Papers by Olympia Hadjiliadis
Stochastics
We consider the problem of quickest detection of an abrupt change when there is uncertainty about... more We consider the problem of quickest detection of an abrupt change when there is uncertainty about the post-change distribution. In particular, we examine this problem in the continuous-time Wiener model where the drift of observations changes from zero to a random drift with a prescribed discrete distribution. We set up the problem as a stochastic optimization in which the objective is to minimize a measure of detection delay subject to a constraint on frequency of false alarms. We design a novel composite stopping rule and prove that it is asymptotically optimal of third order under a weighted Lorden's criterion for detection delay. We also develop the strategy to identify the post-change drift and analyze the conditional identification error asymptotically. Our composite rules are based on CUSUM stopping times, as well as their reaction periods, namely the times between the last reset of the CUSUM statistic process and the CUSUM alarm. The established results shed new light on the performance of CUSUM strategies under model uncertainty and offer strong asymptotic optimality results in this framework.
53rd IEEE Conference on Decision and Control, 2014
We consider the problem of detection of abrupt changes when there is uncertainty about the post-c... more We consider the problem of detection of abrupt changes when there is uncertainty about the post-change distribution. In particular we examine this problem in the prototypical model of continuous time in which the drift of a Wiener process changes at an unknown time from zero to a random value. It is assumed that the change time is an unknown constant while the drift assumed after the change has a Bernoulli distribution with all values of the same sign independent of the process observed. We set up the problem as a stochastic optimization in which the objective is to minimize a measure of detection delay subject to a frequency of false alarm constraint. As a measure of detection delay we consider that of a worst detection delay weighed by the probabilities of the different possible drift values assumed after the change point to which we are able to compute a lower bound amongst the class of all stopping times. Our objective is to then construct low complexity, easy to implement decision rules, that achieve this lower bound exactly, while maintaining the same frequency of false alarms as the family of stopping times. In this effort, we consider a special class of decision rules that are delayed versions of CUSUM algorithm. In this enlarged collection, we are able to construct a family of computationally efficient decision rules that achieve the lower bound with equality, and then choose a best one whose performance is as close to the performance of a stopping time as possible.
Theory of Probability & Its Applications, 2009
and-conditions-of-access.pdf This article may be used for research, teaching and private study pu... more and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
We study a Wiener disorder problem of detecting the minimum of N change-points in N observation c... more We study a Wiener disorder problem of detecting the minimum of N change-points in N observation channels coupled by correlated noises. It is assumed that the observations in each dimension can have different strengths and that the change-points may differ from channel to channel. The objective is the quickest detection of the minimum of the N change-points. We adopt a min-max approach and consider an extended Lorden's criterion, which is minimized subject to a constraint on the mean time to the first false alarm. It is seen that, under partial information of the post-change drifts and a general nonsingular stochastic correlation structure in the noises, the minimum of N cumulative sums (CUSUM) stopping rules is asymptotically optimal as the mean time to the first false alarm increases without bound. We further discuss applications of this result with emphasis on its implications to the efficiency of the decentralized versus the centralized systems of observations which arise in engineering.
We examine the situation when the investor wants to outperform a certain benchmark by actively tr... more We examine the situation when the investor wants to outperform a certain benchmark by actively trading in this asset, typically a stock index. We consider an investor who wants to minimize the expected shortfall in the case he fails to achieve this goal. Using recently developed techniques of Föllmer and Leukert, we can relate this optimal investment problem to option hedging. This allows us to obtain analytical characterizations of the optimal strategy in special cases.
Stochastic Processes and Their Applications, 2009
This paper studies drawdown and drawup processes in a general diffusion model. The main result is... more This paper studies drawdown and drawup processes in a general diffusion model. The main result is a formula for the joint distribution of the running minimum and the running maximum of the process stopped at the time of the first drop of size a. As a consequence, we obtain the probabilities that a drawdown of size a precedes a drawup of size b and vice versa. The results are applied to several examples of diffusion processes, such as drifted Brownian motion, Ornstein-Uhlenbeck process, and Cox-Ingersoll-Ross process.
Methodology and Computing in Applied Probability, 2010
In this work we derive the probability that a rally of a units precedes a drawdown of equal units... more In this work we derive the probability that a rally of a units precedes a drawdown of equal units in a random walk model and its continuous equivalent, a Brownian motion model in the presence of a finite time-horizon. A rally is defined as the difference of the present value of the holdings of an investor and its historical minimum, while the drawdown is defined as the difference of the historical maximum and its present value. We discuss applications of these results in finance and in particular risk management.
The drawdown of an asset is a risk measure defined in terms of the running maximum of the asset's... more The drawdown of an asset is a risk measure defined in terms of the running maximum of the asset's spot price over some period [0, T ]. The asset price is said to have drawn down by at least $K over this period if there exists a time at which the underlying is at least $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.
This work examines the problem of sequential change detection in the constant drift of a Brownian... more 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 Lorden's criterion is proposed. When the possible drifts, assumed after the change, have the same sign, the 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-CUSUM rule is shown to be asymptotically optimal as the frequency of false alarms tends to infinity.
In this work we study drawdowns and drawups of general diffusion processes. The drawdown process ... more In this work we study drawdowns and drawups of general diffusion processes. The drawdown process is defined as the current drop of the process from its running maximum, while the drawup process is defined as the current increase over its running minimum. The drawdown and the drawup are the first hitting times of the drawdown and the drawup processes respectively. In particular, we derive a closed-form formula for the Laplace transform of the probability density of the drawdown of a units when it precedes the drawup of b units. We then separately consider the special case of drifted Brownian motion, for which we derive a closed form formula for the above-mentioned density by inverting the Laplace transform. Finally, we apply the results to a problem of interest in financial risk-management and to the problem of transient signal detection and identification of two-sided changes in the drift of general diffusion processes.
IEEE Transactions on Information Theory, 2009
This work considers the problem of quickest detection with N distributed sensors that receive seq... more 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.
Computing Research Repository, 2008
This work considers the problem of quickest detection with N distributed sensors that receive con... more This work considers the problem of quickest detection with N distributed sensors that receive continuous sequential observations 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, after which they must 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. It is shown that there is no loss of performance of one shot schemes as compared to the centralized case in an extended Lorden minmax sense, since the minimum of N CUSUMs is asymptotically optimal as the mean time between false alarms increases without bound.
This work considers the problem of quickest detection in a coupled system of N sensors, which rec... more This work considers the problem of quickest detection in a coupled system of N sensors, which receive continuous sequential observations from the environment. The signals received by each sensor are assumed to have the same distribution which allows for a general dependence structure in the observations (e.g. continuous-time analogs of Gaussian autoregressive processes). In particular, the signals are modeled through general Itô processes. Linear state space systems commonly used for structural health -monitoring are special cases of the models assumed. Moreover, the observations received in one sensor, after the onset of a signal (time of a change point in the law of the observations), are coupled by those received in another. Yet, the change points in each sensor can be different because whatever causes the signal to start may arrive at different sensors of the system at different times and thus may be felt at only one sensor. The objective in this paper is the optimal detection of the first time in which the system receives a signal. The problem is formulated as a stochastic optimization problem in which is used as a performance index of detection delay an extended Kullback-Leibler divergence criterion. We consider 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.
Computing Research Repository, 2007
Theory of Probability and Its Applications, 2009
Sequential Analysis, 2009
This work compares the performance of all existing 2-CUSUM stopping rules used in the problem of ... more 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.
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Papers by Olympia Hadjiliadis