scholarly journals Weak Convergence of Markov Random Evolutions in a Multidimensional Space

2012 ◽  
Vol 2012 ◽  
pp. 1-19
Author(s):  
Igor V. Samoilenko
Keyword(s):  
Weak Convergence ◽  
Singular Perturbation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Multidimensional Space ◽  
Random Evolution ◽  
Relative Compactness ◽  
Markov Random ◽  
Random Evolutions

We study Markov symmetrical and nonsymmetrical random evolutions in Rn. Weak convergence of Markov symmetrical random evolution to Wiener process and of Markov non-symmetrical random evolution to a diffusion process with drift is proved using problems of singular perturbation for the generators of evolutions. Relative compactness in DRn×Θ[0,∞) of the families of Markov random evolutions is also shown.

Weak convergence of semi-Markov random evolutions in an averaging scheme (martingale approach)

1989 ◽  
Vol 41 (12) ◽  
pp. 1450-1456 ◽  
Author(s):  
A. V. Svishchuk
Keyword(s):  
Weak Convergence ◽  
Martingale Approach ◽  
Markov Random ◽  
Random Evolutions

scholarly journals The ruin problem for a Wiener process with state-dependent jumps

2020 ◽  
Vol 16 (1) ◽  
pp. 13-23
Author(s):  
M. Lefebvre
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Diffusion Process ◽  
Wiener Process ◽  
Jump Diffusion ◽  
Moment Generating Function ◽  
Continuous Part ◽  
State Dependent ◽  
The Moment ◽  
First Time

AbstractLet X(t) be a jump-diffusion process whose continuous part is a Wiener process, and let T (x) be the first time it leaves the interval (0,b), where x = X(0). The jumps are negative and their sizes depend on the value of X(t). Moreover there can be a jump from X(t) to 0. We transform the integro-differential equation satisfied by the probability p(x) := P[X(T (x)) = 0] into an ordinary differential equation and we solve this equation explicitly in particular cases. We are also interested in the moment-generating function of T (x).

A central limit theorem for nonhomogeneous semi-Markov random evolutions

1989 ◽  
Vol 41 (8) ◽  
pp. 912-918
Author(s):  
V. S. Korolyuk ◽  
A. V. Svishchuk
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Markov Random ◽  
Random Evolutions

Limiting representation of continuous semi-Markov random evolutions in the series scheme

1989 ◽  
Vol 41 (11) ◽  
pp. 1267-1274 ◽  
Author(s):  
V. S. Korolyuk ◽  
A. V. Svishchuk
Keyword(s):  
Markov Random ◽  
Series Scheme ◽  
Random Evolutions

Weak convergence of discrete scattering processes

1991 ◽  
Vol 23 (4) ◽  
pp. 733-750 ◽  
Author(s):  
Lajos Horváth
Keyword(s):  
Weak Convergence ◽  
Wiener Process ◽  
Scattering Process

We show that the discrete scattering process converges weakly to a time-changed Wiener process.

scholarly journals Filtering with a limiter

1998 ◽  
Vol 11 (3) ◽  
pp. 289-300 ◽  
Author(s):  
R. Liptser ◽  
P. Muzhikanov
Keyword(s):  
Weak Convergence ◽  
Diffusion Process ◽  
White Noise ◽  
Discrete Time ◽  
Diffusion Approximation ◽  
Conditional Expectation ◽  
Gaussian White Noise ◽  
Sampling Step ◽  
Non Gaussian ◽  
Filtering Problem

We consider a filtering problem for a Gaussian diffusion process observed via discrete-time samples corrupted by a non-Gaussian white noise. Combining the Goggin's result [2] on weak convergence for conditional expectation with diffusion approximation when a sampling step goes to zero we construct an asymptotic optimal filter. Our filter uses centered observations passed through a limiter. Being asymptotically equivalent to a similar filter without centering, it yields a better filtering accuracy in a prelimit case.

On the weak convergence of U-statistic processes, and of the empirical process

1978 ◽  
Vol 83 (2) ◽  
pp. 269-272 ◽  
Author(s):  
R. M. Loynes
Keyword(s):  
Weak Convergence ◽  
Wiener Process ◽  
Empirical Process ◽  
Convergence Result ◽  
The Other ◽  
Direct Argument ◽  
Random Functions ◽  
Martingale Theorem ◽  
The Relationship ◽  
Special Case

1. Summary and introductionIn (5) a weak convergence result for U-statistics was obtained as a special case of a reverse martingale theorem; in (7) Miller and Sen obtained another such result for U-statistics by a direct argument. As they stand these results are not very closely connected, since one is concerned with U-statistics Uk for k ≥ n, while the other deals with Uk for k ≤ n, but if one instead thinks of k as unrestricted and transforms the random functions Xn which enter into one of these results into new functions Yn by setting Yn(t) = tXn(t−1) one finds that the Yn are (aside from variations in interpolated values) just the functions with which the other result is concerned. As the limiting Wiener process W is well-known to have the property that tW(t−1) is another Wiener process it is not too surprising that both results should hold, and part of the purpose of this paper is to provide a general framework within which the relationship between these results will become clear. A second purpose is to illustrate the simplification that the martingale property brings to weak convergence studies; this is shown both in the U-statistic example and in a new proof of the convergence of the empirical process.

On the interior spike solutions for some singular perturbation problems

1998 ◽  
Vol 128 (4) ◽  
pp. 849-874 ◽  
Author(s):  
Juncheng Wei
Keyword(s):  
Weak Convergence ◽  
Singular Perturbation ◽  
Viscosity Solutions ◽  
Sufficient Conditions ◽  
Bounded Smooth Domain ◽  
Neumann Problems ◽  
Weak Convergence Of Measures ◽  
Bounded Smooth ◽  
Necessary And Sufficient ◽  
Perturbation Problems

For some singular perturbed Dirichlet and Neumann problems in a bounded smooth domain, we study solutions which have a spike in the interior. We obtain both necessary and sufficient conditions for the existence of interior spike solutions. We use, among others, the methods of projections and viscosity solutions, weak convergence of measures and Liapunov–Schmidt reduction.

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