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On belated differentiation and a characterization of Henstock-Kurzweil-Ito integrable processes

Tin-Lam Toh, Tuan-Seng Chew (2005)

Mathematica Bohemica

The Henstock-Kurzweil approach, also known as the generalized Riemann approach, has been successful in giving an alternative definition to the classical Itô integral. The Riemann approach is well-known for its directness in defining integrals. In this note we will prove the Fundamental Theorem for the Henstock-Kurzweil-Itô integral, thereby providing a characterization of Henstock-Kurzweil-Itô integrable stochastic processes in terms of their primitive processes.

On bilinear forms based on the resolvent of large random matrices

Walid Hachem, Philippe Loubaton, Jamal Najim, Pascal Vallet (2013)

Annales de l'I.H.P. Probabilités et statistiques

Consider a N × n non-centered matrix 𝛴 n with a separable variance profile: 𝛴 n = D n 1 / 2 X n D ˜ n 1 / 2 n + A n . Matrices D n and D ˜ n are non-negative deterministic diagonal, while matrix A n is deterministic, and X n is a random matrix with complex independent and identically distributed random variables, each with mean zero and variance one. Denote by Q n ( z ) the resolvent associated to 𝛴 n 𝛴 n * , i.e. Q n ( z ) = 𝛴 n 𝛴 n * - z I N - 1 . Given two sequences of deterministic vectors ( u n ) and ( v n ) with bounded Euclidean norms, we study the limiting behavior of the random bilinear form: u n * Q n ( z ) v n ∀ z ∈ ℂ - ℝ + , as the dimensions...

On Billard's Theorem for Random Fourier Series

Guy Cohen, Christophe Cuny (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that Billard's theorem on a.s. uniform convergence of random Fourier series with independent symmetric coefficients is not true when the coefficients are only assumed to be centered independent. We give some necessary or sufficient conditions to ensure the validity of Billard's theorem in the centered case.

On calculation of stationary density of autoregressive processes

Jiří Anděl, Karel Hrach (2000)

Kybernetika

An iterative procedure for computation of stationary density of autoregressive processes is proposed. On an example with exponentially distributed white noise it is demonstrated that the procedure converges geometrically fast. The AR(1) and AR(2) models are analyzed in detail.

On calculation of zeta function of integral matrix

Jiří Janáček (2009)

Mathematica Bohemica

Values of the Epstein zeta function of a positive definite matrix and the knowledge of matrices with minimal values of the Epstein zeta function are important in various mathematical disciplines. Analytic expressions for the matrix theta functions of integral matrices can be used for evaluation of the Epstein zeta function of matrices. As an example, principal coefficients in asymptotic expansions of variance of the lattice point count in the random ball are calculated for some lattices.

On certain Markov processes attached to exponential functionals of Brownian motion: application to Asian options.

Catherine Donati-Martin, Raouf Ghomrasni, Marc Yor (2001)

Revista Matemática Iberoamericana

We obtain a closed formula for the Laplace transform of the first moment of certain exponential functionals of Brownian motion with drift, which gives the price of Asian options. The proof relies on an identity in law between the average on [0,t] of a geometric Brownian motion and the value at time t of a Markov process, for which we can compute explicitly the resolvent.

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