Large sample validity of the binomial distribution for lives with unequal mortality rates

Abstract

Let us consider a group of n lives which are observed during some time or age interval. Suppose that the following conditions are satisfied: 1. The probability of death within the interval considered has the same value q for each person of the group.

2. These lives represent statistically independent observations (with respect to mortality).

     

An effective method for constructing bounds for ruin probabilities for the surplus process perturbed by diffusion

In this paper, we first study orders, valid up to a certain positive initial surplus, between a pair of ruin probabilities resulting from two individual claim size random variables for corresponding continuous time surplus processes perturbed by diffusion. The results are then applied to obtain a smooth upper (lower) bound for the underlying ruin probability; the upper (lower) bound is constructed from exponentially distributed claims, provided that the mean residual lifetime function of the underlying random variable is non-decreasing (non-increasing). Finally, numerical examples are given to illustrate the constructed upper bounds for ruin probabilities with comparisons to some existing ones.     

Asymptotics of ruin probabilities for controlled risk processes in the small claims case

We consider a risk process with the possibility of investment into a risky asset. The aim of the paper is to obtain the asymptotic behaviour of the ruin probability under the optimal investment strategy in the small claims case. In addition we prove convergence of the optimal investment level as the initial capital tends to infinity.     

Hypothesis testing for diffusion processes with continuous observations: Direct computation of large deviation results for error probabilities

We propose statistical tests for deciding between two alternatives for diffusion processes observed continuously over a finite time interval. Our tests emphasize the large deviation aspects, or equivalently, the asymptotic behavior of probabilities of type I and type II errors and the rate at which these probabilities go to zero as the observation time increases. We obtain these rates using direct methods of calculation. We provide specific computational examples for diffusion processes commonly used in finance and show that the error probabilities for these cases go to zero exponentially fast. Applications in finance and economics are discussed.     

Criteria for the Stochastic Ordering of Random Sums,with Actuarial Applications

It is shown that vectors ( S M 1 , … , S Mn ) and ( S' M'1 , …, S' M'n ) of random sums of positive random variables are stochastically ordered by upper orthant dependence, lower orthant dependence, concordance or by the supermodular ordering whenever their corresponding random numbers of terms ( M 1 , … , M n ) and ( M' 1 , … , M' n ) are themselves ordered in this fashion. Actuarial applications of these results are given to different dependence structures for the collective risk model with several classes of business.     

A two-part fractional regression model for the financial leverage decisions of micro,small, medium and large firms

In this paper we examine the following two hypotheses, which traditional theories of capital structure are relatively silent about: (i) the determinants of financial leverage decisions are different for micro, small, medium and large firms; and (ii) the factors that determine whether or not a firm issues debt are different from those that determine how much debt it issues. Using a binary choice model to explain the probability of a firm raising debt and a fractional regression model to explain the relative amount of debt issued, we find strong support for both hypotheses. Confirming recent empirical evidence, we find also that, although larger firms are more likely to use debt, conditional on their having some debt, firm size is negatively related to the proportion of debt used by firms.     

A stochastic differential game for optimal investment of an insurer with regime switching

We introduce a model to discuss an optimal investment problem of an insurance company using a game theoretic approach. The model is general enough to include economic risk, financial risk, insurance risk, and model risk. The insurance company invests its surplus in a bond and a stock index. The interest rate of the bond is stochastic and depends on the state of an economy described by a continuous-time, finite-state, Markov chain. The stock index dynamics are governed by a Markov, regime-switching, geometric Brownian motion modulated by the chain. The company receives premiums and pays aggregate claims. Here the aggregate insurance claims process is modeled by either a Markov, regime-switching, random measure or a Markov, regime-switching, diffusion process modulated by the chain. We adopt a robust approach to model risk, or uncertainty, and generate a family of probability measures using a general approach for a measure change to incorporate model risk. In particular, we adopt a Girsanov transform for the regime-switching Markov chain to incorporate model risk in modeling economic risk by the Markov chain. The goal of the insurance company is to select an optimal investment strategy so as to maximize either the expected exponential utility of terminal wealth or the survival probability of the company in the ‘worst-case’ scenario. We formulate the optimal investment problems as two-player, zero-sum, stochastic differential games between the insurance company and the market. Verification theorems for the HJB solutions to the optimal investment problems are provided and explicit solutions for optimal strategies are obtained in some particular cases.     

Analysis of ruin measures for the classical compound Poisson risk model with dependence

In this paper, we consider an extension to the classical compound Poisson risk model. Historically, it has been assumed that the claim amounts and claim inter-arrival times are independent. In this contribution, a dependence structure between the claim amount and the interclaim time is introduced through a Farlie–Gumbel–Morgenstern copula. In this framework, we derive the integro-differential equation and the Laplace transform (LT) of the Gerber–Shiu discounted penalty function. An explicit expression for the LT of the discounted value of a general function of the deficit at ruin is obtained for claim amounts having an exponential distribution.     

Inequalities for the De Pril approximation to the distribution of the number of policies with claims

In the present paper, we give sufficient conditions for an ordering of De Pril approximations of the distribution of the number of claims in an insurance portfolio of independent policies. Possible extensions are discussed, both for the De Pril approximation and the Kornya approximation. A numerical example is given.     

On the Hoggard–Whalley–Wilmott Equation for the Pricing of Options with Transaction Costs

The Hoggard–Whalley–Wilmott equation is introduced to model portfolios of European type options incorporating transaction costs. The model gives rise to a nonlinear parabolic partial differential equation (PDE), whose nonlinearity reflects the presence of transaction costs. We show analytically the existence of solutions which are not necessarily convex nor concave. Numerical treatments are also given, which are devised to effectively handle an infinite domain and unbounded solutions.      

Ruin probability at a given time for a model with liabilities of the fractional Brownian motion type: A partial differential equation approach

In this paper we study the ruin probability at a given time for liabilities of diffusion type, driven by fractional Brownian motion with Hurst exponent in the range (0.5, 1). Using fractional Itô calculus we derive a partial differential equation the solution of which provides the ruin probability. An analytical solution is found for this equation and the results obtained by this approach are compared with the results obtained by Monte-Carlo simulation.