In the paper, we discuss convergence properties and Voronovskaja type theorem for bivariate $q$-Bernstein polynomials for a function analytic in the polydisc $D_{R_{1}}\times D_{R_{2}}=\{z\in C\colon \vert z\vert <R_{1}\} \times \{ z\in C\colon \vert z\vert <R_{1}\}$ for arbitrary fixed $q>1$. We give quantitative Voronovskaja type estimates for the bivariate $q$-Bernstein polynomials for $q>1$. In the univariate case the similar results were obtained by S. Ostrovska: $q$-Bernstein polynomials and their iterates. J. Approximation Theory 123 (2003), 232–255. and S. G. Gal: Approximation by Complex Bernstein and Convolution Type Operators. Series on Concrete and Applicable Mathematics 8. World Scientific, New York, 2009.
The paper deals with a class of discrete-time stochastic control processes under a discounted optimality criterion with random discount rate, and possibly unbounded costs. The state process { x_{t} } and the discount process { \alpha_{t} } evolve according to the coupled difference equations x_{t+1}=F(x_{t},\alpha _{t},a_{t},\xi _{t}), \alpha_{t+1}=G(\alpha _{t},\eta _{t}) where the state and discount disturbance processes { \xi _{t} } and { \eta _{t} } are sequences of i.i.d. random variables with densities \rho^{\xi } and \rho^{\eta } respectively. The main objective is to introduce approximation algorithms of the optimal cost function that lead up to construction of optimal or nearly optimal policies in the cases when the densities \rho ^{\xi } and \rho ^{\eta } are either known or unknown. In the latter case, we combine suitable estimation methods with control procedures to construct an asymptotically discounted optimal policy.
In this paper, we show two applications of the Banach's Fixed-Point Theorem: first, to approximate the ultimate ruin probability in the classical risk model or Cramér-Lundberg model when claim sizes have some arbitrary continuous distribution and second, we propose an algorithm based in this theorem and some conditions to guarantee the continuity of the ruin probability with respect to the weak metric (Kantorovich). In risk theory literature, there is no methodology based in the Banach's Fixed-Point Theorem to calculate the ruin probability. Numerical results in this paper, guarantee a good approximation to the analytic solution of the ruin probability problem. Finally, we present numerical examples when claim sizes have distribution light and heavy-tailed.
In the present paper, using a Picard type method of approximation, we investigate the global existence of mild solutions for a class of Ito type stochastic differential equations whose coefficients satisfy conditions more general than the Lipschitz and linear growth ones.
The aim of this paper is to present some ideas how to relax the notion of the optimal solution of the stochastic optimization problem. In the deterministic case, ε-minimal solutions and level-minimal solutions are considered as desired relaxations. We call them approximative solutions and we introduce some possibilities how to combine them with randomness. Relations among random versions of approximative solutions and their consistency are presented in this paper. No measurability is assumed, therefore, treatment convenient for nonmeasurable objects is employed.
The expression of aquaporins (AQPs ) in the fetal porcine urinary tract and its relation to gestational age has not been established. Tissue samples from the renal pelvis, ureter, bladder and urethra were obtained from porcine fetuses. Samples were examined by RT-PCR (AQPs 1-11 ), QPCR (AQPs positive on RT-PCR), and immunohistochemistry. Bladder samples were additionally examined by Western blotting. RNA was extracted from 76 tissue samples obtained from 19 fetuses. Gestational age was 60 (n=11) or 100 days (n=8). PCR showed that AQP1, 3, 9 and 11 mRNA was expressed in all locations. The expression of AQP3 increased significantly at all four locations with gestational age, whereas AQP11 significantly decreased. AQP1 expression increased in the ureter, bladder and urethra. AQP9 mRNA expression increased in the urethra and bladder, but decreased in the ureter. AQP5 was expressed only in the urethra. Immunohistochemistry showed AQP1 staining in sub-urothelial vessels at all locations. Western blotting analysis confirmed increased AQP1 protein levels in bladder samples during gestation. Expression levels of AQP1, 3, 5, 9 and 11 in the urinary tract change during gestation, and further studies are needed to provide insights into normal and pathophysiological water handling mechanisms in the fetus., L. K. Jakobsen, K. F. Trelborg, P. S. Kingo, S. Høyer, K.-E. Andersson, J. C. Djurhuus, R. Nørregaard, L. H. Olsen., and Seznam literatury