Weak Convergence Rate of SDE with Stochastic Integration by Trapezoidal Rule

2013 ◽  
Author(s):  
Chao Zheng
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Trapezoidal Rule ◽  
Stochastic Integration ◽  
Weak Convergence Rate

scholarly journals On the weak convergence rate of an exponential Euler scheme for SDEs governed by coefficients with superlinear growth

Bernoulli ◽  
2021 ◽  
Vol 27 (1) ◽  
pp. 312-347
Author(s):  
Mireille Bossy ◽  
Jean-François Jabir ◽  
Kerlyns Martínez
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Euler Scheme ◽  
Superlinear Growth ◽  
Weak Convergence Rate

scholarly journals The Weak Convergence Rate of Two Semi-Exact Discretization Schemes for the Heston Model

Risks ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 23
Author(s):  
Annalena Mickel ◽  
Andreas Neuenkirch
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Asset Price ◽  
Stochastic Volatility Model ◽  
Heston Model ◽  
Smoothness Assumption ◽  
Volatility Model ◽  
Discretization Schemes ◽  
Weak Convergence Rate ◽  
Weak Error

Inspired by the article Weak Convergence Rate of a Time-Discrete Scheme for the Heston Stochastic Volatility Model, Chao Zheng, SIAM Journal on Numerical Analysis 2017, 55:3, 1243–1263, we studied the weak error of discretization schemes for the Heston model, which are based on exact simulation of the underlying volatility process. Both for an Euler- and a trapezoidal-type scheme for the log-asset price, we established weak order one for smooth payoffs without any assumptions on the Feller index of the volatility process. In our analysis, we also observed the usual trade off between the smoothness assumption on the payoff and the restriction on the Feller index. Moreover, we provided error expansions, which could be used to construct second order schemes via extrapolation. In this paper, we illustrate our theoretical findings by several numerical examples.

scholarly journals Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-20 ◽  
Author(s):  
Somyot Plubtieng ◽  
Wanna Sriprad
Keyword(s):  
Banach Space ◽  
Weak Convergence ◽  
Convergence Rate ◽  
Minimization Problem ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Convex Minimization Problem ◽  
Mann Iteration ◽  
Strong And Weak Convergence

We prove strong and weak convergence theorems for a new resolvent of maximal monotone operators in a Banach space and give an estimate of the convergence rate of the algorithm. Finally, we apply our convergence theorem to the convex minimization problem. The result present in this paper extend and improve the corresponding result of Ibaraki and Takahashi (2007), and Kim and Xu (2005).

On pair and tuple formation under independent Poisson or renewal arrival processes

2004 ◽  
Vol 41 (4) ◽  
pp. 1138-1144 ◽  
Author(s):  
K. Borovkov
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Formation Process ◽  
Short Proof ◽  
Probabilistic Approach ◽  
Convergence Result ◽  
Pair Formation ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Arrival Processes

We present several results refining and extending those of Neuts and Alfa on weak convergence of the pair-formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward, and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair-formation process to a Poisson process actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuple formation).

scholarly journals The Trapezoidal Rule for Computing Cauchy Principal Value Integral on Circle

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Jin Li
Keyword(s):  
Singular Point ◽  
Theoretical Analysis ◽  
Convergence Rate ◽  
Error Estimate ◽  
A Priori ◽  
Trapezoidal Rule ◽  
Cauchy Principal ◽  
Numerical Examples ◽  
Cauchy Principal Value ◽  
Principal Value

The composite trapezoidal rule for the computation of Cauchy principal value integral with the singular kernelcot((x-s)/2)is discussed. Our study is based on the investigation of the pointwise superconvergence phenomenon; that is, when the singular point coincides with some a priori known point, the convergence rate of the trapezoidal rule is higher than what is globally possible. We show that the superconvergence rate of the composite trapezoidal rule occurs at middle of each subinterval and obtain the corresponding superconvergence error estimate. Some numerical examples are provided to validate the theoretical analysis.

scholarly journals A New Simplified Weak Second-Order Scheme for Solving Stochastic Differential Equations with Jumps

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 224
Author(s):  
Yang Li ◽  
Yaolei Wang ◽  
Taitao Feng ◽  
Yifei Xin
Keyword(s):  
Differential Equations ◽  
Convergence Rate ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
Second Order ◽  
Trapezoidal Rule ◽  
Order Convergence ◽  
Integration By Parts Formula ◽  
Order Scheme ◽  
Second Order Convergence

In this paper, we propose a new weak second-order numerical scheme for solving stochastic differential equations with jumps. By using trapezoidal rule and the integration-by-parts formula of Malliavin calculus, we theoretically prove that the numerical scheme has second-order convergence rate. To demonstrate the effectiveness and the second-order convergence rate, three numerical experiments are given.

The abscissa of convergence of the Laplace transform

1992 ◽  
Vol 29 (02) ◽  
pp. 353-362 ◽  
Author(s):  
Peter Hall ◽  
Jozef L. Teugels ◽  
Ann Vanmarcke
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Laplace Transform ◽  
Sample Size ◽  
Convergence Result ◽  
Optimal Convergence Rate ◽  
Optimal Convergence ◽  
Abscissa Of Convergence ◽  
The Laplace Transform ◽  
The Mean

Assume that we want to estimate – σ, the abscissa of convergence of the Laplace transform. We show that no non-parametric estimator of σ can converge at a faster rate than (log n)–1, where n is the sample size. An optimal convergence rate is achieved by an estimator of the form where xn = O(log n) and is the mean of the sample values overshooting xn. Under further parametric restrictions this (log n)–1 phenomenon is also illustrated by a weak convergence result.

On pair and tuple formation under independent Poisson or renewal arrival processes

2004 ◽  
Vol 41 (04) ◽  
pp. 1138-1144 ◽  
Author(s):  
K. Borovkov
Keyword(s):  
Weak Convergence ◽  
Convergence Rate ◽  
Formation Process ◽  
Short Proof ◽  
Probabilistic Approach ◽  
Convergence Result ◽  
Pair Formation ◽  
Poisson Processes ◽  
Renewal Processes ◽  
Arrival Processes

We present several results refining and extending those of Neuts and Alfa on weak convergence of the pair-formation process when arrivals follow two independent Poisson processes. Our results are obtained using a different, more straightforward, and apparently simpler probabilistic approach. Firstly, we give a very short proof of the fact that the convergence of the pair-formation process to a Poisson process actually holds in total variation (with a bound for convergence rate). Secondly, we extend the result of the theorem to the case of multiple labels: there are d independent arrival Poisson processes, and we are looking at the epochs when d-tuples are formed. Thirdly, we extend the original (weak convergence) result to the case when arrivals follow independent renewal processes (this extension is also valid for the d-tuple formation).

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