This paper evaluates vulnerable American put options under jump–diffusion assumptions on the underlying asset and the assets of the counterparty. Sudden shocks on the asset prices are described as a compound Poisson process. Analytical pricing formulae of vulnerable European put options and vulnerable twice-exercisable European put options are derived. Employing the two-point Geske and Johnson method, we derive an approximate analytical pricing formula of vulnerable American put options under jump–diffusions. Numerical simulations are performed for investigating the impacts of jumps and default risk on option prices.

We consider a single server system accepting two types of retrial customers, which arrive according to two independent Poisson streams. The service station can handle at most one customer, and in case of blocking, type i customer, i=1, 2, is routed to a separate type i orbit queue of infinite capacity. Customers from the orbits try to access the server according to the constant retrial policy. We consider coupled orbit queues, and thus, when both orbit queues are non-empty, the orbit queue i tries to re-dispatch a blocked customer of type i to the main service station after an exponentially distributed time with rate μi. If an orbit queue empties, the other orbit queue changes its re-dispatch rate from μi to $\mu_{i}^{\ast}$. We consider both exponential and arbitrary distributed service requirements, and show that the probability generating function of the joint stationary orbit queue length distribution can be determined using the theory of Riemann (–Hilbert) boundary value problems. For exponential service requirements, we also investigate the exact tail asymptotic behavior of the stationary joint probability distribution of the two orbits with either an idle or a busy server by using the kernel method. Performance metrics are obtained, computational issues are discussed and a simple numerical example is presented.

We propose two random network models for complex networks, which are treelike and always grow very fast. One is the uniform model and the other is the preferential attachment model, and both of them depends on a parameter 0<p<1. We first briefly discuss the network sizes, each of which can be corresponding to a supercritical branching process. And then we mainly study the degree distributions of both models. The asymptotic degree distribution of the first one with any parameter 0<p<1 is a geometric distribution with parameter 1/2, whereas that of the second one, which depends on p, can be uniquely determined by a functional equation of its probability generating function.

In this paper, we propose a stochastic method to project the public pension fund in the public pension system (PPS). For this we introduce the stochastic differential equations for the three parts: the premium revenue, the benefit expenditure, and the fund process. From these we show that the solution of the aggregated fund process is the sum of log-normals, which is approximated as one log-normal for the analytic result. Related to the parameter estimations, we implement the moment matching in the first moment. For the second moment, we apply the extreme value method following Parkinson. In order to follow Parkinson, we take the maximum and the minimum range of the fund amount based on the various sensitivity result as well as the baseline one from the deterministic projection result. In this reason, it is naturally to maintain the close interrelation with the deterministic projection result, which is very important since it is still key result in the actuarial valuation of the PPS.

On an incomplete financial market, the stocks are modeled as pure jump processes subject to defaults. The exponential utility maximization problem is investigated characterizing the value function in term of Backward Stochastic Differential Equations (BSDEs), driven by pure jump processes. In general, in this setting, there is no unique solution. This is the reason why, the value function is proven to be the limit of a sequence of processes. Each of them is the solution of a Lipschitz BSDE and it corresponds to the value function associated with a subset of bounded admissible strategies. Given a representation of the jump processes driving the model, the aim of this note is to give a recursive backward scheme for the value function of the initial problem.

There are n customers that need to be served by m parallel servers (n≥m). Customer i will only wait in queue for an exponentially distributed time with rate λi before departing the system. The service time on server j is exponentially distributed with rate μj for all customers, and upon completion of service of customer i a positive reward ri is earned. The non-preemptive problem is to choose, after each service completion, which currently in queue customer to serve next. The preemptive problem is to decide when to preempt a service, and to choose, after each service completion or preemption, which currently in queue customer to serve next. The objective of both problems is to maximize the expected total return. We give conditions under which a list policy is optimal for both problems.

The knowledge gradient (KG) policy was originally proposed for online ranking and selection problems but has recently been adapted for use in online decision-making in general and multi-armed bandit problems (MABs) in particular. We study its use in a class of exponential family MABs and identify weaknesses, including a propensity to take actions which are dominated with respect to both exploitation and exploration. We propose variants of KG which avoid such errors. These new policies include an index heuristic, which deploys a KG approach to develop an approximation to the Gittins index. A numerical study shows this policy to perform well over a range of MABs including those for which index policies are not optimal. While KG does not take dominated actions when bandits are Gaussian, it fails to be index consistent and appears not to enjoy a performance advantage over competitor policies when arms are correlated to compensate for its greater computational demands.