A novel version for three-player gambler's ruin problem

For calculating the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. However, the martingale approach is not appropriate to determine the ruin time of asymmetric game. In general, ruin probabilities in both cases, that is, symmetric and asymmetric games as well as the expected ruin time for the asymmetric games are still need to be calculated. The current work is also about three-player gambler’s ruin problem with some extensions. We provide expressions for the ruin time with or without ties when all the players have equal or unequal initial fortunes. Finally, the validity of the asymmetric game is also tested through a Monte Carlo simulation study.
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For the expected ruin time of the classic three-player symmetric game, Sandell derived a general formula by introducing an appropriate martingale and stopping time. For the case of asymmetric game, the martingale approach is not valid to determine the ruin time. In general, the ruin probabilities for both cases, i.e. symmetric and asymmetric game and expected ruin time for asymmetric game are still awaiting to be solved for this game. The current work is also about three-player gambler’s ruin problem with some extensions as well. We provide expressions for the ruin time with (without) ties when all the players have equal (unequal) initial fortunes. Finally, the validity of asymmetric game is also tested through a Monte Carlo simulation study.

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.

Parisian excursion of a Levy process is defined as the excursion of the process below or above a pre-defined barrier continuously exceeding a certain time length. In this thesis, we study classical and Parisian type of ruin problems, as well as Parisian excursions of collective risk processes generalized on the classical Cramer-Lundberg risk model. We consider that claim sizes follow mixed exponential distributions and that the main focus is claim arrival process converging to an inverse Gaussian process. By this convergence, there are infinitely many and arbitrarily small claim sizes over any finite time interval. The results are obtained through Gerber-Shiu penalty function employed in an infinitesimal generator and inverting corresponding Laplace transform applied to the generator. In Chapter 3, the classical collective risk process under the Cram´er-Lundberg risk model framework is introduced, and probabilities of ruin with claim sizes following exponential distribution and a combination of exponential distributions are also studied. In Chapter 4, we focus on a surplus process with the total claim process converging to an inverse Gaussian process. The classical probability of ruin and the joint distribution of ruin time, overshoot and initial capital are given. This joint distribution could provide us with probabilities of ruin given different initial capitals in any finite time horizon. In Chapter 5, the classical ruin problem is extended to Parisian type of ruin, which requires that the length of excursions of the surplus process continuously below zero reach a predetermined time length. The joint law of the first excursion above zero and the first excursion under zero is studied. Based on the result, the Laplace transform of Parisian ruin time and formulae of probability of Parisian type of ruin with different initial capitals are obtained. Considering the asymptotic properties of claim arrival process, we also propose an approximation of the probability of Parisian type of ruin when the initial capital converges to infinity. In Chapter 6, we generalize the surplus process to two cases with total claim process still following an inverse Gaussian process. The first generalization is the case of variable premium income, in which the insurance company invests previous surplus and collects interest. The probability of survival and numerical results are given. The second generalization is the case in which capital inflow is also modelled by a stochastic process, i.e. a compound Poisson process. The explicit formula of the probability of ruin is provided.

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.

In the classical ruin problem two players with finite initial capitals play until one of them is ruined. It is often useful in analyzing this problem, for instance in finding the probability that a specific player is ruined and the expected duration of the game, to interpret the game as a random walk on the line with absorbing barriers. We will here treat a ruin problem which can be interpreted as a random walk on a polygon. We start with two simple cases.

In this paper, we have fitted two heavy tailed distributions viz the Weibull distribution and the Burr XII distribution to a set of Motor insurance claim data. As it is known, the probability of ruin is obtained as a solution to an integro differential equation, general solution of which leads to what is known as the Pollaczek-Khinchin Formula for the probability of ultimate ruin. In case, the claim severity is distributed as the above two mentioned distributions, and Pollaczek-Khinchin formula cannot be used to evaluate the probability of ruin through inversion of their Laplace transform since the Laplace Transforms themselves don’t have closed form expression. However, an approximation to the probability of ultimate ruin in such cases can be obtained by the Pollaczek-Khinchin formula through simulation and one crucial step in this simulation is to simulate from the corresponding Equilibrium distribution of the claim severity distribution. The paper lays down methodologies to simulate from the Equilibrium distribution of Burr XII distribution and Weibull distribution and has used them to obtain an approximation to the probability of ultimate ruin through Pollaczek-Khinchin formula by Monte Carlo simulation. An attempt has also been made to obtain numerical values to the probability function for the number of claims until ruin in case of zero initial surplus under these claim severity distributions and this in turn necessitates the computation of the convolutions of these distributions. The paper makes a preliminary effort to address this issue. All the computations are done under the assumption of the Classical Risk Model.

We compute ruin probabilities, in both infinite-time and finite-time, for a Gambler's Ruin problem with both catastrophes and windfalls in addition to the customary win/loss probabilities. For constant transition probabilities, the infinite-time ruin probabilities are derived using difference equations. Finite-time ruin probabilities of a system having constant win/loss probabilities and variable catastrophe/windfall probabilities are determined using lattice path combinatorics. Formulae for expected time till ruin and the expected duration of gambling are also developed. The ruin probabilities (in infinite-time) for a system having variable win/loss/catastrophe probabilities but no windfall probability are found. Finally, the infinite-time ruin probabilities of a system with variable win/loss/catastrophe/windfall probabilities are determined.

The effect of inflation of premium income and claims size distribution, but not of free reserves, on the probability of ruin of an insurer is studied.An interesting similarity between this problem and the ruin problem in an experience-rated scheme is exhibited. This similarity allows the deduction of parallel results for the two problems in later sections.It is shown that the probability of ruin is always increased when the (constant) inflation rate is increased.The distribution of aggregate claims under inflationary conditions is described and used to calculate an upper bound on the ruin probability. Some numerical examples show that this bound is not always sharp enough to be practically useful. It is also shown, however, that this bound can be used to construct an approximation of the effect of inflation on ruin probability.It is shown that if inflation occurs at a constant rate, then ruin is certain, irrespective of the smallness of that rate and of the largeness of initial free reserves and the safety margin in the premium. The corresponding result for experiencerated schemes is that a practical and “intuitively reasonable” experience-rating scheme leads eventually to certain ruin.Finally, a simple modification of the techniques of the paper is made in order to bring investment income into account.

The gambler's ruin problem with n players and asymmetric play

We present Monte Carlo (MC) simulation studies of phase separation in binary (AB) mixtures with bond-disorder that is introduced in two different ways: (i) at randomly selected lattice sites and (ii) at regularly selected sites. The Ising model with spin exchange (Kawasaki) dynamics represents the segregation kinetics in conserved binary mixtures. We find that the dynamical scaling changes significantly by varying the number of disordered sites in the case where bond-disorder is introduced at the randomly selected sites. On the other hand, when we introduce the bond-disorder in a regular fashion, the system follows the dynamical scaling for the modest number of disordered sites. For a higher number of disordered sites, the evolution morphology illustrates a lamellar pattern formation. Our MC results are consistent with the Lifshitz-Slyozov power-law growth in all the cases.

A Monte Carlo (MC) simulation study has been done in order to investigate the effects of line- edge-roughness (LER) induced by either 1P1E (single-patterning and single-etching) or 2P2E (double-patterning and double-etching) on fully- depleted silicon-on-insulator (FDSOI) tri-gate metal- oxide-semiconductor field-effect transistors (MOSFETs). Three parameters for characterizing the LER profile (i.e., root-mean square deviation (σ), correlation length (ζ), and fractal dimension (D)) are extracted from the image-processed scanning electron microscopy (SEM) image for each photolithography method. It is experimentally verified that two parameters (i.e., σ and D) are almost the same in each case, but the correlation length in the 2P2E case is longer than that in the 1P1E case. The 2P2E-LER- induced VTH variation in FDSOI tri-gate MOSFETs is smaller than the 1P1E-LER-induced VTH variation. The total random variation in VTH, however, is very dependent on the other major random variation sources, such as random dopant fluctuation (RDF) and work-function variation (WFV).

We report a comprehensive Monte Carlo (MC) simulation study of the vapor-to-droplet transition in Lennard-Jones fluid confined to a spherical container with repulsive walls, which is a case study system to investigate homogeneous nucleation. The focus is made on the application of a modified version of the ghost field method (Vishnyakov, A.; Neimark, A. V. J. Chem. Phys. 2003, 119, 9755) to calculate the nucleation barrier. This method allows one to build up a continuous trajectory of equilibrium states stabilized by the ghost field potential, which connects a reference droplet with a reference vapor state. Two computation schemes are employed for free energy calculations, direct thermodynamic integration along the constructed trajectory and umbrella sampling. The nucleation barriers and the size dependence of the surface tension are reported for droplets containing from 260 to 2000 molecules. The MC simulation study is complemented by a review of the simulation methods applied to computing the nucleation barriers and a detailed analysis of the vapor-to-droplet transition by means of the classical nucleation theory.

The time-dependent virtual waiting time in a M/G/1 queue converges to a proper limit when the traffic intensity is less than one. In this paper we give precise rates on the speed of this convergence when the service time distribution has a heavy regularly varying tail.The result also applies to the classical ruin problem. We obtain the exact rate of convergence for the ruin probability after time t for the case where claims arrive according to a Poisson process and claim sizes are heavy tailed.Our result supplements similar theorems on exponential convergence rates for relaxation times in queueing theory and ruin probabilities in risk theory.

The time-dependent virtual waiting time in a M/G/1 queue converges to a proper limit when the traffic intensity is less than one. In this paper we give precise rates on the speed of this convergence when the service time distribution has a heavy regularly varying tail. The result also applies to the classical ruin problem. We obtain the exact rate of convergence for the ruin probability after time t for the case where claims arrive according to a Poisson process and claim sizes are heavy tailed. Our result supplements similar theorems on exponential convergence rates for relaxation times in queueing theory and ruin probabilities in risk theory.