Mean-square rate of convergence of orthogonal series

1988 ◽  
Vol 39 (5) ◽  
pp. 493-497
Author(s):  
A. I. Stepanets
Keyword(s):  
Rate Of Convergence ◽  
Orthogonal Series ◽  
Mean Square

scholarly journals Mean square rate of convergence for random walk approximation of forward-backward SDEs

2020 ◽  
Vol 52 (3) ◽  
pp. 735-771
Author(s):  
Christel Geiss ◽  
Céline Labart ◽  
Antti Luoto
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Backward Stochastic Differential Equation ◽  
Stochastic Equations ◽  
Terminal Condition ◽  
Probabilistic Methods ◽  
Mean Square ◽  
Finite Difference Equations ◽  
Backward Sdes

AbstractLet (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

Strong Convergence Analysis of Split-Step θ-Scheme for Nonlinear Stochastic Differential Equations with Jumps

2016 ◽  
Vol 8 (6) ◽  
pp. 1004-1022 ◽  
Author(s):  
Xu Yang ◽  
Weidong Zhao
Keyword(s):  
Differential Equations ◽  
Theoretical Result ◽  
Strong Convergence ◽  
Stochastic Differential Equations ◽  
Rate Of Convergence ◽  
Numerical Experiments ◽  
Mean Square ◽  
Nonlinear Stochastic Differential Equations ◽  
Mean Square Convergence ◽  
The Mean

AbstractIn this paper, we investigate the mean-square convergence of the split-step θ-scheme for nonlinear stochastic differential equations with jumps. Under some standard assumptions, we rigorously prove that the strong rate of convergence of the split-step θ-scheme in strong sense is one half. Some numerical experiments are carried out to assert our theoretical result.

The rate of convergence of (C, 1)-means of orthogonal series

1970 ◽  
Vol 22 (2) ◽  
pp. 230-239 ◽  
Author(s):  
�. A. Storozhenko ◽  
Yu. M. Shmandin
Keyword(s):  
Rate Of Convergence ◽  
Orthogonal Series

On the rate of convergence for extremes of mean square differentiable stationary normal processes

1997 ◽  
Vol 34 (4) ◽  
pp. 908-923 ◽  
Author(s):  
Marie F. Kratz ◽  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Covariance Function ◽  
Mean Square ◽  
Normal Process ◽  
Distribution Of The Maximum ◽  
Image Position ◽  
High Level

Let ξ (t); t ≧ 0 be a normalized continuous mean square differentiable stationary normal process with covariance function r(t). Further, let and set . We give bounds which are roughly of order Τ –δ for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by ξ (t) in the interval [0, T]. The results assume that r(t) and r′(t) decay polynomially at infinity and that r″ (t) is suitably bounded. For the number of upcrossings it is in addition assumed that r(t) is non-negative.

Estimating Functionals of a Stochastic Process

1997 ◽  
Vol 29 (1) ◽  
pp. 249-270 ◽  
Author(s):  
Jacques Istas ◽  
Catherine Laredo
Keyword(s):  
Stochastic Process ◽  
Finite Number ◽  
Rate Of Convergence ◽  
Mean Square Error ◽  
A Priori ◽  
Mean Square ◽  
Hölder Condition ◽  
Quadratic Mean ◽  
Sampling Points ◽  
The Mean

The problem of estimating the integral of a stochastic process from observations at a finite number N of sampling points has been considered by various authors. Recently, Benhenni and Cambanis (1992) studied this problem for processes with mean 0 and Hölder index K + ½, K ; ℕ These results are here extended to processes with arbitrary Hölder index. The estimators built here are linear in the observations and do not require the a priori knowledge of the smoothness of the process. If the process satisfies a Hölder condition with index s in quadratic mean, we prove that the rate of convergence of the mean square error is N2s+1 as N goes to ∞, and build estimators that achieve this rate.

On the rate of convergence for extremes of mean square differentiable stationary normal processes

1997 ◽  
Vol 34 (04) ◽  
pp. 908-923 ◽  
Author(s):  
Marie F. Kratz ◽  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Covariance Function ◽  
Mean Square ◽  
Normal Process ◽  
Distribution Of The Maximum ◽  
Image Position ◽  
High Level

Let ξ (t); t ≧ 0 be a normalized continuous mean square differentiable stationary normal process with covariance function r(t). Further, let and set . We give bounds which are roughly of order Τ –δ for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by ξ (t) in the interval [0, T]. The results assume that r(t) and r′(t) decay polynomially at infinity and that r ″ (t) is suitably bounded. For the number of upcrossings it is in addition assumed that r(t) is non-negative.

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