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32
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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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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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 14 (3 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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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
Ódor: On an integrodifferential transform on the sphere
 Studia Sci. Math. Hungar
"... Abstract. In a recent paper the authors have proved that a convex body K ⊂ Rd, d ≥ 2, containing the origin 0 in its interior, is symmetric with respect to 0 if and only if Vd−1(K ∩ H ′) ≥ Vd−1(K ∩ H) for all hyperplanes H, H ′ such that H and H ′ are parallel and H ′ ∋ 0 (Vd−1 is (d − 1)–measure) ..."
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Abstract. In a recent paper the authors have proved that a convex body K ⊂ Rd, d ≥ 2, containing the origin 0 in its interior, is symmetric with respect to 0 if and only if Vd−1(K ∩ H ′) ≥ Vd−1(K ∩ H) for all hyperplanes H, H ′ such that H and H ′ are parallel and H ′ ∋ 0 (Vd−1 is (d − 1)–measure). For the proof the authors have employed a new type of integro–differential transform, that lets to correspond to a sufficiently nice function f on Sd−1 the function R (1) f, where (R(1) f)(ξ) = R (∂f/∂ψ)dη — with ξ ∈ S d−1 as pole and ψ as geographic latitude — and Sd−1∩ξ ⊥ have determined the null–space of the operator R (1). In this paper we extend the definition to any integer m ≥ 1, defining (R(m) f)(ξ) analogously as for m = 1, but using ∂mf/∂ψ m rather than ∂f/∂ψ. (The case m = 0 is the spherical Radon transformation (Funk transformation).) We investigate the null–space of the operator R (m): up to a summand of finite dimension, it consists of the even (odd) functions in the domain of the operator, for m odd (even). For the proof we use spherical harmonics. 1.
On Markovian Traffic with Applications to TES Processes
, 1993
"... Markov processes arc an important ingredient in a variety of stochastic applications. Notable instances include queueing systems and traffic processes offered to them. This paper is concerned with Markovian traffic, i.e., traffic processes whose interarrival times (separating the time points of di ..."
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Cited by 1 (1 self)
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Markov processes arc an important ingredient in a variety of stochastic applications. Notable instances include queueing systems and traffic processes offered to them. This paper is concerned with Markovian traffic, i.e., traffic processes whose interarrival times (separating the time points of discrete arrivals) form a realvalued Markov chain. As such this paper aims to cxtcnd the classical results of renewal traffic, where interarriva] times are assumed to be independent, identically distributed. Following traditional renewal theory, three functions are addressed: the probability of the number of arrivals in a given interval, the corresponding mean number, and the probability of the times of future arrivals. The paper derives integral equations for these functions in the transform domain. These arc then specialized to a subclass, TES +, of a versatile class of random sequences, called TES (TransformExpan&SampIe), consisting of marginally uniform autoregressivc schemes with modu]oi reduction, followed by various transformations. TES models arc designed to simultaneously capture both firstorder and secondorder statistics of empirical records, and consequently can produce highfidelity models. Two theoretical solutions for TES + traffic functions are rived: an operatorbased solution and a matric solution, both in the transform domain. A special case, permitting the conversion of the integral equations to differential equations, is illustrated and solved. Finally, the results are applied to obtain instructive closedform representations for two measures of traffic burstincss: peakedness and index of dispersion, elucidating the relationship between them.
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...
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