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Vyacheslav Girko
May 31, 2011

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Mounir Zili
May 31, 2011

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Henri Schurz
April 17, 2011

### Abstract

Almost sure asymptotic stability of trivial solution and almost sure convergence of stochastic Theta methods applied to bilinear systems of ordinary stochastic differential equations (SDEs) of Itô-type in are proven. For this purpose, we prove and exploit a convergence theorem for non-negative semi-martingale decompositions, and verify a practical criteria based on the uniform boundedness of nonrandom eigenvalues related to certain matrix systems in any dimension d . We do not assume commutativity or simultaneous diagonalizability of drift and diffusion parts as many other authors, neither we restrict our analysis and applicability to only 2D or 3D cases nor to uniform step sizes (since the problem of adequate stochastic test equations cannot be solved within non-anticipative Itô calculus). However, an example of 2D diagonal-noised systems illustrates our approach. The discrete time systems of stochastic Theta methods are driven by L 2 -martingales (i.e. martingale differences, not necessarily Gaussian) and can be interpreted as nonautonomous discretizations (e.g. with variable step sizes or dependence on time).

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Walid Horrigue, Elias Ould Saïd
April 17, 2011

### Abstract

Let ( T, C, X ) be a vector of random variables (rvs) where T, C and X are the interest variable, a right censoring rv and a covariate, respectively. In this paper, we study the kernel conditional quantile estimation in the dependent case and when the covariable takes values in an infinite-dimension space. An estimator of the conditional quantile is given and, under some regularity conditions, among which the small-ball probability for the covariate, its uniform strong convergence with rates is established.

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Soumia Kharfouchi, Karima Kimouche
April 17, 2011

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In the present paper, we establish the central limit theorem for the spatial bilinear process (SBL) noted SBL d ( p, q, P, Q ) under stationary conditions.

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Fuke Wu, Xuerong Mao, Peter E. Kloeden
April 20, 2011

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By the continuous and discrete nonnegative semimartingale convergence theorems, this paper investigates conditions under which the Euler–Maruyama (EM) approximations of stochastic functional differential equations (SFDEs) can share the almost sure exponential stability of the exact solution. Moreover, for sufficiently small stepsize, the decay rate as measured by the Lyapunov exponent can be reproduced arbitrarily accurately.

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Evgeny Ivanko
May 2, 2011

### Abstract

In this article we propose a simple method to estimate the complexity of a finite word written over a finite alphabet. We use the notion of subword complexity (which is equal to the number of different subwords in the word) as a starting point and show the computation difficulties connected with the usage of subword complexity. To avoid them we propose a new simple measure of a word's complexity, which turned out to be not only easy to compute, but also more precise than classical subword complexity.

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Ionuţ Florescu, Ciprian A. Tudor
April 20, 2011

### Abstract

We consider a stochastic volatility model where the volatility process is a fractional Brownian motion. We estimate the memory parameter of the volatility from discrete observations of the price process. We use criteria based on Malliavin calculus in order to characterize the asymptotic normality of the estimators.