Yiqiang Q. Zhao

In this paper, we study the asymptotic behavior of the maximum queue length in a fluid queue driven by a self-similar Gaussian process with non-stationary increments. We derive almost sure and in-probability convergence of the normalized maximum queue length, described by a function [ ], where [ ] is the Hurst parameter. We obtain sharp tail bounds for the distribution of the queue length, using Malliavin calculus, which explicitly depend on [ ] and the drift parameter [ ]. Our results generalize classical results on fractional Brownian motion (fBm) to broader classes such as sub-fractional Brownian motion (sub-fBm) and bi-fractional Brownian motion (bi-fBm). The key techniques introduced in this framework include Gaussian supremum localization, path-penalty large deviations, and metric entropy. Systems that exhibit long-range dependence and self-similarity are developed based on these results.

In this paper, we apply techniques of Malliavin–Skorokhod calculus for Lévy processes to study the short-time asymptotics of the vanilla option price in the at-the-money (ATM), in-the-money (ITM) and out-of-the-money (OTM) scenarios, under a Lévy stochastic volatility model with infinite activity jumps.

Mean-field limits have been used now as a standard tool in approximations, including for networks with a large number of nodes. Statistical inference on mean-filed models has attracted more attention recently mainly due to the rapid emergence of data-driven systems. However, studies reported in the literature have been mainly limited to continuous models. In this paper, we initiate a study of statistical inference on discrete mean-field models (or jump processes) in terms of a well-known and extensively studied model, known as the power-of-L, or the supermarket model, to demonstrate how to deal with new challenges in discrete models. We focus on system parameter estimation based on the observations of system states at discrete time epochs over a finite period. We show that by harnessing the weak convergence results developed for the supermarket model in the literature, an asymptotic inference scheme based on an approximate least squares estimation can be obtained from the mean-field limiting equation. Also, by leveraging the law of large numbers alongside the central limit theorem, the consistency of the estimator and its asymptotic normality can be established when the number of servers and the number of observations go to infinity. Moreover, numerical results for the power-of-two model are provided to show the efficiency and accuracy of the proposed estimator.

In this paper, we study an optimal online resource reservation problem in a simple communication network. The network is composed of two compute nodes linked by a local communication link. The system operates in discrete time; at each time slot, the administrator reserves resources for servers before the actual job requests are known. A cost is incurred for the reservations made. Then, after the client requests are observed, jobs may be transferred from one server to the other to best accommodate the demands by incurring an additional transport cost. If certain job requests cannot be satisfied, there is a violation that engenders a cost to pay for each of the blocked jobs. The goal is to minimize the overall reservation cost over finite horizons while maintaining the cumulative violation and transport costs under a certain budget limit. To study this problem, we first formalize it as a repeated game against nature where the reservations are drawn randomly according to a sequence of probability distributions that are derived from an online optimization problem over the space of allowable reservations. We then propose an online saddle-point algorithm for which we present an upper bound for the associated K-benchmark regret together with an upper bound for the cumulative constraint violations. Finally, we present numerical experiments where we compare the performance of our algorithm with those of simple deterministic resource allocation policies.

We tackle in this paper an online network resource allocation problem with job transfers. The network is composed of many servers connected by communication links. The system operates in discrete time; at each time slot, the administrator reserves resources at servers for future job requests, and a cost is incurred for the reservations made. Then, after receptions, the jobs may be transferred between the servers to best accommodate the demands. This incurs an additional transport cost. Finally, if a job request cannot be satisfied, there is a violation that engenders a cost to pay for the blocked job. We propose a randomized online algorithm based on the exponentially weighted method. We prove that our algorithm enjoys a sub-linear in time regret, which indicates that the algorithm is adapting and learning from its experiences and is becoming more efficient in its decision-making as it accumulates more data. Moreover, we test the performance of our algorithm on artificial data and compare it against a reinforcement learning method where we show that our proposed method outperforms the latter.

In this article we show that Theorem 2 in Lie et al. (2023) is incorrect. Since Wombat Exchange, a decentralized exchange, is built upon Lie et al. (2023) and Theorem 2 is fundamental to Wombat Finance, we show that an undesirable phenomenon, which we call the robbed withdrawal, can happen as a consequence.