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215
Extracting macroscopic dynamics: model problems and algorithms
 NONLINEARITY
, 2004
"... In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of macroscopic, or coarsegrained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic ..."
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Cited by 51 (8 self)
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In many applications, the primary objective of numerical simulation of timeevolving systems is the prediction of macroscopic, or coarsegrained, quantities. A representative example is the prediction of biomolecular conformations from molecular dynamics. In recent years a number of new algorithmic approaches have been introduced to extract effective, lowerdimensional, models for the macroscopic dynamics; the starting point is the full, detailed, evolution equations. In many cases the effective lowdimensional dynamics may be stochastic, even when the original starting point is deterministic. This review surveys a number of these new approaches to the problem of extracting effective dynamics, highlighting similarities and differences between them. The importance of model problems for the evaluation of these new approaches is stressed, and a number of model problems are described. When the macroscopic dynamics is stochastic, these model problems are either obtained through a clear separation of timescales, leading to a stochastic effect of the fast dynamics on the slow dynamics, or by considering high dimensional ordinary differential equations which, when projected onto a low dimensional subspace, exhibit stochastic behaviour through the presence of a broad frequency spectrum. Models whose stochastic microscopic behaviour leads to deterministic macroscopic dynamics are also introduced. The algorithms we overview include SVDbased methods for nonlinear problems, model reduction for linear control systems, optimal prediction techniques, asymptoticsbased mode elimination, coarse timestepping methods and transferoperator based methodologies.
Fully nonparametric estimation of scalar diffusion models
 Econometrica
, 2003
"... We propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process. We show how to identify the drift and diffusion function in situations where one or the other functi ..."
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Cited by 48 (7 self)
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We propose a functional estimation procedure for homogeneous stochastic differential equations based on a discrete sample of observations and with minimal requirements on the data generating process. We show how to identify the drift and diffusion function in situations where one or the other function is considered a nuisance parameter. The asymptotic behavior of the estimators is examined as the observation frequency increases and as the time span lengthens. We prove almost sure consistency and weak convergence to mixtures of normal laws, where the mixing variates depend on the chronological local time of the underlying diffusion process, that is the random time spent by the process in the vicinity of a generic spatial point. The estimation method and asymptotic results apply to both stationary and nonstationary recurrent processes.
On the optimal stopping problem for onedimensional diffusions, 2002. Working Paper (http://www.stat.columbia.edu/ ˜ik/DAYKAR.pdf
"... A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the ..."
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Cited by 36 (2 self)
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A new characterization of excessive functions for arbitrary one–dimensional regular diffusion processes is provided, using the notion of concavity. It is shown that excessivity is equivalent to concavity in some suitable generalized sense. This permits a characterization of the value function of the optimal stopping problem as “the smallest nonnegative concave majorant of the reward function ” and allows us to generalize results of Dynkin and Yushkevich for standard Brownian motion. Moreover, we show how to reduce the discounted optimal stopping problems for an arbitrary diffusion process to an undiscounted optimal stopping problem for standard Brownian motion. The concavity of the value functions also leads to conclusions about their smoothness, thanks to the properties of concave functions. One is thus led to a new perspective and new facts about the principle of smooth–fit in the context of optimal stopping. The results are illustrated in detail on a number of non–trivial, concrete optimal stopping problems, both old and new.
Feedback Control of Quantum State Reduction
, 2004
"... Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for th ..."
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Cited by 35 (3 self)
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Feedback control of quantum mechanical systems must take into account the probabilistic nature of quantum measurement. We formulate quantum feedback control as a problem of stochastic nonlinear control by considering separately a quantum filtering problem and a state feedback control problem for the filter. We explore the use of stochastic Lyapunov techniques for the design of feedback controllers for quantum spin systems and demonstrate the possibility of stabilizing one outcome of a quantum measurement with unit probability.
Multiscale stochastic volatility asymptotics
 SIAM J. MULTISCALE MODELING AND SIMULATION
, 2003
"... In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the pri ..."
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Cited by 28 (11 self)
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In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales. The classical BlackScholes formula gives the price of call options when the underlying is a geometric Brownian motion with a constant volatility. The underlying might be the price of a stock or an index say and a constant volatility corresponds to a fixed standard deviation for the random fluctuations in the returns of the underlying. Modern market phenomena makes it important to analyze the situation when this volatility is not fixed but rather is heterogeneous and varies with time. In previous work, see for instance [5], we considered the situation when the volatility is fast mean reverting. Using a singular perturbation expansion we derived an approximation for option prices. We also provided a calibration method using observed option prices as represented by the socalled term structure of implied volatility. Our analysis of market data, however, shows the need for introducing also a slowly varying factor in the model for the stochastic volatility. The combination of regular and singular perturbations approach that we set forth in this paper deals with this case. The resulting approximation is still independent of the particular details of the volatility model and gives more flexibility in the parametrization of the
Modeling a simple choice task: stochastic dynamics of mutually inhibitory neural groups
 Stochastics & Dynamics
, 2001
"... We describe the dynamical and bifurcational behavior of two mutually inhibitory, leaky, neural units subject to external stimulus, random noise, and ‘priming biases. ’ The model describes a simple forced choice experiment and accounts for varying levels of expectation and control. By projecting the ..."
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Cited by 22 (12 self)
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We describe the dynamical and bifurcational behavior of two mutually inhibitory, leaky, neural units subject to external stimulus, random noise, and ‘priming biases. ’ The model describes a simple forced choice experiment and accounts for varying levels of expectation and control. By projecting the model’s dynamics onto slow manifolds, using judicious linear approximations, and solving for onedimensional (reduced) probability densities, analytical estimates are developed for reaction time distributions and shown to compare satisfactorily with ‘full ’ numerical data. A sensitivity analysis is performed and the effects of parameters assessed. The predictions are also compared with behavioral data. These results may help correlate lowdimensional models of stochastic neural networks with cognitive test data, and hence assist in parameter choices and model building. 1 Introduction and
Fluctuation properties of the TASEP with periodic initial configuration
, 2006
"... We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and det ..."
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Cited by 21 (13 self)
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We consider the joint distributions of particle positions for the continuous time totally asymmetric simple exclusion process (TASEP). They are expressed as Fredholm determinants with a kernel defining a signed determinantal point process. We then consider certain periodic initial conditions and determine the kernel in the scaling limit. This result has been announced first in a letter by one of us [27] and here we provide a selfcontained derivation. Connections to last passage directed percolation and random matrices are also briefly discussed.
Optimal liquidity trading
 Rev. Finance
, 2005
"... A liquidity trader wishes to trade a Þxed number of shares within a certain time horizon and to minimize the mean and variance of the costs of trading. Explicit formulas for the optimal trading strategies show that riskaverse liquidity traders reduce their order sizes over time and execute a higher ..."
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Cited by 17 (0 self)
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A liquidity trader wishes to trade a Þxed number of shares within a certain time horizon and to minimize the mean and variance of the costs of trading. Explicit formulas for the optimal trading strategies show that riskaverse liquidity traders reduce their order sizes over time and execute a higher fraction of their total trading volume in early periods when price volatility or liquidity increases. In the presence of transaction fees, numerical simulations suggest that traders want to trade more frequently when price volatility goes up or liquidity declines. In the multiasset case, price effects across assets have a substantial impact on trading behavior, as does continuoustime trading. JEL classiÞcation: