A Remark on the Principle of Zero Utility

Let u(x) be a utility function, i.e., a function with u′(x)>0, u″(x)<0 for all x. If S is a risk to be insured (a random variable), the premium P = P(x) is obtained as the solution of the equation which is the condition that the premium is fair in terms of utility. It is clear that an affine transformation of u generates the same principle of premium calculation. To avoid this ambiguity, one can standardize the utility function in the sense that for an arbitrarily chosen point y. Alternatively, one can consider the risk aversion which is the same for all affine transformations of a utility function. Given the risk aversion r(x), the standardized utility function can be retrieved from the formula It is easily verified that this expression satisfies (2) and (3). The following lemma states that the greater the risk aversion the greater the premium, a result that does not surpris

Three Methods to Calculate the Probability of Ruin

The first method, essentially due to GOOVAERTS and DE VYLDER, uses the connection between the probability of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability, of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not neede

Optimal Dividends in the Dual Model with Diffusion

In the dual model, the surplus of a company is a Lévy process with sample paths that are skip-free downwards. In this paper, the aggregate gains process is the sum of a shifted compound Poisson process and an independent Wiener process. By means of Laplace transforms, it is shown how the expectation of the discounted dividends until ruin can be calculated, if a barrier strategy is applied, and how the optimal dividend barrier can be determined. Conditions for optimality are discussed and several numerical illustrations are given. Furthermore, a family of models is analysed where the individual gain amount distribution is rescaled and compensated by a change of the Poisson paramete

Martingale Approach to Pricing Perpetual American Options

The method of Esscher transforms is a tool for valuing options on a stock, if the logarithm of the stock price is governed by a stochastic process with stationary and independent increments. The price of a derivative security is calculated as the expectation, with respect to the risk-neutral Esscher measure, of the discounted payoffs. Applying the optional sampling theorem we derive a simple, yet general formula for the price of a perpetual American put option on a stock whose downward movements are skip-free. Similarly, we obtain a formula for the price of a perpetual American call option on a stock whose upward movements are skip-free. Under the classical assumption that the stock price is a geometric Brownian motion, the general perpetual American contingent claim is analysed, and formulas for the perpetual down-and-out call option and Russian option are obtained. The martingale approach avoids the use of differential equations and provides additional insight. We also explain the relationship between Samuelson's high contact condition and the first order condition for optimalit

On the Probability and Severity of Ruin

In the usual model of the collective risk theory, we are interested in the severity of ruin, as well as its probability. As a quantitative measure, we propose G(u, y), the probability that for given initial surplus u ruin will occur and that the deficit at the time of ruin will be less than y, and the corresponding density g(u, y). First a general answer in terms of the transform is obtained. Then, assuming that the claim amount distribution is a combination of exponential distributions, we determine g; here the roots of the equation that defines the adjustment coefficient play a central role. An explicit answer is also given in the case in which all claims are of constant siz

A Note on the Dividends-Penalty Identity and the Optimal Dividend Barrier

For a general class of risk models, the dividends-penalty identity is derived by probabilistic reasoning. This identity is the key for understanding and determining the optimal dividend barrier, which maximizes the difference between the expected present value of all dividends until ruin and the expected discounted value of a penalty at ruin (which is typically a function of the deficit at ruin). As an illustration, the optimal barrier is calculated in two classical models, for different penalty functions and a variety of parameter value

Maximizing Dividends without Bankruptcy

Consider the classical compound Poisson model of risk theory, in which dividends are paid to the shareholders according to a barrier strategy. Let b* be the level of the barrier that maximizes the expectation of the discounted dividends until ruin. This paper is inspired by Dickson and Waters (2004). They point out that the shareholders should be liable to cover the deficit at ruin. Thus, they consider b0 , the level of the barrier that maximizes the expectation of the difference between the discounted dividends until ruin and the discounted deficit at ruin. In this paper, b* and b0 are compared, when the claim amount distribution is exponential or a combination of exponential

Risk Theory with the Gamma Process

The aggregate claims process is modelled by a process with independent, stationary and nonnegative increments. Such a process is either compound Poisson or else a process with an infinite number of claims in each time interval, for example a gamma process. It is shown how classical risk theory, and in particular ruin theory, can be adapted to this model. A detailed analysis is given for the gamma process, for which tabulated values of the probability of ruin are provide

Antibiotic Therapy of Infections Due to Pseudomonas aeruginosa in Normal and Granulocytopenic Mice: Comparison of Murine and Human Pharmacokinetics

An effort was made to elucidate the limits of drug-activity tests in small animals. Human plasma kinetics of gentamicin, netilmicin, ticarcillin, ceftazidime, and ceftriaxone were approximated in normal and in granulocytopenic mice infected with various strains of Pseudomonas aeruginosa in the thigh muscle or intraperitoneally. The effect of such dosing on bacterial time-kill curves and on survival was compared with the effect of identical amounts of drug given as a single-bolus injection. With β-lactams, a highly significant superiority of fractionated dosing (simulated human kinetics) over bolus injections (murine plasma kinetics) was demonstrated, whereas with aminoglycosides it was a single-bolus injection that tended to be more active. Thus, when tested in conventional small-animal models, aminoglycoside activity may be overestimated, whereas β-lactam activity may be underestimated in respect to humans. These differences found in vivo most probably reflect the different pharmacodynamics between aminoglycosides and β-lactam drugs (time-kill curves, dose-response curves, and postantibiotic effect) similar to those previously observed in vitr

The Wiener process with drift between a linear retaining and an absorbing barrier

The Wiener process with constant drift is modified by a time-dependent retaining barrier that increases at a constant rate and by an absorbing barrier at zero. Explicit expressions in terms of series expansions are derived for the Laplace transform and the probability density function of the time of absorption.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/24175/1/0000434.pd