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21
On the American option problem
 Math. Finance
, 2005
"... We show how the changeofvariable formula with local time on curves derived recently in [17] can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium repre ..."
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Cited by 17 (7 self)
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We show how the changeofvariable formula with local time on curves derived recently in [17] can be used to prove that the optimal stopping boundary for the American put option can be characterized as the unique solution of a nonlinear integral equation arising from the early exercise premium representation. This settles the question raised in [15] (dating back to [13]). 1.
ON THE NUMERICAL EVALUATION OF FREDHOLM DETERMINANTS
, 804
"... Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical ..."
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Cited by 10 (5 self)
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Abstract. Some significant quantities in mathematics and physics are most naturally expressed as the Fredholm determinant of an integral operator, most notably many of the distribution functions in random matrix theory. Though their numerical values are of interest, there is no systematic numerical treatment of Fredholm determinants to be found in the literature. Instead, the few numerical evaluations that are available rely on eigenfunction expansions of the operator, if expressible in terms of special functions, or on alternative, numerically more straightforwardly accessible analytic expressions, e.g., in terms of Painlevé transcendents, that have masterfully been derived in some cases. In this paper we close the gap in the literature by studying projection methods and, above all, a simple, easily implementable, general method for the numerical evaluation of Fredholm determinants that is derived from the classical Nyström method for the solution of Fredholm equations of the second kind. Using Gauss–Legendre or Clenshaw– Curtis as the underlying quadrature rule, we prove that the approximation error essentially behaves like the quadrature error for the sections of the kernel. In particular, we get exponential convergence for analytic kernels, which are typical in random matrix theory. The application of the method to the distribution functions of the Gaussian unitary ensemble (GUE), in the bulk and the edge scaling limit, is discussed in detail. After extending the method to systems of integral operators, we evaluate the twopoint correlation functions of the more recently studied Airy and Airy 1 processes. Key words. Fredholm determinant, Nyström’s method, projection method, trace class operators, random
Wijland: Thermodynamic formalism for systems with Markov dynamics
 J. Stat. Phys
, 2008
"... The thermodynamic formalism allows one to access the chaotic properties of equilibrium and outofequilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not sui ..."
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Cited by 8 (2 self)
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The thermodynamic formalism allows one to access the chaotic properties of equilibrium and outofequilibrium systems, by deriving those from a dynamical partition function. The definition that has been given for this partition function within the framework of discrete time Markov chains was not suitable for continuous time Markov dynamics. Here we propose another interpretation of the definition that allows us to apply the thermodynamic formalism to continuous time. We also generalize the formalism –a dynamical Gibbs ensemble construction– to a whole family of observables and their associated large deviation functions. This allows us to make the connection between the thermodynamic formalism and the observable involved in the muchstudied fluctuation theorem. We illustrate our approach on various physical systems: random walks, exclusion processes, an Ising model and the contact process. In the latter cases, we identify a signature of the occurrence of dynamical phase transitions. We show that this signature can already be unraveled using the simplest dynamical ensemble one could define, based on the number of configuration changes a system has undergone over an asymptotically large time window. 1 1
The support of the equilibrium measure for a class of external fields on a finite interval
 Pacific J. Math
"... We investigate the support of the equilibrium measure associated with a class of nonconvex, nonsmooth external fields on a finite interval. Such equilibrium measures play an important role in various branches of analysis. In this paper we obtain a sufficient condition which ensures that the support ..."
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Cited by 4 (2 self)
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We investigate the support of the equilibrium measure associated with a class of nonconvex, nonsmooth external fields on a finite interval. Such equilibrium measures play an important role in various branches of analysis. In this paper we obtain a sufficient condition which ensures that the support consists of at most two intervals. This is applied to external fields of the form −c sign(x)x  α with c>0, α ≥ 1 and x ∈ [−1, 1]. If α is an odd integer, these external fields are smooth, and for this case the support was studied before by Deift, Kriecherbauer and McLaughlin, and by Damelin and Kuijlaars. 1. Introduction. In recent years, equilibrium measures with external fields have found an increasing number of applications in a variety of areas. We refer to [2, 3, 4, 5, 8, 10, 14, 15] for these relations, ranging from classical topics as
Integration through transients for Brownian particles under steady shear
 J. Phys.: Condens. Matter
, 2005
"... Abstract. Starting from the microscopic Smoluchowski equation for interacting Brownian particles under stationary shearing, exact expressions for shear– dependent steady–state averages, correlation and structure functions, and susceptibilities are obtained, which take the form of generalized Green–K ..."
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Abstract. Starting from the microscopic Smoluchowski equation for interacting Brownian particles under stationary shearing, exact expressions for shear– dependent steady–state averages, correlation and structure functions, and susceptibilities are obtained, which take the form of generalized Green–Kubo relations. They require integration of transient dynamics. Equations of motion with memory effects for transient density fluctuation functions are derived from the same microscopic starting point. We argue that the derived formal expressions provide useful starting points for approximations in order to describe the stationary non–equilibrium state of steadily sheared dense colloidal dispersions.
Nonlocal Electrodynamics of Linearly Accelerated Systems
, 2008
"... The measurement of an electromagnetic radiation field by a linearly accelerated observer is discussed. The nonlocality of this process is emphasized. The nonlocal theory of accelerated observers is briefly described and the consequences of this theory are illustrated using a concrete example involvi ..."
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The measurement of an electromagnetic radiation field by a linearly accelerated observer is discussed. The nonlocality of this process is emphasized. The nonlocal theory of accelerated observers is briefly described and the consequences of this theory are illustrated using a concrete example involving the measurement of an incident pulse of radiation by an observer that experiences uniform acceleration during a limited interval of time. 1
The Run Probabilities Of TES Processes
 Stochastic Models
, 1994
"... The run statistics of a discretetime, realvalued stochastic process are the statistics of process excursions above a given level. As such they are a special case of first passage times (hitting times) in discrete time. The study of run probabilities is motivated by applications such as compressed ..."
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The run statistics of a discretetime, realvalued stochastic process are the statistics of process excursions above a given level. As such they are a special case of first passage times (hitting times) in discrete time. The study of run probabilities is motivated by applications such as compressed video, where a random and autocorrelated sequence of compressed frames arrives deterministically at a finite buffer, and the loss probability of consecutive frames (runs) constitutes a better measure of service quality than simple loss probabilities. This paper studies the run probabilities of a subclass of TES processes. A uniform TES process is a modulo1 autoregressive stochastic process, uniform on [0; 1); general TES processes are obtained by transforming a basic TES process to ones with general marginals. The paper develops an integral equation in the generating function of the run probabilities of TES processes. An exact matrix solution of theoretical interest is obtained, but the sol...
On a Singular Integrodifferential Equation arising from a Linearised Free Surface Problem
, 2005
"... A problem of linear surface waves discussed by Forbes [6] initially gave rise to a singular integrodifferential equation over the real line. We have been able to transform this integrodifferential equation into a linear second order differential equation whose solution has been found explicitly in t ..."
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A problem of linear surface waves discussed by Forbes [6] initially gave rise to a singular integrodifferential equation over the real line. We have been able to transform this integrodifferential equation into a linear second order differential equation whose solution has been found explicitly in terms of the sine and cosine integral functions. Furthermore, we have been able to recover known results of physical
Vacuum Electrodynamics of Accelerated Systems: Nonlocal Maxwell’s Equations
, 2008
"... The nonlocal electrodynamics of accelerated systems is discussed in connection with the development of Lorentzinvariant nonlocal field equations. Nonlocal Maxwell’s equations are presented explicitly for certain linearly accelerated systems. In general, the field equations remain nonlocal even afte ..."
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The nonlocal electrodynamics of accelerated systems is discussed in connection with the development of Lorentzinvariant nonlocal field equations. Nonlocal Maxwell’s equations are presented explicitly for certain linearly accelerated systems. In general, the field equations remain nonlocal even after accelerated motion has ceased. PACS numbers: 03.30.+p, 11.10.Lm
THE BOUNDARIES OF THE SOLUTIONS OF THE LINEAR VOLTERRA INTEGRAL EQUATIONS WITH CONVOLUTION KERNEL
"... Abstract. Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of a ..."
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Abstract. Some boundaries about the solution of the linear Volterra integral equations of the second type with unit source term and positive monotonically increasing convolution kernel were obtained in Ling, 1978 and 1982. A method enabling the expansion of the boundary of the solution function of an equation in this type was developed in I. Özdemir and Ö. F. Temizer, 2002. In this paper, by using the method in Özdemir and Temizer, it is shown that the boundary of the solution function of an equation in the same form can also be expanded under different conditions than those that they used. 1.