This article investigates the optimal hedging problem of the European contingent claims written on non-tradable assets. We assume that the risky assets satisfy jump diffusion models with a common jump process which reflects the correlated jump risk. The non-tradable asset and jump risk lead to an incomplete financial market. Hence, the cross-hedging method will be used to reduce the potential risk of the contingent claims seller. First, we obtain an explicit closed-form solution for the locally risk-minimizing hedging strategies of the European contingent claims by using the Föllmer–Schweizer decomposition. Then, we consider the hedging for a European call option as a special case. The value of the European call option under the minimal martingale measure is derived by the Fourier transform method. Next, some semi-closed solution formulae of the locally risk-minimizing hedging strategies for the European call option are obtained. Finally, some numerical examples are provided to illustrate the sensitivities of the optimal hedging strategies. By comparing the optimal hedging strategies when the underlying asset is a non-tradable asset or a tradable asset, we find that the liquidity risk has a significant impact on the optimal hedging strategies.

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

This paper considers the customers’ equilibrium and socially optimal joining–balking behavior in a single-server Markovian queue with a single working vacation and Bernoulli interruptions. The model is motivated by practical service systems where the service rate can be adjusted according to whether or not the system is empty. Specifically, we focus on a single-server queue in which the server's service rate is reduced from a regular to a lower one when the system becomes empty. This lower rate period is called a working vacation for the server which may represent that part of the service facility is under a maintenance process or works on other non-queueing job, or simply for saving the energy (for a machine server case). In this paper, we assume that the working vacation period is terminated after a random period or with probability p after serving a customer in a non-empty system. Such a system is called a queue with single working vacation and Bernoulli interruptions. Customers are strategic and can make choice of joining or balking based on different levels of system information. We consider four scenarios: fully observable, almost observable, almost unobservable, and fully unobservable queue cases. Under a reward-cost structure, we analyze the customer's equilibrium and social-optimal strategies. In addition, the effects of system parameters on optimal strategies are illustrated by numerical examples.

Ranked set sampling (RSS) and some of its variants are sampling designs that are applied widely in different areas. When the underlying population contains different subpopulations, we can use stratified ranked set sampling (SRSS) which combines the advantages of stratification with RSS. In the present paper, we consider the information content of SRSS in terms of extropy measure. Some results using stochastic orders properties are obtained. The effect of imperfect ranking on discrimination information is analytically investigated. It is proved that discrimination information between the perfect SRSS and simple random sampling (SRS) data sets performs better than that of between the imperfect SRSS and SRS data sets.

Reliability properties associated with the classic models of systems with age replacement have been a usual topic of research. Most previous works have checked the aging properties of the lifetime of the working units using stochastic comparisons among the systems with age replacement at different times. However, from a practical point of view, it would also be interesting to deduce to which aging classes the lifetime of the system belongs, making use of the aging properties of the lifetime of its working units. The first part of this article deals with this problem. Further along, stochastic orderings are established between the systems with replacement at the same time using several stochastic comparisons among the lifetimes of their working units. In addition, the lifetimes of two systems with age replacement are compared as well. This is performed assuming stochastic orderings between the number of replacement until failure, and the lifetimes of their working units conditioned to be less or equal than the replacement time. Similar comparisons are accomplished considering two systems with age replacement where the replacements occur at a random time. Illustrative examples are presented throughout the paper.

New computing and communications paradigms will result in traffic loads in information server systems that fluctuate over much broader ranges of time scales than current systems. In addition, these fluctuation time scales may only be indirectly known or even be unknown. However, we should still be able to accurately design and manage such systems. This paper addresses this issue: we consider an M/M/1 queueing system operating in a random environment (denoted M/M/1(R)) that alternates between HIGH and LOW phases, where the load in the HIGH phase is higher than in the LOW phase. Previous work on the performance characteristics of M/M/1(R) systems established fundamental properties of the shape of performance curves. In this paper, we extend monotonicity results to include convexity and concavity properties, provide a partial answer to an open problem on stochastic ordering, develop new computational techniques, and include boundary cases and various degenerate M/M/1(R) systems. The basis of our results are novel representations for the mean number in system and the probability of the system being empty. We then apply these results to analyze practical aspects of system operation and design; in particular, we derive the optimal service rate to minimize mean system cost and provide a bias analysis of the use of customer-level sampling to estimate time-stationary quantities.

Nakamoto doublespend strategy, described in Bitcoin foundational article, leads to total ruin with positive probability. The simplest strategy that avoids this risk incorporates a stopping threshold when success is unlikely. We compute the exact profitability and the minimal double spend that is profitable for this strategy. For a given amount of the transaction, we determine the minimal number of confirmations to be requested by the recipient that makes the double-spend strategy non-profitable. This number of confirmations is only 1 or 2 for average transactions and for a small relative hashrate of the attacker. This is substantially lower than the original Nakamoto number, which is about six confirmations and is widely used. Nakamoto analysis is only based on the success probability of the attack instead of on a profitability analysis that we carry out.

We bound the error for the normal approximation of the number of triangles in the Erdős–Rényi random graph with respect to the Kolmogorov metric. Our bounds match the best available Wasserstein bounds obtained by Barbour et al. [(1989). A central limit theorem for decomposable random variables with applications to random graphs. Journal of Combinatorial Theory, Series B 47: 125–145], resolving a long-standing open problem. The proofs are based on a new variant of the Stein–Tikhomirov method—a combination of Stein's method and characteristic functions introduced by Tikhomirov [(1976). The rate of convergence in the central limit theorem for weakly dependent variables. Vestnik Leningradskogo Universiteta 158–159, 166].

This paper considers optimal admission and routing control in multi-class service systems in which customers can either receive quality regular service which is subject to congestion or can receive congestion-free but less desirable service at an alternative service station, which we call the self-service station. We formulate the problem within the Markov decision process framework and focus on characterizing the structure of dynamic optimal policies which maximize the expected long-run rewards. For this, value function and sample path arguments are used. The congestion sensitivity of customers is modeled with class-independent holding costs at the regular service station. The results show how the admission rewards of customer classes affect their priorities at the regular and self-service stations. We explore that the priority for regular service may not only depend on regular service admission rewards of classes but also on the difference between regular and self-service admission rewards. We show that optimal policies have monotonicity properties, regarding the optimal decisions of individual customer classes such that they divide the state space into three connected regions per class.

Consider a multidimensional risk model, in which an insurer simultaneously confronts m (m ≥ 2) types of claims sharing a common non-stationary and non-renewal arrival process. Assuming that the claims arrival process satisfies a large deviation principle and the claim-size distributions are heavy-tailed, asymptotic estimates for two common types of ruin probabilities for this multidimensional risk model are obtained. As applications, we give two examples of the non-stationary point process: a Hawkes process and a Cox process with shot noise intensity, and asymptotic ruin probabilities are obtained for these two examples.

The new reliability notion describing the remaining lifetime is introduced for items with monotonically increasing degradation. We consider the remaining lifetime of an item (to be called, the predicted remaining lifetime) after its degradation reaches the predetermined level. The prediction is executed at inception of an item into operation. For the nonhomogeneous stochastic processes of degradation, this characteristic depends on the random first passage time of a degradation process. Some properties of the predicted remaining lifetime and the corresponding stochastic comparisons are discussed for items from homogeneous and heterogeneous populations, and a generalization to the case of the n-component coherent system is outlined. The problem of regime switching is described, and the new notion of the corresponding virtual age after the switching is proposed.

In this paper, we consider multi-state coherent systems that can be regarded as a series/parallel/recurrent connection of multi-state modules with binary/multi-state components. The multi-state (survival) signatures of such systems are presented in terms of multi-state (survival) signatures of related modules based on the structures. For a recurrent structure, the multi-state survival signature of the structure is also needed. The results established here are finally illustrated with a number of examples.

We revisit an old topic in algorithms, the deterministic walk on a finite graph which always moves toward the nearest unvisited vertex until every vertex is visited. There is an elementary connection between this cover time and ball-covering (metric entropy) measures. For some familiar models of random graphs, this connection allows the order of magnitude of the cover time to be deduced from first passage percolation estimates. Establishing sharper results seems a challenging problem.

Standard server assignment policies for multi-server queueing stations include the noncollaborative policy, where the servers work in parallel on different jobs; and the fully collaborative policy, where the servers work together on the same job. However, if each job can be decomposed into subtasks with no precedence relationships, then we consider a form of server coordination named task assignment where the servers work in parallel on different subtasks of the same job. We identify the task assignment policy that maximizes the long-run average throughput of a queueing station with finite internal buffers when blocked servers can be idled or reassigned to either replace or collaborate with other servers on unblocked subtasks. We then compare the server coordination policies and show that the task assignment is best when the servers are highly specialized; otherwise, the fully collaborative or noncollaborative policies are preferable depending on whether the synergy level among the servers is high or not. We also provide numerical results that quantify our previous comparison. Finally, we address buffer allocation in longer lines where there are precedence relationships between some of the tasks, and present numerical results that suggest our comparisons for one queueing station generalize to longer lines.

This article discusses the stochastic behavior and reliability properties for the inactivity times of failed components in coherent systems under double monitoring. A mixture representation of reliability function is obtained for the inactivity times of failed components, and some stochastic comparison results are also established. Furthermore, some sufficient conditions are developed in terms of the aging properties of the inactivity times of failed components. Finally, some numerical examples are presented to illustrate the theoretical results.