Mean-square rate of convergence of orthogonal series

1988 ◽  
Vol 39 (5) ◽  
pp. 493-497
Author(s):  
A. I. Stepanets
2020 ◽  
Vol 52 (3) ◽  
pp. 735-771
Author(s):  
Christel Geiss ◽  
Céline Labart ◽  
Antti Luoto

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.


2016 ◽  
Vol 8 (6) ◽  
pp. 1004-1022 ◽  
Author(s):  
Xu Yang ◽  
Weidong Zhao

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.


1970 ◽  
Vol 22 (2) ◽  
pp. 230-239 ◽  
Author(s):  
�. A. Storozhenko ◽  
Yu. M. Shmandin

1997 ◽  
Vol 34 (4) ◽  
pp. 908-923 ◽  
Author(s):  
Marie F. Kratz ◽  
Holger Rootzén

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.


1997 ◽  
Vol 29 (1) ◽  
pp. 249-270 ◽  
Author(s):  
Jacques Istas ◽  
Catherine Laredo

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.


1997 ◽  
Vol 34 (04) ◽  
pp. 908-923 ◽  
Author(s):  
Marie F. Kratz ◽  
Holger Rootzén

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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