ratio. In particular, for $z={a}/{2}$ , in terms of the likelihood ratio order, the distribution of the duration is maximized over $p \in [0,1]$ by $p={\frac{1}{2}}$ . The case of odd a is also considered in terms of the usual stochastic order. Furthermore, as a limit, the first exit time of Brownian motion is briefly discussed.

On the duration of a gambler's ruin problem

Consider a gambler who on each bet either wins 1 with probability p or loses 1 with probability q=1−p, with the results of successive bets being independent. The gambler will stop betting when they are either up k or down k. Letting N be the number of bets made, we show that N is a new better than used random variable. Moreover, we show that if k is even then N/2 has an increasing failure rate, and if k is odd then (N+1)/2 has an increasing failure rate.

Solution to the N-player gambler’s ruin using recursions based on multigraphs

This paper studies the N-player gambler’s ruin problem where two players are involved in an even-money bet during each round. The objective is to solve for each player’s final placing probabilities given their initial wealths, as well as the expected time until ruin. Multigraphs were constructed to model the transitions between chip states. Linear systems were constructed based on these graphs. Solutions for the placing probabilities of each player were obtained using these linear systems. A numerical algorithm is developed to solve the general N-player gambler’s ruin for any positive integer chip total. Expected time until ruin for any initial state can be solved using absorbing Markov chains. The solution leads to exact values and equities in this model depend on the exact number, not just proportion, of chips each player holds. Compared with the usual lattice model, the multigraph model uses smaller matrices in computations, yielding considerable savings in the number of operations and program run time.

Some global topological properties of a free boundary problem appearing in a two dimensional controlled ruin problem

AbstractIn this paper a two-dimensional Brownian motion (modeling the endowment of two companies), absorbed at the boundary of the positive quadrant, with controlled drift, is considered. The volatilities of the Brownian motions are different. We control the drifts of these processes and allow that both drifts add up to the maximal value of one. Our target is to choose the strategy in a way, s.t. the probability that both companies survive is maximized. It turns out that the state space of the problem is divided into two sets. In one set the first company gets the full drift, and in the other set the second one. We describe some topological properties of these sets and their separating curve.

Techniques for Computing the Probability of a Gambler’s Ruin with Applications to Playing Roulette

An interest in gambling has greatly increased over the last few decades with the more common use of slot machines and online gambling, especially sports betting. A concern that has been publically raised is addiction and eventual gambler’s ruin (loss of all money). In this paper we provide a solution to the Gambler’s Ruin problem in regards to roulette. We compute the probability of a gamblers ruin with applications to the various betting opportunities playing roulette by determining the W+1 roots of the relevant polynomials and from there determine the probability of a gamblers ruin. We find situations where the payoff becomes higher, the probability of ruin becomes lower. Lower goals of gain are associated with a lower probability of ruin and larger bets and larger odds payoff also increase the probability of ruin.

Cumulative Parisian ruin in finite and infinite time horizons for a renewal risk process with exponential claims

The Parisian ruin time, which is the first time the insurer's surplus process has an excursion below level zero that exceeds a prescribed time length, has been extensively analyzed in recent years mainly in the Lévy model and its special cases. However, the cumulative Parisian ruin time, which is the first time the total time spent by the surplus process below level zero exceeds a certain time length, has been rarely considered in the literature. In this paper, we study the cumulative Parisian ruin problem in a renewal risk model with general interclaim times and exponential claims. Explicit formulas for the infinite-time cumulative Parisian ruin probability is first derived under a deterministic Parisian clock and then under an Erlang clock, where the latter case can also serve as an approximation of the former. The finite-time cumulative Parisian ruin probability is subsequently analyzed as well when the time horizon is another Erlang random variable. Our formulas are applied in various numerical examples where the interclaim times follow gamma, Weibull, or Pareto distribution. Consequently, we demonstrate that the choice of the interclaim distribution does have a significant impact on the cumulative Parisian ruin probabilities when one deviates from the exponential assumption.

Parisian ruin with random deficit-dependent delays for spectrally negative Lévy processes

We consider an interesting natural extension to the Parisian ruin problem under the assumption that the risk reserve dynamics are given by a spectrally negative Lévy process. The distinctive feature of this extension is that the distribution of the random implementation delay windows' lengths can depend on the deficit at the epochs when the risk reserve process turns negative, starting a new negative excursion. This includes the possibility of an immediate ruin when the deficit hits a certain subset. In this general setting, we derive a closed-form expression for the Parisian ruin probability and the joint Laplace transform of the Parisian ruin time and the deficit at ruin.

A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process

Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P, we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation principles in an insurance market without arbitrage opportunities are discussed in [26] and [27], respectively.

The symmetric 4-Player gambler’s problem with unequal initial stakes

Abstract This research advances the 4-player gambler’s ruin problem for the case of arbitrary initial stakes. The aim of the research is attained by offering simple expressions using the difference equation approach and thus providing closed form solution to the problem. Moreover, the existing technique of Chang [A game with four players, Statist. Probab. Lett. 23(2) (1995), 111–115] dealing with equal initial stakes is demonstrated as a sub-case of the newly devised scheme. The legitimacy of the proposed formulation is further verified by considering various parametric settings.

The Equity-Based Modeling for Two-Player Gambler's Ruin Problem

In the context of the classical two-player gambler's ruin problem, the winning probabilities and initial stakes are pre-decided. If a player (who is in financial crisis) starts with less amount than his/her opponent in the symmetric game, has more chances to be ruined. Besides, a player (based on previous record data) with more winning probability than his/her competitor, has fewer chances to be ruined. We observe that most of the time, usually a weaker player is not fully willing to make a contest with a strong player. To give a fair chance to fight back for a weaker player and to develop the audience's interest, equity-based modeling is required. In this research, we propose some new equity-based models for the game of two players. In this way, we advocate the weaker player (with less winning probability or less amount to start the game) is motivated to participate in the contest because of a fair chance to make a comeback. The working methodology of newly proposed schemes is executed by deriving general expressions of the ruin probabilities for mathematical evaluation along with observing the ruin times, and then findings are compared with the results of a classic two-player game. Hence, the prime objectives related to the study are achieved by taking diverse parametric settings in the favor of equity-based modeling.

Finite-time ruin probabilities using bivariate Laguerre series

In this paper, we revisit the finite-time ruin probability in the classical compound Poisson risk model. Traditional general solutions to finite-time ruin problems are usually expressed in terms of infinite sums involving the convolutions related to the claim size distribution and their integrals, which can typically be evaluated only in special cases where the claims follow exponential or (more generally) mixed Erlang distribution. We propose to tackle the partial integro-differential equation satisfied by the finite-time ruin probability and develop a new approach to obtain a solution in terms of bivariate Laguerre series as a function of the initial surplus level and the time horizon for a large class of light-tailed claim distributions. To illustrate the versatility and accuracy of our proposed method which is easy to implement, numerical examples are provided for claim amount distributions such as generalized inverse Gaussian, Weibull and truncated normal where closed-form convolutions are not available in the literature.

Parisian excursion with capital injection for drawdown reflected Lévy insurance risk process

ABSTRACT This paper discusses Parisian ruin problem under drawdown with capital injection when the underlying source of randomness of the surplus is modeled by a general Lévy insurance risk process. Capital injection is provided at the first instance the surplus drops below the drawdown level that is a pre-specified function of its current maximum. The capital is continuously injected to keep the surplus above the drawdown level until either it goes above its current maximum or a Parisian type ruin occurs, which is announced at the first time the surplus process has undergone an excursion below its current maximum for an independent exponential period of time consecutively since the most recent drawdown time. Some distributional identities concerning this excursion are presented. Firstly, we give the Parisian ruin probability and the joint Laplace transform (possibly killed at the first passage time above a fixed level for the surplus process) of the ruin time, the surplus position at ruin, and the total capital injection at ruin. Secondly, we obtain the q-potential measure of the surplus process killed at Parisian ruin. Finally, we give the expected present value of the total discounted capitals injected up to the Parisian ruin time. The results are derived using recent developments in fluctuation and excursion theory of spectrally negative Lévy process and are presented semi-explicitly in terms of the scale function of the Lévy process. Some numerical examples are given to facilitate the analysis of the impact of initial surplus and frequency of observation periods to the ruin probability and to the expected total discounted capital injection.