An explicit formula for the probability of nonruin of an insurance company in a finite time interval is derived, assuming Poisson claim arrivals, any continuous joint distribution of the claim amounts and any nonnegative, increasing real function representing its premium income. The formula is compact and expresses the nonruin probability in terms of Appell polynomials. An example, illustrating its numerical convenience, is also given in the case of inverted Dirichlet-distributed claims and a linearly increasing premium-income function.

We drive a car along a street towards our destination and look for an available parking place without turning around. Each parking place is associated with a loss which decreases with the distance of the parking place from our destination. Assume that the states (empty or filled) of the parking places form a Markov chain. We want to find an optimal parking strategy to minimize the expected loss. A curious example is constructed and two sufficient conditions for the existence of the threshold-type optimal parking strategy are given.

A gambler, with an initial fortune less than 1, wants to buy a house which sells today for 1. Due to inflation, the price of the house tomorrow will be 1 + α, where α is a nonnegative constant, and will continue to go up at this rate, becoming (1 + α) n on the nth day. Once each day, he can stake any amount of fortune in his possession, but no more than he possesses, on a primitive casino. It is well known that, in a subfair primitive casino without the presence of inflation, the gambler should play boldly. The presence of inflation would motivate the gambler to recognize the time value of his fortune and to try to reach his goal as quickly as possible; intuitively, we would conjecture that the gambler should again play boldly. However, in this note we will show that, unexpectedly, bold play is not necessarily optimal.

It is shown that the method of deriving bounds on the rate of convergence for birth–death processes developed by Zeifman can be effectively applied to stochastic models of chemical reactions.