Kac Theorem Research Articles

A numerical method for pricing spread options on LIBOR rates with a PDE model

In this paper we present a new numerical method for solving a Black–Scholes type of model for pricing a class of interest rate derivatives: spread options on LIBOR rates. The interest rates are assumed to follow the recently introduced LIBOR Market Model. The Feynman–Kac theorem provides a PDE model for the spread option pricing problem which is initially posed in an unbounded domain. After a localization procedure and the consideration of appropriate boundary conditions in a bounded domain, we propose a Crank–Nicholson characteristic time discretization scheme combined with a Lagrange piecewise quadratic finite element for the spatial discretization. In order to illustrate the performance of the PDE model and the numerical methods, we present a real example of spread option pricing.

Mod-Poisson Convergence in Probability and Number Theory

Building on earlier work introducing the notion of “mod-Gaussian” convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of “mod-Poisson” convergence. We show in particular how it occurs naturally in analytic number theory in the classical Erdős– Kac Theorem. In fact, this case reveals deep connections and analogies with conjectures concerning the distribution of L functions on the critical line, which belong to the mod-Gaussian framework, and with analogues over finite fields, where it can be seen as a zero-dimensional version of the Katz–Sarnak philosophy in the “large conductor” limit.

The Global Statistics of Return Times: Return Time Dimensions Versus Generalized Measure Dimensions

We investigate the relations holding among generalized dimensions of invariant measures in dynamical systems and similar quantities defined by the scaling of global averages of powers of return times. Because of a heuristic use of Kac theorem, these latter have been used in place of the former in numerical and experimental investigations; to mark this distinction, we call them return time dimensions. We derive a full set of inequalities linking measure and return time dimensions and we comment on their optimality with the aid of two maps due to von Neumann -- Kakutani and to Gaspard -- Wang. We conjecture the behavior of return time dimensions in a typical system. We only assume ergodicity of the dynamical system under investigation.

ON YOUNG TOWERS ASSOCIATED WITH INFINITE MEASURE PRESERVING TRANSFORMATIONS

For a σ-finite measure preserving dynamical system (X, μ, T), we formulate necessary and sufficient conditions for a Young tower (Δ, ν, F) to be a (measure theoretic) extension of the original system. Because F is pointwise dual ergodic by construction, one immediate consequence of these conditions is that the Darling–Kac theorem carries over from F to T. One advantage of the Darling–Kac theorem in terms of Young towers is that sufficient conditions can be read off from the tail behavior and we illustrate this with relevant examples. Furthermore, any two Young towers with a common factor T, have return time distributions with tails of the same order.

Two new proofs of the Erdös–Kac Theorem, with bound on the rate of convergence, by Stein's method for distributional approximations

AbstractIn this paper, we apply Stein's method for distributional approximations to prove a quantitative form of the Erdös–Kac Theorem. We obtain our best bound on the rate of convergence, on the order of log log log n (log log n)−1/2, by making an intermediate Poisson approximation; we believe that this approach is simpler and more probabilistic than others, and we also obtain an explicit numerical value for the constant implicit in the bound. Different ways of applying Stein's method to prove the Erdös–Kac Theorem are discussed, including a Normal approximation argument via exchangeable pairs, where the suitability of a Poisson approximation naturally suggests itself.

STATISTICS OF PRIME DIVISORS IN FUNCTION FIELDS

We show that the prime divisors of a random polynomial in 𝔽q[t] are typically "Poisson distributed". This result is analogous to the result in ℤ of Granville [1]. Along the way, we use a sieve developed by Granville and Soundararajan [2] to give a simple proof of the Erdös–Kac theorem in the function field setting. This approach gives stronger results about the moments of the sequence {ω(f)}f∈𝔽q[t] than was previously known, where ω(f) is the number of prime divisors of f.

Entropy-energy decomposition from nonequilibrium work trajectories

We derive expressions for the equilibrium entropy and energy changes in the context of the Jarzynski equality relating nonequilibrium work to equilibrium free energy. The derivation is based on a stochastic path integral technique that reweights paths at different temperatures. Stochastic dynamics generated by either a Langevin equation or a Metropolis Monte Carlo scheme are treated. The approach enables the entropy-energy decomposition from trajectories evolving at a single-temperature and does not require simulations or measurements at two or more temperatures. Both finite difference and analytical formulae are derived. Testing is performed on a prototypical model system and the method is compared with existing thermodynamic integration and thermodynamic perturbation approaches for entropy-energy decomposition. The new formulae are also put in the context of more general, dynamics-independent expressions that derive from either a fluctuation theorem or the Feynman-Kac theorem.

Prime analogues of the Erdős–Kac theorem for elliptic curves

Let E / Q be an elliptic curve. For a prime p of good reduction, let E ( F p ) be the set of rational points defined over the finite field F p . We denote by ω ( # E ( F p ) ) , the number of distinct prime divisors of # E ( F p ) . We prove that the quantity (assuming the GRH if E is non-CM) ω ( # E ( F p ) ) − log log p log log p distributes normally. This result can be viewed as a “prime analogue” of the Erdős–Kac theorem. We also study the normal distribution of the number of distinct prime factors of the exponent of E ( F p ) .

Mean-field description of a dilute s– d system

The Feynman–Kac theorem and Bogolyubov inequality are applied to obtain a lower bound and an upper bound to the free energy of the s– d Hamiltonian with locally smeared interactions between electrons and impurities. The two bounds, which express in terms of the free energy of impurities in a mean field and electrons in a field of barriers and wells localized at the impurity sites, are almost equal if the impurity concentration is sufficiently small or s– d coupling sufficiently weak.

Reconstructing complex materials via effective grain shapes.

We introduce a powerful method based on integral geometry and the Kac theorem for the spectrum of the Laplace operator to define the effective shape of an inclusion in a system made up of a distribution of arbitrarily shaped constituents. Reconstructing the microstructure using the effective inclusion shape leads to an excellent match to the percolation thresholds and to the mechanical and transport properties across all phase fractions. Use of the equivalent shape in effective medium formulations leads to good predictions. The method is verified for a sedimentary rock sample.

Consistency conditions for affine term structure models

Affine term structural models (ATSM) are widely applied for pricing of bonds and interest rate derivatives but the consistency of ATSM when the short rate, r, is unbounded from below remains essentially an open question. First, the standard approach to ATSM uses the Feynman–Kac theorem which is easily applicable only when r is bounded from below. Second, if the tuple of state variables belongs to the region where r is positive, the bond price should decrease in any state variable for which the corresponding coefficient in the formula for r is positive; the bond price should also decrease as the time to maturity increases. In the paper, sufficient conditions for the application of the Feynman–Kac formula, and monotonicity of the bond price are derived, for wide classes of affine term structure models in the pure diffusion case. Necessary conditions for the monotonicity are obtained as well. The results can be generalized for jump-diffusion processes.

A new approach to Kac's theorem on representations of valued quivers

By using the Ringel-Hall algebra approach we first present a new proof of Kac's theorem for species over a finite field. The method used here is quite different from earlier techniques. In [SV] Sevenhant and Van den Bergh have discovered an interesting relation between a conjecture of Kac on representations of quivers and the structure of the Ringel-Hall algebras in some special cases. We second show that this relation still holds for species.

Regenerative processes in supercooled liquids and glasses

The mode-coupling equations used to study glasses and supercooled liquids define the underlying regenerative processes represented by an indicator function Z( t). Such a process is a special case of an alternating renewal process, and it introduces in a natural way a stochastic two level system. In terms of the fundamental Z-process one can define several other processes, such as a local time process H(t)= ∫ 0 t Z(u) du and its inverse process T(t)= sup{u : H(u)⩽t} . At the critical point T c these processes have ergodic limits when t→∞ given by the stable additive process Y a ( t) and its inverse process X a ( t), where a is the critical exponent. These processes are selfsimilar, and the latter is given by the Mittag-Leffler distribution. The appearance of these limit processes, which is a consequence of the Darling–Kac theorem, is the generic reason for the universal predictions of the mode-coupling theory, and are observed in many glassforming systems. We also find a similar behaviour for the α-relaxation function but for the initial behaviour at t→0, and the limit processes are in this case given by Y 1− b and X 1− b , where b is the von Schweidler exponent. This also implies that the relaxation function belongs to the domain of attraction of the stable distribution with the characteristic function exp(− t b ).

Geometry of the Moment Map for Representations of Quivers

We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its Marsden–Weinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deformed preprojective algebras. In an appendix we use deformed preprojective algebras to give a simple proof of much of Kac's Theorem on representations of quivers in characteristic zero.

Integral Geometry and Real Zeros of Thue–Morse Polynomials

We study the average number of intersecting points of a given curve with random hyperplanes in an n-dimensional Euclidean space. As noticed by A. Edelman and E. Kostlan, this problem is closely linked to finding the average number of real zeros of random polynomials. They showed that a real polynomial of degree n has on average log n + O(1) real zeros (M. Kac's theorem). This result leads us to the following problem: given a real sequence (α k ) k ∊N, study the average where ρ(fn) is the number of real zeros of fn(X) = α0+α1X+ … + αnXn. We give theoretical results for the Thue—Morse polynomials and numerical evidence for other polvnomials

Infinite-volume limit of continuous n-particle quantum systems

The Feynman–Kac theorem is applied in order to establish the infinite-volume limit behaviour of the free energy per particle of continuous n-particle quantum systems with bounded separable 2-body interactions defined in the configuration space of particle positions. The mean-field character of such systems is demonstrated. A similar technique is applied to n-particle quantum systems with separable interactions defined in the space of particle momenta and spins. Three examples of systems with separable interactions are given and solved, one of which deals with an electron gas interacting with localized impurity spins in a dilute magnetic alloy (DMA) and extension of Kondo’s resistivity formula for DMA to temperatures close to 0 K. Most of the results are generalizations or more detailed presentations of those published earlier.

Variations on a theme by Mark Kac

Mark Kac's theorem on the mean recurrence time in a stationary stochastic process in discrete time with discrete states is taken as the starting point for a series of variations, most of which are formulated in terms of 0–1 processes. Whereas the original theorem deals with the mean recurrence time of a given state under the condition that the state is realized at time 0, this condition is dropped in part of the variations; two others refer to the variance of the recurrence time and two to the Poincare cycle of a dynamical system. Most variations consist in inequalities and formal identities for the mean first-arrival time and subsequent recurrence times for the given state.

A return time invariant for finitary isomorphisms

AbstractPoincare's recurrence theorem says that, given a measurable subset of a space on which a finite measure-preserving transformation acts, almost every point of the subset returns to the subset after a finite number of applications of the transformation. Moreover, Kac's recurrence theorem refines this result by showing that the average of the first return times to the subset over the subset is at most one, with equality in the ergodic case. In particular, the first return time function to any measurable set is integrable. By considering the supremum over all p ≥ 1 for which the first return time function is p-integrable for all open sets, we obtain a number for each almost-topological dynamical system, which we call the return time invariant. It is easy to show that this invariant is non-decreasing under finitary homomorphism. We use the invariant to construct a continuum number of countable state Markov shifts with a given entropy (and hence measure-theoretically isomorphic) which are pairwise non-finitarily isomorphic.

On an elaboration of M. Kac's theorem concerning eigenvalues of the Laplacian in a region with randomly distributed small obstacles

We removem-balls of centersw 1,...,w m with the same radius α/m from a bounded domain Ω inR 3 with smooth boundary γ. Let μ k (α/m;w(m)) denote thek-th eigenvalue of the Laplacian in Ω/m-balls under the Dirichlet condition. We consider μ k (α/m;w(m)) as a random variable on a probability space (w 1,...,w m)∈Ω × ... × Ω and we examine a precise behaviour of μ k (α/m;w(m)) asm → ∞. We give an elaboration of. M. Kac's theorem.

Two definitions of a determinant and proof of the Szeg�-Kac theorem

Two definitions of a determinant and proof of the Szeg�-Kac theorem