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85
NonEquilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
, 1999
"... . We study the statistical mechanics of a finitedimensional nonlinear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two differ ..."
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Cited by 53 (14 self)
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. We study the statistical mechanics of a finitedimensional nonlinear Hamiltonian system (a chain of anharmonic oscillators) coupled to two heat baths (described by wave equations). Assuming that the initial conditions of the heat baths are distributed according to the Gibbs measures at two different temperatures we study the dynamics of the oscillators. Under suitable assumptions on the potential and on the coupling between the chain and the heat baths, we prove the existence of an invariant measure for any temperature difference, i.e., we prove the existence of steady states. Furthermore, if the temperature difference is sufficiently small, we prove that the invariant measure is unique and mixing. In particular, we develop new techniques for proving the existence of invariant measures for random processes on a noncompact phase space. These techniques are based on an extension of the commutator method of H ormander used in the study of hypoelliptic differential operators. 1. Intr...
Optimal Stopping of Controlled Jump Diffusion Processes: A Viscosity Solution Approach
 Journal of Mathematical Systems, Estimation and Control
, 1998
"... This paper concerns the optimal stopping time problem in a finite horizon of a controlled jump diffusion process. We prove that the value function is continuous and is a viscosity solution of the integrodifferential variational inequality arising from the associated dynamic programming. We also esta ..."
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Cited by 41 (1 self)
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This paper concerns the optimal stopping time problem in a finite horizon of a controlled jump diffusion process. We prove that the value function is continuous and is a viscosity solution of the integrodifferential variational inequality arising from the associated dynamic programming. We also establish comparison principles, which yield uniqueness results. Moreover, the viscosity solution approach allows us to extend maximum principles for linear parabolic integrodifferential operators in C 0 ([0; T ] \Theta IR n ) and to obtain C 1;2 ([0; T ) \Theta IR n ) existence result for the associated Cauchy problem in the nondegenerate case. Key words: stochastic control, viscosity solutions, jumpdiffusion processes AMS Subject Classifications: 93E20, 49L25, 60J75 1 Introduction In this paper, we investigate the optimal stopping time problem of a controlled jump diffusion process and the associated Bellman variational inequality. Let us briefly recall the stochastic background f...
Stochastic Game Theory: Adjustment to Equilibrium Under Noisy Directional Learning
, 1999
"... This paper presents a dynamic model in which agents adjust their decisions in the direction of higher payoffs, subject to random error. This process produces a probability distribution of players' decisions whose evolution over time is determined by the FokkerPlanck equation. The dynamic process is ..."
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Cited by 26 (13 self)
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This paper presents a dynamic model in which agents adjust their decisions in the direction of higher payoffs, subject to random error. This process produces a probability distribution of players' decisions whose evolution over time is determined by the FokkerPlanck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This "logit equilibrium" forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data.
Optimal filtering of jump diffusions: extracting latent states from asset prices
, 2007
"... This paper provides a methodology for computing optimal filtering distributions in discretely observed continuoustime jumpdiffusion models. Although it has received little attention, the filtering distribution is useful for estimating latent states, forecasting volatility and returns, computing mo ..."
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Cited by 23 (5 self)
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This paper provides a methodology for computing optimal filtering distributions in discretely observed continuoustime jumpdiffusion models. Although it has received little attention, the filtering distribution is useful for estimating latent states, forecasting volatility and returns, computing model diagnostics such as likelihood ratios, and parameter estimation. Our approach combines timediscretization schemes with Monte Carlo methods to compute the optimal filtering distribution. Our approach is very general, applying in multivariate jumpdiffusion models with nonlinear characteristics and even nonanalytic observation equations, such as those that arise when option prices are available. We provide a detailed analysis of the performance of the filter, and analyze four applications: disentangling jumps from stochastic volatility, forecasting realized volatility, likelihood based model comparison, and filtering using both option prices and underlying returns.
Stochastic Differential Systems With Memory. Theory, Examples And Applications
 Ustunel, Progress in Probability, Birkhauser
, 1996
"... this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic timelag model with Gaussian noise, and the c ..."
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Cited by 22 (9 self)
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this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic timelag model with Gaussian noise, and the classical "heatbath" model of R. Kubo, modeling the motion of a "large" molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde's). We then establish pathwise existence and uniqueness of solutions to these classes of sfde's under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde's is not covered by classical results of Protter, Metivier and Pellaumail and DoleansDade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from 0. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel
A particle migrating randomly on a sphere
 J. Theoretical Prob
, 1997
"... Consider a particle moving on the surface of the unit sphere in R 3 and heading towards a specific destination with a constant average speed, but subject to random deviations. The motion is modeled as a diffusion with drift restricted to the surface of the sphere. Expressions are set down for variou ..."
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Cited by 21 (11 self)
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Consider a particle moving on the surface of the unit sphere in R 3 and heading towards a specific destination with a constant average speed, but subject to random deviations. The motion is modeled as a diffusion with drift restricted to the surface of the sphere. Expressions are set down for various characteristics of the process including expected travel time to a cap, the limiting distribution, the likelihood ratio and some estimates for parameters appearing in the model. KEY WORDS: Drift; great circle path; likelihood ratio; poleseeking; skew product; spherical Brownian motion; stochastic differential equation; travel time. 1.
Convergence of discretized stochastic (interest rate) processes with stochastic drift term
 Appl. Stochastic Models Data Anal
, 1998
"... For applications in finance, we study the stochastic differential equation dXs = (2βXs+δs)ds+g(Xs)dBs with β a negative real number, g a continuous function vanishing at zero which satisfies a Hölder condition and δ a measurable and adapted stochastic process such that ∫ t 0 δudu < ∞ a.e. for all t ..."
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Cited by 19 (0 self)
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For applications in finance, we study the stochastic differential equation dXs = (2βXs+δs)ds+g(Xs)dBs with β a negative real number, g a continuous function vanishing at zero which satisfies a Hölder condition and δ a measurable and adapted stochastic process such that ∫ t 0 δudu < ∞ a.e. for all t ∈ IR + and which may have a random correlation with the process X itself. In this paper, we concentrate on the Euler discretization scheme for such processes and we study the convergence in L1supnorm and in H1norm towards the solution of the stochastic differential equation with stochastic drift term. We also check the order of strong convergence. KEY WORDS Stochastic differential equation stochastic drift term Hölder condition Euler discretization scheme strong convergence 1.1. Aim of the present study 1.
Applied Stochastic Processes and Control for JumpDiffusions: Modeling, Analysis and Computation
 Analysis and Computation, SIAM Books
, 2007
"... Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions usi ..."
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Cited by 18 (7 self)
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Abstract. An applied compact introductory survey of Markov stochastic processes and control in continuous time is presented. The presentation is in tutorial stages, beginning with deterministic dynamical systems for contrast and continuing on to perturbing the deterministic model with diffusions using Wiener processes. Then jump perturbations are added using simple Poisson processes constructing the theory of simple jumpdiffusions. Next, markedjumpdiffusions are treated using compound Poisson processes to include random marked jumpamplitudes in parallel with the equivalent Poisson random measure formulation. Otherwise, the approach is quite applied, using basic principles with no abstractions beyond Poisson random measure. This treatment is suitable for those in classical applied mathematics, physical sciences, quantitative finance and engineering, but have trouble getting started with the abstract measuretheoretic literature. The approach here builds upon the treatment of continuous functions in the regular calculus and associated ordinary differential equations by adding nonsmooth and jump discontinuities to the model. Finally, the stochastic optimal control of markedjumpdiffusions is developed, emphasizing the underlying assumptions. The survey concludes with applications in biology and finance, some of which are canonical, dimension reducible problems and others are genuine nonlinear problems. Key words. Jumpdiffusions, Wiener processes, Poisson processes, random jump amplitudes, stochastic differential equations, stochastic chain rules, stochastic optimal control AMS subject classifications. 60G20, 93E20, 93E03 1. Introduction. There
Estimation of Continuous Time Models for Stock Returns and Interest Rates
 MACROECONOMIC DYNAMICS
, 1997
"... Efficient Method of Moments (EMM) is used to estimate and test continuous time diffusion models for stock returns and interest rates. For stock returns, a fourstate, twofactor diffusion with one state observed can account for the dynamics of the daily return on the S&P composite index, 19271987. ..."
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Cited by 16 (2 self)
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Efficient Method of Moments (EMM) is used to estimate and test continuous time diffusion models for stock returns and interest rates. For stock returns, a fourstate, twofactor diffusion with one state observed can account for the dynamics of the daily return on the S&P composite index, 19271987. This contrasts with results indicating that discretetime, stochastic volatility models cannot explain these dynamics. For interest rates, a trivariate yield factor model is estimated from weekly, 19621995, Treasury rates. The yield factor model is sharply rejected, although extensions permitting convexities in the local variance come closer to fitting the data.
Atlas models of equity markets
 Ann. Appl. Probab
, 2005
"... Atlastype models are constantparameter models of uncorrelated stocks for equity markets with a stable capital distribution, in which the growth rates and variances depend on rank. The simplest such model assigns the same, constant variance to all stocks; zero rate of growth to all stocks but the s ..."
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Cited by 15 (7 self)
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Atlastype models are constantparameter models of uncorrelated stocks for equity markets with a stable capital distribution, in which the growth rates and variances depend on rank. The simplest such model assigns the same, constant variance to all stocks; zero rate of growth to all stocks but the smallest; and positive growth rate to the smallest, the Atlas stock. In this paper we study the basic properties of this class of models, as well as the behavior of various portfolios in their midst. Of particular interest are portfolios that do not contain the Atlas stock. 1. Introduction. Size