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Testing ContinuousTime Models of the Spot Interest Rate
 Review of Financial Studies
, 1996
"... Different continuoustime models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuoustime model by discrete approximations, even though the data are rec ..."
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Cited by 302 (10 self)
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Different continuoustime models for interest rates coexist in the literature. We test parametric models by comparing their implied parametric density to the same density estimated nonparametrically. We do not replace the continuoustime model by discrete approximations, even though the data are recorded at discrete intervals. The principal source of rejection of existing models is the strong nonlinearity of the drift. Around its mean, where the drift is essentially zero, the spot rate behaves like a random walk. The drift then meanreverts strongly when far away from the mean. The volatility is higher when away from the mean. The continuoustime financial theory has developed extensive tools to price derivative securities when the underlying traded asset(s) or nontraded factor(s) follow stochastic differential equations [see Merton (1990) for examples]. However, as a practical matter, how to specify an appropriate stochastic differential equation is for the most part an unanswered question. For example, many different continuoustime The comments and suggestions of Kerry Back (the editor) and an anonymous referee were very helpful. I am also grateful to George Constantinides,
The Variational Formulation of the FokkerPlanck Equation
 SIAM J. Math. Anal
, 1999
"... The FokkerPlanck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of timedependent systems in which randomness plays a role. In this paper, ..."
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Cited by 286 (22 self)
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The FokkerPlanck equation, or forward Kolmogorov equation, describes the evolution of the probability density for a stochastic process associated with an Ito stochastic differential equation. It pertains to a wide variety of timedependent systems in which randomness plays a role. In this paper, we are concerned with FokkerPlanck equations for which the drift term is given by the gradient of a potential. For a broad class of potentials, we construct a timediscrete, iterative variational scheme whose solutions converge to the solution of the FokkerPlanck equation. The major novelty of this iterative scheme is that the time step is governed by the Wasserstein metric on probability measures. This formulation enables us to reveal an appealing, and previously unexplored, relationship between the FokkerPlanck equation and the associated free energy functional. Namely, we demonstrate that the dynamics may be regarded as a gradient flux, or a steepest descent, for the free energy wi...
The physics of optimal decision making: A formal analysis of models of performance in twoalternative forced choice tasks
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Adaptive Greedy Approximations
"... The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the ..."
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Cited by 187 (0 self)
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The problem of optimally approximating a function with a linear expansion over a redundant dictionary of waveforms is NPhard. The greedy matching pursuit algorithm and its orthogonalized variant produce suboptimal function expansions by iteratively choosing dictionary waveforms that best match the function's structures. A matching pursuit provides a means of quickly computing compact, adaptive function approximations. Numerical experiments show that the approximation errors from matching pursuits initially decrease rapidly, but the asymptotic decay rate of the errors is slow. We explain this behavior by showing that matching pursuits are chaotic, ergodic maps. The statistical properties of the approximation errors of a pursuit can be obtained from the invariant measure of the pursuit. We characterize these measures using group symmetries of dictionaries and by constructing a stochastic differential equation model. We derive a notion of the coherence of a signal with respect to a dict...
HighOrder Collocation Methods for Differential Equations with Random Inputs
 SIAM Journal on Scientific Computing
"... Abstract. Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling met ..."
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Cited by 180 (9 self)
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Abstract. Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementations of stochastic Galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations. On the other hand, the traditional sampling methods, e.g., Monte Carlo methods, are straightforward to implement, but they do not offer convergence as fast as stochastic Galerkin methods. In this paper, a highorder stochastic collocation approach is proposed. Similar to stochastic Galerkin methods, the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence. However, the numerical implementation of stochastic collocation is trivial, as it requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods. The computational cost of the collocation methods depends on the choice of the collocation points, and we present several feasible constructions. One particular choice, based on sparse grids, depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs. Numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods. Key words. collocation methods, stochastic inputs, differential equations, uncertainty quantification
An introduction to collective intelligence
 Handbook of Agent technology. AAAI
, 1999
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Regulating Stock Externalities Under Uncertainty
 Journal of Environmental Economics and Management
, 2000
"... Using a simple analytical model incorporating benefits of a stock, costs of adjusting the stock, and uncertainty in costs, we uncover several important principles governing the choice of pricebased policies (e.g., taxes) relative to quantitybased policies (e.g., tradable permits) for controlling st ..."
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Cited by 132 (20 self)
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Using a simple analytical model incorporating benefits of a stock, costs of adjusting the stock, and uncertainty in costs, we uncover several important principles governing the choice of pricebased policies (e.g., taxes) relative to quantitybased policies (e.g., tradable permits) for controlling stock externalities. Applied to the problem of greenhouse gases and climate change, we find that a pricebased instrument generates several times the expected net benefits of a quantity instrument. As in Weitzman (1974), the relative slopes of the marginal benefits and costs of controlling the externality continue to be critical determinants of the efficiency of prices relative to quantities, with flatter marginal benefits and steeper marginal costs favoring prices. But some important adjustments for dynamic effects are necessary, including correlation of cost shocks across time, discounting, stock decay, and the rate of benefits growth. We also demonstrate an important link between instrume...
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 111 (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.
Realtime kinetics of gene activity in individual bacteria
 Cell
, 2005
"... Protein levels have been shown to vary substantially between individual cells in clonal populations. In prokaryotes, the contribution to such fluctuations from the inherent randomness of gene expression has largely been attributed to having just a few transcriptsofthecorrespondingmRNAs.Bycontrast, e ..."
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Cited by 106 (0 self)
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Protein levels have been shown to vary substantially between individual cells in clonal populations. In prokaryotes, the contribution to such fluctuations from the inherent randomness of gene expression has largely been attributed to having just a few transcriptsofthecorrespondingmRNAs.Bycontrast, eukaryotic studies tend to emphasize chromatin remodeling and burstlike transcription. Here, we study singlecell transcription in Escherichia coli by measuring mRNA levels in individual living cells. The results directly demonstrate transcriptional bursting, similar to that indirectly inferred for eukaryotes.We also measure mRNA partitioning at cell division and correlate mRNA and protein levels in single cells. Partitioning is approximately binomial, and mRNAprotein correlations are weaker earlier in the cell cycle, where cell division has recently randomized the relative concentrations. Our methods further extend proteinbased approaches by counting the integervalued number of transcript with singlemolecule resolution. This greatly facilitates kinetic interpretations in terms of the integervalued random processes that produce the fluctuations.
Statistical Mechanics of Dissipative Particle Dynamics
, 1995
"... The stochastic differential equations corresponding to the updating algorithm of Dissipative Particle Dynamics (DPD), and the corresponding FokkerPlanck equation are derived. It is shown that a slight modification to the algorithm is required before the Gibbs distribution is recovered as the statio ..."
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Cited by 91 (1 self)
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The stochastic differential equations corresponding to the updating algorithm of Dissipative Particle Dynamics (DPD), and the corresponding FokkerPlanck equation are derived. It is shown that a slight modification to the algorithm is required before the Gibbs distribution is recovered as the stationary solution to the FokkerPlanck equation. The temperature of the system is then directly related to the noise amplitude by means of a fluctuationdissipation theorem. However, the correspondingly modified, discrete DPD algorithm is only found to obey these predictions if the length of the timestep is sufficiently reduced. This indicates the importance of time discretisation in DPD. Recently, Hoogerbrugge and Koelman have introduced a new method for simulating hydrodynamic behavior which has been coined Dissipative Particle Dynamics (DPD)[1],[2]. This technique was conceived as an improvement over conventional molecular dynamics MD in order to describe complex hydrodynamic behavior with c...