Searcharxiv⌕ Search

arXiv subjects

Bernardo D'Auria

Publications and source records attributed to Bernardo D'Auria.

27 records · Page 2Linked to original sources
arXiv2019

A short note on "Anticipative portfolio optimization"

In 1996, Pikovsky and Karatzas did one of the earliest studies on portfolio optimization problems in presence of insider information. They were able to successfully show that the knowledge of the stock price at future time is an insider information with associated unbounded value. However when the insider information only gives an interval containing the future value of the stock price, they could not prove that the value of the information is finite. They made a conjecture of this result and tried to convince about its validity by showing some numerical approximations. Instead of applying more sophisticated results, such as Shannons information theory, we show that their conjecture holds true by giving a simple proof of the finiteness of the value of the insider information for this case.

math.PR↗
arXiv2019

Optimal stopping times for a class of Ito diffusion bridges

The scope of this paper is to study the optimal stopping problems associated to a stochastic process, which may represent the gain of an investment, for which information on the final value is available a priori. This information may proceed, for example, from insider trading or from pinning at expiration of stock options. We solve and provide explicit solutions to these optimization problems. As special case, we discuss different processes whose optimal barrier has the same shape as the optimal barrier of the Brownian bridge. So doing we provide a catalogue of alternatives to the Brownian bridge which in practice could be better adapted to the data. Moreover, we investigate if, for any given (decreasing) curve, there exists a process with this curve as optimal barrier. This provides a model for the optimal liquidation time, i.e. the optimal time at which the investor should liquidate a position in order to maximize the gain.

math.PR↗
arXiv2016

Proportional switching in FIFO networks

We consider a family of discrete time multihop switched queueing networks where each packet moves along a fixed route. In this setting, BackPressure is the canonical choice of scheduling policy; this policy has the virtues of possessing a maximal stability region and not requiring explicit knowledge of traffic arrival rates. BackPressure has certain structural weaknesses because implementation requires information about each route, and queueing delays can grow super-linearly with route length. For large networks, where packets over many routes are processed by a queue, or where packets over a route are processed by many queues, these limitations can be prohibitive. In this article, we introduce a scheduling policy for FIFO networks, the Proportional Scheduler, which is based on the proportional fairness criterion. We show that, like BackPressure, the Proportional Scheduler has a maximal stability region and does not require explicit knowledge of traffic arrival rates. The Proportional Scheduler has the advantage that information about the network's route structure is not required for scheduling, which substantially improves the policy's performance for large networks. For instance, packets can be routed with only next-hop information and new nodes can be added to the network with only knowledge of the scheduling constraints.

math.PR↗
arXiv2015

Sojourn time in a single server queue with threshold service rate control

We study the sojourn time in a queueing system with a single exponential server, serving a Poisson stream of customers in order of arrival. Service is provided at low or high rate, which can be adapted at exponential inspection times. When the number of customers in the system is above a given threshold, the service rate is upgraded to the high rate, and otherwise, it is downgraded to the low rate. The state dependent changes in the service rate make the analysis of the sojourn time a challenging problem, since the sojourn time now also depends on future arrivals. We determine the Laplace transform of the stationary sojourn time and describe a procedure to compute all moments as well. First we analyze the special case of continuous inspection, where the service rate immediately changes once the threshold is crossed. Then we extend the analysis to random inspection times. This extension requires the development of a new methodological tool, that is "matrix generating functions". The power of this tool is that it can also be used to analyze generalizations to phase-type services and inspection times.

math.PR↗
arXiv2015

Pure threshold strategies for a two-node tandem network under partial information

In a two node tandem network, customers decide to join or balk by maximizing a given profit function whose costs are proportional to the sojourn time they spend at each queue. Assuming that their choices are taken without knowing the complete state of the system, we show that a pure threshold equilibrium policy exists. In particular we analyze the case when the partial information consists in informing the arrival customers of the total number of users in the network.

math.PR↗
arXiv2014

Closed queueing networks under congestion: non-bottleneck independence and bottleneck convergence

We analyze the behavior of closed product-form queueing networks when the number of customers grows to infinity and remains proportionate on each route (or class). First, we focus on the stationary behavior and prove the conjecture that the stationary distribution at non-bottleneck queues converges weakly to the stationary distribution of an ergodic, open product-form queueing network. This open network is obtained by replacing bottleneck queues with per-route Poissonian sources whose rates are determined by the solution of a strictly concave optimization problem. Then, we focus on the transient behavior of the network and use fluid limits to prove that the amount of fluid, or customers, on each route eventually concentrates on the bottleneck queues only, and that the long-term proportions of fluid in each route and in each queue solve the dual of the concave optimization problem that determines the throughputs of the previous open network.

math.PR↗
arXiv2011

Two-sided reflected Markov-modulated Brownian motion with applications to fluid queues and dividend payouts

In this paper we study a reflected Markov-modulated Brownian motion with a two sided reflection in which the drift, diffusion coefficient and the two boundaries are (jointly) modulated by a finite state space irreducible continuous time Markov chain. The goal is to compute the stationary distribution of this Markov process, which in addition to the complication of having a stochastic boundary can also include jumps at state change epochs of the underlying Markov chain because of the boundary changes. We give the general theory and then specialize to the case where the underlying Markov chain has two states. Moreover, motivated by an application of optimal dividend strategies, we consider the case where the lower barrier is zero and the upper barrier is subject to control. In this case we generalized earlier results from the case of a reflected Brownian motion to the Markov modulated case.

math.PR↗
arXiv2010

First passage process of a Markov additive process, with applications to reflection problems

In this paper we consider the first passage process of a spectrally negative Markov additive process (MAP). The law of this process is uniquely characterized by a certain matrix function, which plays a crucial role in fluctuation theory. We show how to identify this matrix using the theory of Jordan chains associated with analytic matrix functions. Importantly, our result also provides us with a technique, which can be used to derive various further identities. We then proceed to show how to compute the stationary distribution associated with a one-sided reflected (at zero) MAP for both the spectrally positive and spectrally negative cases as well as for the two sided reflected Markov-modulated Brownian motion; these results can be interpreted in terms of queues with MAP input.

math.PR↗