scholarly journals Convergence of Locally Square Integrable Martingales to a Continuous Local Martingale

2011 ◽  
Vol 2011 ◽  
pp. 1-34
Author(s):  
Andriy Yurachkivsky
Keyword(s):  
Gaussian Processes ◽  
Continuous Functions ◽  
Local Martingale ◽  
Initial Value ◽  
Finite Dimensional ◽  
Independent Increments ◽  
Integrable Martingale ◽  
Probability Spaces ◽  
Uniformly Continuous

Let for each be an -valued locally square integrable martingale w.r.t. a filtration (probability spaces may be different for different ). It is assumed that the discontinuities of are in a sense asymptotically small as and the relation holds for all , row vectors , and bounded uniformly continuous functions . Under these two principal assumptions and a number of technical ones, it is proved that the 's are asymptotically conditionally Gaussian processes with conditionally independent increments. If, moreover, the compound processes converge in distribution to some , then a sequence () converges in distribution to a continuous local martingale with initial value and quadratic characteristic , whose finite-dimensional distributions are explicitly expressed via those of .

scholarly journals Lattices of uniformly continuous functions

2013 ◽  
Vol 160 (1) ◽  
pp. 50-55 ◽  
Author(s):  
Félix Cabello Sánchez ◽  
Javier Cabello Sánchez
Keyword(s):  
Continuous Functions ◽  
Uniformly Continuous

Higher Dimensional Spaces of Functions on the Spectrum of a Uniform Algebra

2007 ◽  
Vol 50 (1) ◽  
pp. 3-10
Author(s):  
Richard F. Basener
Keyword(s):  
Continuous Functions ◽  
Uniform Algebra ◽  
Finite Dimensional ◽  
Higher Dimensional ◽  
Spaces Of Continuous Functions ◽  
The Family ◽  
Continuous Complex ◽  
Finite Dimensional Case ◽  
Complex Valued ◽  
Shed Light

AbstractIn this paper we introduce a nested family of spaces of continuous functions defined on the spectrum of a uniform algebra. The smallest space in the family is the uniform algebra itself. In the “finite dimensional” case, from some point on the spaces will be the space of all continuous complex-valued functions on the spectrum. These spaces are defined in terms of solutions to the nonlinear Cauchy–Riemann equations as introduced by the author in 1976, so they are not generally linear spaces of functions. However, these spaces do shed light on the higher dimensional properties of a uniform algebra. In particular, these spaces are directly related to the generalized Shilov boundary of the uniform algebra (as defined by the author and, independently, by Sibony in the early 1970s).

On the moments of some first-passage times

1985 ◽  
Vol 22 (02) ◽  
pp. 461-466
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Continuous Function ◽  
Exponential Type ◽  
First Passage Time ◽  
Sequential Estimation ◽  
Passage Time ◽  
Positive Continuous Function ◽  
First Passage ◽  
Finite Dimensional ◽  
Independent Increments ◽  
Passage Times

Let {Xt } t≧0 (t may be discrete or continuous) be a random process whose finite-dimensional distributions are of exponential type. The first-passage time inf{t:Xt ≧f(t)}, where f(t) is a positive, continuous function, such that f(t)= o(t) as t↑∞, is considered. The problem of finiteness of its moments is solved for both the case that {Xt } t≧0 has stationary independent increments as well as the case in which no assumptions are made about stationarity and independence for the increments of the process. Applications to sequential estimation are also given.

scholarly journals Limit theorems for hitting times of 1-dimensional generalized diffusions

1998 ◽  
Vol 152 ◽  
pp. 1-37
Author(s):  
Matsuyo Tomisaki ◽  
Makoto Yamazato
Keyword(s):  
Limit Theorems ◽  
Bessel Functions ◽  
Diffusion Processes ◽  
Hitting Times ◽  
Finite Dimensional ◽  
Independent Increments ◽  
Starting Point ◽  
Self Similar ◽  
The Mean ◽  
The One

Abstract.Limit theorems are obtained for suitably normalized hitting times of single points for 1-dimensional generalized diffusion processes as the hitting points tend to boundaries under an assumption which is slightly stronger than that the existence of limits γ + 1 of the ratio of the mean and the variance of the hitting time. Laplace transforms of limit distributions are modifications of Bessel functions. Results are classified by the one parameter {γ}, each of which is the degree of corresponding Bessel function. In case the limit distribution is degenerate to one point, by changing the normalization, we obtain convergence to the normal distribution. Regarding the starting point as a time parameter, we obtain convergence in finite dimensional distributions to self-similar processes with independent increments under slightly stronger assumption.

Inversion of the Initial Value for a Time-Fractional Diffusion-Wave Equation by Boundary Data

2020 ◽  
Vol 20 (1) ◽  
pp. 109-120 ◽  
Author(s):  
Suzhen Jiang ◽  
Kaifang Liao ◽  
Ting Wei
Keyword(s):  
Inverse Problem ◽  
Wave Equation ◽  
Fractional Diffusion ◽  
Initial Value ◽  
Diffusion Wave ◽  
One Dimensional ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Dimensional Approximation ◽  
Continuation Technique

AbstractIn this study, we consider an inverse problem of recovering the initial value for a multi-dimensional time-fractional diffusion-wave equation. By using some additional boundary measured data, the uniqueness of the inverse initial value problem is proven by the Laplace transformation and the analytic continuation technique. The inverse problem is formulated to solve a Tikhonov-type optimization problem by using a finite-dimensional approximation. We test four numerical examples in one-dimensional and two-dimensional cases for verifying the effectiveness of the proposed algorithm.

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