Construction of semimartingales from pieces by the method of excursion point processes
We define a class of step cocycles (which are coboundaries) for irrational rotations of the unit circle and give conditions for their approximation by smooth and real analytic coboundaries. The transfer functions of the approximating (smooth and real analytic) coboundaries are close (in the supremum norm) to the transfer functions of the original ones. This result makes it possible to construct smooth and real analytic cocycles which are ergodic, ergodic and squashable (see [Aaronson, Lemańczyk,...
In this article, we study the approximation of a probability measure on by its empirical measure interpreted as a random quantization. As error criterion we consider an averaged th moment Wasserstein metric. In the case where , we establish fine upper and lower bounds for the error, ahigh resolution formula. Moreover, we provide a universal estimate based on moments, a Pierce type estimate. In particular, we show that quantization by empirical measures is of optimal order under weak assumptions....
Using the concept of Hellinger integrals, necessary and sufficient conditions are established for the contiguity of two sequences of distributions of Poisson point processes with an arbitrary state space. The distribution of logarithm of the likelihood ratio is shown to be infinitely divisible. The canonical measure is expressed in terms of the intensity measures. Necessary and sufficient conditions for the LAN-property are formulated in terms of the corresponding intensity measures.
The continuity of the solutions of difference and algebraic coupled Riccati equations for the discrete-time Markovian jump linear quadratic control problem as a function of coefficients is verified. The line of reasoning goes through the use of the minimum property formulated analogously to the one for coupled continuous Riccati equations presented by Wonham and a set of comparison theorems.
Let be a Brownian motion, and let be the space of all continuous periodic functions with period 1. It is shown that the set of all such that the stochastic convolution , does not have a modification with bounded trajectories, and consequently does not have a continuous modification, is of the second Baire category.
Let denote the generator of the rotation group in the space , where denotes the unit circle. We show that the stochastic Cauchy problem where is a standard Brownian motion and is fixed, has a weak solution if and only if the stochastic convolution process has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all...
We study γk(x2, …, xk; t), the k-fold renormalized self-intersection local time for brownian motion in R1. Our main result says that γk(x2, …, xk; t) is continuously differentiable in the spatial variables, with probability 1.
We investigate the state feedback stabilization, in the sense of weak solution, of nonlinear stochastic systems when the drift is quadratic in the control and the diffusion term is affine in the control. Based on the generalised stochastic Lyapunov theorem, we derive the necessary conditions and the sufficient conditions, respectively, for the global asymptotic stabilization in probability by a continuous feedback explicitly computed. The interest of this work is that the existing control methods...