Trading cookies with a random walk

The mathematical problem of determining a gambler’s risk of ruin involves analyzing decisions of only one agent, namely the “gambler”. In this work we consider an extension that introduces two additional players, so called “sellers”. These two new agents can boost the probability of success for the gambler by selling to him (using a jargon borrowed from the theory of excited random walks) a “cookie” which is used to increase the probability of moving forward in the next step. The generalized gambler’s ruin scenario considers an excited random walk on a finite interval of integer line with two “cookie store” locations and unlimited supply of cookies at each. Each time when the buyer (walker) visits a store location, he has an opportunity to decide whether he is willing to consume the cookie or not. We wish to determine the equilibrium prices and cookie store locations in a formal game associated with this generalized gambler’s ruin scenario

Ruin problems in the generalized Erlang (n) risk model

For actuarial applications we consider the Sparre–Andersen risk model when the interclaim times are Generalized Erlang(n) distributed. Unlike the standard Erlang(n) case, the roots of the generalized Lundberg’s equation with positive real parts can be multiple. This has a significant impact in the formulae for ruin probabilities that have to be found. We start by addressing the problem of solving an integro–differential equation that is satisfied by the survival probability, as well as other probabilities related, and present a method to solve such equation. This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and then establishing a one-one relation between them and the solutions of the integro–differential equation mentioned above. We first study the cases when all the roots are single and when there are roots with higher multiplicity. Secondly, we show that it is possible to have double roots but no higher multiplicity. Also, we show that the number of double roots depend on the choice of the parameters of the generalized Erlang(n) distribution, with a maximum number depending on n being even or odd. Afterwards, we extend our findings above for the computation of the distribution of the maximum severity of ruin as well as, considering an interest force, to the study the expected discounted future dividends, prior to ruin. Our findings show an alternative and more general method to the one provided by Albrecher et al. (2005), by considering a general claim amount distributioninfo:eu-repo/semantics/publishedVersio

Further developments in the Erlang(n) risk process

For actuarial aplications, we consider the Sparre–Andersen risk model when the interclaim times are Erlang(n) distributed. We first address the problem of solving an integro-differential equation that is satisfied by the survival probability and other probabilities, and show an alternative and improved method to solve such equations to that presented by Li (2008). This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and establishing a one–one relation between them and the solutions of the integro-differential equation mentioned before. Afterwards, we apply our findings above in the computation of the distribution of the maximum severity of ruin. This computation depends on the non-ruin probability and on the roots of the fundamental Lundberg’s equation. We illustrate and give explicit formulae for Erlang(3) interclaim arrivals with exponentially distributed single claim amounts and Erlang(2) interclaim times with Erlang(2) claim amounts. Finally, considering an interest force, we consider the problem of calculating the expected discounted dividends prior to ruin, finding an integro-differential equation that they satisfy and solving it. Numerical examples are also provided for illustrationinfo:eu-repo/semantics/publishedVersio

Further developments in the Erlang(n) risk process

For actuarial aplications, we consider the Sparre–Andersen risk model when the interclaim times are Erlang(n) distributed. We first address the problem of solving an integro-differential equation that is satisfied by the survival probability and other probabilities, and show an alternative and improved method to solve such equations to that presented by Li (2008). This is done by considering the roots with positive real parts of the generalized Lundberg’s equation, and establishing a one–one relation between them and the solutions of the integro-differential equation mentioned before. Afterwards, we apply our findings above in the computation of the distribution of the maximum severity of ruin. This computation depends on the non-ruin probability and on the roots of the fundamental Lundberg’s equation. We illustrate and give explicit formulae for Erlang(3) interclaim arrivals with exponentially distributed single claim amounts and Erlang(2) interclaim times with Erlang(2) claim amounts. Finally, considering an interest force, we consider the problem of calculating the expected discounted dividends prior to ruin, finding an integro-differential equation that they satisfy and solving it. Numerical examples are also provided for illustration

On the Dual Risk Models

Abstract This thesis focuses on developing and computing ruin-related quantities that are potentially measurements for the dual risk models which was proposed to describe the annuity-type businesses from the perspective of the collective risk theory in 1950’s. In recent years, the dual risk models are revisited by many researchers to quantify the risk of the similar businesses as the annuity-type businesses. The major extensions included in this thesis consist of two aspects: the first is to search for new ruin-related quantities that are potentially indices of the risk for well-established dual models; the other aspect is to generalize the settings of the dual models instead of the ruin quantities. There are four separate articles in this thesis, in which the first (Chapter 2) and the last (Chapter 5) belong to the first type of extensions while the others (Chapter 3 and Chapter 4) belong to the generalizations of the dual models. The first article (Chapter 2) studies the discounted moments of the surplus at the time of the last jump before ruin for the compound Poisson dual risk model. The idea comes from that the ruin of the compound Poisson dual models is caused by absence of positive jumps within a period with length being propotional to the surplus at the time of the last jump. As a quantity related to a non-stopping time, the explicit expression of the target quantity is obtained through integro-differential equations. The second article (Chapter 3) investigate the Sparre-Andersen dual risk models in which the epochs are independently, identically distributed generalized Erlang-n random variables. An important difference between this model and some other models such as the Erlang-n dual risk models is that the roots to the generalized Lundberg’s equation are not necessarily distinct. By taking the multiple roots into account, the explicit expressions of the Laplace transform of the time to ruin and expected discounted aggregate dividends under the threshold strategy and exponential distributed revenues are derived. The third article (Chapter 4) revisits the the dual Lévy risk model. The target ruin quantity is the expected discounted aggregate dividends paid up to ruin under the threshold dividend strategy. The explicit expression is obtained in terms of the q-scale functions through constructing a new dividend strategy having the target ruin quantity converging to that under the threshold strategy. Also, the optimality of the threshold strategy among all the absolutely continuous stategies when evaluating the target quantity as a value function is discussed. The fourth article (Chapter 5) initiate the study of the Parisian ruin problem for the general dual Lévy risk models. Unlike the regular ruin for the dual models, the deficit at Parisian ruin is not necessarily equal to zero. Hence we introduce the Gerber-Shiu expected discounted penalty function (EPDF) at the Parisian ruin and obtain an explicit expression for this function. Keywords: Sparre-Andersen dual models, expected discounted aggregate dividends, dual Levy risk models, Parisian ruin, Gerber-Shiu function ii

Synchronous vs. asynchronous dynamics of diffusion-controlled reactions

An analytical method based on the classical ruin problem is developed to compute the mean reaction time between two walkers undergoing a generalized random walk on a 1d lattice. At each time step, either both walkers diffuse simultaneously with probability $p$ (synchronous event) or one of them diffuses while the other remains immobile with complementary probability (asynchronous event). Reaction takes place through same site occupation or position exchange. We study the influence of the degree of synchronicity $p$ of the walkers and the lattice size $N$ on the global reaction's efficiency. For odd $N$, the purely synchronous case ($p=1$) is always the most effective one, while for even $N$, the encounter time is minimized by a combination of synchronous and asynchronous events. This new parity effect is fully confirmed by Monte Carlo simulations on 1d lattices as well as for 2d and 3d lattices. In contrast, the 1d continuum approximation valid for sufficiently large lattices predicts a monotonic increase of the efficiency as a function of $p$. The relevance of the model for several research areas is briefly discussed.Comment: 21 pages (including 12 figures and 4 tables), uses revtex4.cls, accepted for publication in Physica