Olympia Hadjiliadis

One Shot Schemes for Decentralized Quickest Change Detection

This work considers the problem of quickest detection with N distributed sensors that receive sequential observations either in discrete or in continuous time from the environment. These sensors employ cumulative sum (CUSUM) strategies and communicate to a central fusion center by one shot schemes. One shot schemes are schemes in which the sensors communicate with the fusion center only once, via which they signal a detection. The communication is clearly asynchronous and the case is considered in which the fusion center employs a minimal strategy, which means that it declares an alarm when the first communication takes place. It is assumed that the observations received at the sensors are independent and that the time points at which the appearance of a signal can take place are different. Both the cases of the same and different signal distributions across sensors are considered. It is shown that there is no loss of performance of one shot schemes as compared to the centralized case in an extended Lorden min-max sense, since the minimum of N CUSUMs is asymptotically optimal as the mean time between false alarms increases without bound. In the case of different signal distributions the optimal threshold parameters are explicitly computed.

Optimal and Asymptotically Optimal CUSUM Rules for Change Point Detection in the Brownian Motion Model with Multiple Alternatives

This work examines the problem of sequential change detection in the constant drift of a Brownian motion in the case of multiple alternatives. As a performance measure an extended Lordens criterion is proposed. When the possible drifts, assumed after the change, have the same sign, the \hbox{CUSUM} rule, designed to detect the smallest in absolute value drift, is proven to be the optimum. If the drifts have opposite signs, then a specific 2-\hbox{CUSUM} rule is shown to be asymptotically optimal as the frequency of false alarms tends to infinity.

Maximum Drawdown Insurance

The drawdown of an asset is a risk measure defined in terms of the running maximum of the assets spot price over some period [0, T]. The asset price is said to have drawn down by at least

Drawdowns preceding rallies in the Brownian motion model

K over this period if there exists a time at which the underlying is at least

Stochastic Modeling and Fair Valuation of Drawdown Insurance

K below its maximum-to-date. We introduce insurance against a large realization of maximum drawdown and a novel way to hedge the liability incurred by underwriting this insurance. Our proposed insurance pays a fixed amount should the maximum drawdown exceed some fixed threshold over a specified period. The need for this drawdown insurance would diminish should markets rise before they fall. Consequently, we propose a second kind of cheaper maximum drawdown insurance that pays a fixed amount contingent on the drawdown preceding a drawup. We propose double barrier options as hedges for both kinds of insurance against large maximum drawdowns. In fact for the second kind of insurance we show that the hedge is model-free. Since double barrier options do not trade liquidly in all markets, we examine the assumptions under which alternative hedges using either single barrier options or standard vanilla options can be used.

Quickest Detection in Coupled Systems

We study drawdowns and rallies of Brownian motion. A rally is defined as the difference of the present value of the Brownian motion and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. This paper determines the probability that a drawdown of a units precedes a rally of b units. We apply this result to examine stock market crashes and rallies in the geometric Brownian motion model.

Online Algorithms for Classification of Urban Objects in 3D Point Clouds

This paper studies the stochastic modeling of market drawdown events and the fair valuation of insurance contracts based on drawdowns. We model the asset drawdown process as the current relative distance from the historical maximum of the asset value. We first consider a vanilla insurance contract whereby the protection buyer pays a constant premium over time to insure against a drawdown of a pre-specified level. This leads to the analysis of the conditional Laplace transform of the drawdown time, which will serve as the building block for drawdown insurance with early cancellation or drawup contingency. For the cancellable drawdown insurance, we derive the investor’s optimal cancellation timing in terms of a two-sided first passage time of the underlying drawdown process. Our model can also be applied to insure against a drawdown by a defaultable stock. We provide analytic formulas for the fair premium and illustrate the impact of default risk.

A Comparison of 2-CUSUM Stopping Rules for Quickest Detection of Two-Sided Alternatives in a Brownian Motion Model

This work considers the problem of quickest detection of signals in a coupled system of N sensors, which receive continuous sequential observations from the environment. It is assumed that the signals, which are modeled a general Itô processes, are coupled across sensors, but that their onset times may differ from sensor to sensor. The objective is the optimal detection of the first time at which any sensor in the system receives a signal. The problem is formulated as a stochastic optimization problem in which an extended average Kullback-Leibler divergence criterion is used as a measure of detection delay, with a constraint on the mean time between false alarms. The case in which the sensors employ cumulative sum (CUSUM) strategies is considered, and it is proved that the minimum of N CUSUMs is asymptotically optimal as the mean time between false alarms increases without bound.

Quickest detection in a system with correlated noise

The current technology in stationary laser range-scanning enables high-resolution acquisition of 3D data in a sequential fashion. Traditionally, range scans are processed offline after acquisition, which significantly slows down the procedure. In this work we alleviate this limitation by developing low-complexity, online detection and classification algorithms. These algorithms are innovative in that they classify points into 5 distinct classes (vegetation, vertical, horizontal, car and curb regions) and robustly determine the level of the ground without requiring any prior training or parameter estimation. To construct these algorithms we extract cleverly chosen summary statistics which significantly reduce the dimensionality of the data. This reduction enables us to contrast the different classes by appropriately chosen Markov models and then use online techniques to detect a transition from one Markov model to the other. The identification of the ground level is further achieved by taking advantage of statistical properties of the distribution of the summary statistics. Our algorithms also use contextual cues to verify the existence of specific classes of objects. All our algorithms take advantage of the sequential nature of data acquisition by running in parallel and labeling points on-the-fly. Thus, these algorithms can be potentially integrated with the scanners hardware and provide the foundation for the construction of high-resolution 3D data scanners that classify data as acquired. We have run experiments using complex urban range scans and have evaluated the classification accuracy against ground-truth.

Quickest detection in the Wiener disorder problem with post-change uncertainty

Abstract This work compares the performance of all existing 2-CUSUM stopping rules used in the problem of sequential detection of a change in the drift of a Brownian motion in the case of two-sided alternatives. As a performance measure, an extended Lorden criterion is used. According to this criterion, the optimal stopping rule is an equalizer rule. This paper compares the performance of the modified drift harmonic mean 2-CUSUM equalizer rules with the performance of the best classical 2-CUSUM equalizer rule whose threshold parameters are chosen so that equalization is achieved. This comparison is made possible through the derivation of a closed-form formula for the expected value of a general classical 2-CUSUM stopping rule.