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350
A Quartet of Semigroups for Model Specification, Robustness, Prices of Risk, and Model Detection
 Journal of the European Economic Association
, 2003
"... A representative agent fears that his model, a continuous time Markov process with jump and diffusion components, is misspecified and therefore uses robust control theory to make decisions. Under the decision maker’s approximating model, that cautious behavior puts adjustments for model misspecifica ..."
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Cited by 47 (17 self)
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A representative agent fears that his model, a continuous time Markov process with jump and diffusion components, is misspecified and therefore uses robust control theory to make decisions. Under the decision maker’s approximating model, that cautious behavior puts adjustments for model misspecification into market prices for risk factors. We use a statistical theory of detection to quantify how much model misspecification the decision maker should fear, given his historical data record. A semigroup is a collection of objects connected by something like the law of iterated expectations. The law of iterated expectations defines the semigroup for a Markov process, while similar laws define other semigroups. Related semigroups describe (1) an approximating model; (2) a model misspecification adjustment to the continuation value in the decision maker’s Bellman equation; (3) asset prices; and (4) the behavior of the model detection statistics that we use to calibrate how much robustness the decision maker prefers. Semigroups 2, 3, and 4 establish a tight link between the market price of uncertainty and a bound on the error in statistically discriminating between an approximating and a worst case model.
Large deviation principles and complete equivalence and nonequivalence results for pure and mixed ensembles,
 J. Stat. Phys.
, 2000
"... We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of twodimensional fluid motion, quasigeostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ens ..."
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Cited by 46 (13 self)
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We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of twodimensional fluid motion, quasigeostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates are characterized as the solutions of an unconstrained minimization problem in which the constraint in the first problem is replaced by a Lagrange multiplier. The analysis of equivalence and nonequivalence of ensembles reduces to the following question in global optimization. What are the relationships between the set of solutions of the constrained minimization problem that characterizes microcanonical equilibrium macrostates and the set of solutions of the unconstrained minimization problem that characterizes canonical equilibrium macrostates? In general terms, our main result is that a necessary and sufficient condition for equivalence of ensembles to hold at the level of equilibrium macrostates is that it holds at the level of thermodynamic functions, which is the case if and only if the microcanonical entropy is concave. The necessity of this condition is new and has the following striking formulation. If the microcanonical entropy is not concave at some value of its argument, then the ensembles are nonequivalent in the sense that the corresponding set of microcanonical equilibrium macrostates is disjoint from 999 00224715Â00Â12000999 18.00Â0 2000 Plenum Publishing Corporation
Recursive Robust Estimation and Control Without Commitment
, 2006
"... In a Markov decision problem with hidden state variables, a posterior distribution serves as a state variable and Bayes ’ law under an approximating model gives its law of motion. A decision maker expresses fear that his model is misspecified by surrounding it with a set of alternatives that are nea ..."
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Cited by 44 (9 self)
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In a Markov decision problem with hidden state variables, a posterior distribution serves as a state variable and Bayes ’ law under an approximating model gives its law of motion. A decision maker expresses fear that his model is misspecified by surrounding it with a set of alternatives that are nearby when measured by their expected log likelihood ratios (entropies). Martingales represent alternative models. A decision maker constructs a sequence of robust decision rules by pretending that a sequence of minimizing players choose increments to a martingale and distortions to the prior over the hidden state. A risk sensitivity operator induces robustness to perturbations of the approximating model conditioned on the hidden state. Another risk sensitivity operator induces robustness to the prior distribution over the hidden state. We use these operators to extend the approach of Hansen and Sargent (1995) to problems that contain hidden states. 1
A Variational Representation for Positive Functionals of Infinite Dimensional Brownian Motion
"... A variational representation for positive functionals of a Hilbert space valued Wiener process (W ()) is proved. This representation is then used to prove a large deviations principle for the family {G # (W ())}#>0 where G # is an appropriate family of measurable maps from the Wiener space ..."
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Cited by 43 (7 self)
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A variational representation for positive functionals of a Hilbert space valued Wiener process (W ()) is proved. This representation is then used to prove a large deviations principle for the family {G # (W ())}#>0 where G # is an appropriate family of measurable maps from the Wiener space to some Polish space. Key Words: Large deviations, Laplace principle, stochastic control, cylindrical Brownian motion, stochastic evolution equations, infinite dimensional stochastic calculus. # This research supported in part by the National Science Foundation (NSFDMI9812857) and the University of Notre Dame Faculty Research Program. + This research was supported in part by the National Science Foundation (NSFDMS9704426) and the Army Research O#ce (DAAD199910223). 1 1 Introduction The theory of large deviations is one of the classical areas in probability and statistics (see for example [23, 7, 6, 13, 11]). The book [10] develops an approach to this topic that is based on proving the...
Dynamic importance sampling for queueing networks
, 2005
"... Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a prio ..."
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Cited by 39 (11 self)
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Importance sampling is a technique that is commonly used to speed up Monte Carlo simulation of rare events. However, little is known regarding the design of efficient importance sampling algorithms in the context of queueing networks. The standard approach, which simulates the system using an a priori fixed change of measure suggested by large deviation analysis, has been shown to fail in even the simplest network setting (e.g., a twonode tandem network). Exploiting connections between importance sampling, differential games, and classical subsolutions of the corresponding Isaacs equation, we show how to design and analyze simple and efficient dynamic importance sampling schemes for general classes of networks. The models used to illustrate the approach include dnode tandem Jackson networks and a twonode network with feedback, and the rare events studied are those of large queueing backlogs, including total population overflow and the overflow of individual buffers.
LARGE DEVIATIONS FOR INFINITE DIMENSIONAL STOCHASTIC DYNAMICAL SYSTEMS
"... The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we sho ..."
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Cited by 37 (6 self)
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The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process. 1. Introduction. Small
Law of large number limits for manyserver queues
, 2007
"... Abstract. This work considers a manyserver queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system is represented in terms of a process that describes the total number of customers in the system, as well a ..."
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Cited by 35 (4 self)
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Abstract. This work considers a manyserver queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system is represented in terms of a process that describes the total number of customers in the system, as well as a measurevalued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterised as the unique solution to a coupled pair of integral equations, which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, in the timehomogeneous setting, the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these
Fluid limits of manyserver queues with reneging.
 Ann. Appl. Prob.
, 2010
"... Abstract. This work considers a manyserver queueing system in which impatient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service excee ..."
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Cited by 34 (3 self)
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Abstract. This work considers a manyserver queueing system in which impatient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service exceeds the patience time. The dynamics of the system is represented in terms of a pair of measurevalued processes, one that keeps track of the waiting times of the customers in queue and the other that keeps track of the amounts of time each customer being served has been in service. Under mild assumptions, essentially only requiring that the service and reneging distributions have densities, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is shown to be the unique solution of a coupled pair of deterministic integral equations that admits an explicit representation. In addition, a fluid limit for the virtual waiting time process is also established. This paper extends previous work by Kaspi and Ramanan, which analyzed the model in the absence of reneging. A strong motivation for understanding performance in the presence of reneging arises from models of call centers.
Asymptotic stability of the Wonham filter: ergodic and nonergodic signals
 SIAM J. Control Optim
"... Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the ..."
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Cited by 33 (15 self)
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Abstract. Stability problem of the Wonham filter with respect to initial conditions is addressed. The case of ergodic signals is revisited in view of a gap in the classic work of H. Kunita (1971). We give new bounds for the exponential stability rates, which do not depend on the observations. In the nonergodic case, the stability is implied by identifiability conditions, formulated explicitly in terms of the transition intensities matrix and the observation structure. Key words. Nonlinear filtering, stability, Wonham filter
Axiomatic foundations of multiplier preferences
, 2007
"... This paper axiomatizes the robust control criterion of multiplier preferences introduced by Hansen and Sargent (2001). The axiomatization relates multiplier preferences to other classes of preferences studied in decision theory. Some properties of multiplier preferences are generalized to the broade ..."
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Cited by 33 (3 self)
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This paper axiomatizes the robust control criterion of multiplier preferences introduced by Hansen and Sargent (2001). The axiomatization relates multiplier preferences to other classes of preferences studied in decision theory. Some properties of multiplier preferences are generalized to the broader class of variational preferences, recently introduced by Maccheroni, Marinacci and Rustichini (2006). The paper also establishes a link between the parameters of the multiplier criterion and the observable behavior of the agent. This link enables measurement of the parameters on the basis of observable choice data and provides a useful tool for applications. I am indebted to my advisor Eddie Dekel for his continuous guidance, support, and encouragement. I am grateful to Peter Klibanoff and Marciano Siniscalchi for many discussions which resulted in significant improvements of the paper. I would also like to thank Jeff Ely and Todd Sarver for helpful comments and suggestions. This project started after a very stimulating conversation with Tom Sargent and was further shaped by conversations with Lars Hansen. All errors are my own.