scholarly journals Exact simulation of multidimensional reflected Brownian motion

2018 ◽  
Vol 55 (1) ◽  
pp. 137-156
Author(s):  
Jose Blanchet ◽  
Karthyek Murthy
Keyword(s):  
Brownian Motion ◽  
Information Structure ◽  
Piecewise Linear ◽  
Simulation Method ◽  
Uniform Norm ◽  
Piecewise Linear Approximation ◽  
Reflected Brownian Motion ◽  
Proposal Distribution ◽  
Exact Simulation ◽  
Simulation Techniques

AbstractWe present the first exact simulation method for multidimensional reflected Brownian motion (RBM). Exact simulation in this setting is challenging because of the presence of correlated local-time-like terms in the definition of RBM. We apply recently developed so-called ε-strong simulation techniques (also known as tolerance-enforced simulation) which allow us to provide a piecewise linear approximation to RBM with ε (deterministic) error in uniform norm. A novel conditional acceptance–rejection step is then used to eliminate the error. In particular, we condition on a suitably designed information structure so that a feasible proposal distribution can be applied.

scholarly journals Simulating Nonlinear Dynamics of a 3D Crystal Lattice of Metals

2021 ◽  
Vol 2131 (3) ◽  
pp. 032092
Author(s):  
A S Semenov ◽  
M N Semenova ◽  
Yu V Bebikhov ◽  
P V Zakharov ◽  
E A Korznikova
Keyword(s):  
Nonlinear Dynamics ◽  
Crystal Lattice ◽  
Mathematical Simulation ◽  
Piecewise Linear ◽  
Experimental Studies ◽  
Piecewise Linear Approximation ◽  
Calculation Error ◽  
Simulation Techniques ◽  
Welch Method ◽  
Computationally Intensive

Abstract Oscillations of crystal lattices determine important material properties such as thermal conductivity, heat capacity, thermal expansion, and many others; therefore, their study is an urgent and important problem. Along with experimental studies of the nonlinear dynamics of a crystal lattice, effective computer simulation techniques such as ab initio simulation and the molecular dynamics method are widely used. Mathematical simulation is less commonly used since the calculation error there can reach 10 %. Herewith, it is the least computationally intensive. This paper describes the process and results of mathematical simulation of the nonlinear dynamics of a 3D crystal lattice of metals using the Lennard-Jones potential in the MatLab software package, which is well-proven for solving technical computing problems. The following main results have been obtained: 3D distribution of atoms over the computational cell has been plotted, proving the possibility of displacement to up to five interatomic distances; the frequency response has been evaluated using the Welch method with a relative RMS error not exceeding 30 %; a graphical dependence between the model and the reference cohesive energy data for a metal HCP cell has been obtained with an error of slightly more than 3 %; an optimal model for piecewise-linear approximation has been calculated, and its 3D interpolation built. All studies performed show good applicability of mathematical simulation to the problems of studying dynamic processes in crystal physics.

Boundary crossing probability for Brownian motion

2001 ◽  
Vol 38 (1) ◽  
pp. 152-164 ◽  
Author(s):  
Klaus Pötzelberger ◽  
Liqun Wang
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Simulation Method ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Approximation Errors ◽  
Crossing Probabilities ◽  
Asymptotic Upper Bound

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n2).

scholarly journals Exact Simulation for Diffusion Bridges: An Adaptive Approach

2014 ◽  
Vol 51 (02) ◽  
pp. 346-358
Author(s):  
Hongsheng Dai
Keyword(s):  
Brownian Motion ◽  
Sampling Methods ◽  
Simulation Method ◽  
New Method ◽  
Sampling Techniques ◽  
Rejection Sampling ◽  
Exact Simulation ◽  
Adaptive Approach

Exact simulation approaches for a class of diffusion bridges have recently been proposed based on rejection sampling techniques. The existing rejection sampling methods may not be practical owing to small acceptance probabilities. In this paper we propose an adaptive approach that improves the existing methods significantly under certain scenarios. The idea of the new method is based on a layered process, which can be simulated from a layered Brownian motion with reweighted layer probabilities. We will show that the new exact simulation method is more efficient than existing methods theoretically and via simulation.

Boundary crossing probability for Brownian motion

2001 ◽  
Vol 38 (01) ◽  
pp. 152-164 ◽  
Author(s):  
Klaus Pötzelberger ◽  
Liqun Wang
Keyword(s):  
Brownian Motion ◽  
Piecewise Linear ◽  
Approximation Error ◽  
Simulation Method ◽  
Boundary Crossing ◽  
Boundary Crossing Probability ◽  
Crossing Probability ◽  
Approximation Errors ◽  
Crossing Probabilities ◽  
Asymptotic Upper Bound

Wang and Pötzelberger (1997) derived an explicit formula for the probability that a Brownian motion crosses a one-sided piecewise linear boundary and used this formula to approximate the boundary crossing probability for general nonlinear boundaries. The present paper gives a sharper asymptotic upper bound of the approximation error for the formula, and generalizes the results to two-sided boundaries. Numerical computations are easily carried out using the Monte Carlo simulation method. A rule is proposed for choosing optimal nodes for the approximating piecewise linear boundaries, so that the corresponding approximation errors of boundary crossing probabilities converge to zero at a rate of O(1/n 2).

scholarly journals Exact Simulation for Diffusion Bridges: An Adaptive Approach

2014 ◽  
Vol 51 (2) ◽  
pp. 346-358 ◽  
Author(s):  
Hongsheng Dai
Keyword(s):  
Brownian Motion ◽  
Sampling Methods ◽  
Simulation Method ◽  
New Method ◽  
Sampling Techniques ◽  
Rejection Sampling ◽  
Exact Simulation ◽  
Adaptive Approach

Exact simulation approaches for a class of diffusion bridges have recently been proposed based on rejection sampling techniques. The existing rejection sampling methods may not be practical owing to small acceptance probabilities. In this paper we propose an adaptive approach that improves the existing methods significantly under certain scenarios. The idea of the new method is based on a layered process, which can be simulated from a layered Brownian motion with reweighted layer probabilities. We will show that the new exact simulation method is more efficient than existing methods theoretically and via simulation.

Moderate deviation principle for multivalued stochastic differential equations

2019 ◽  
Vol 20 (03) ◽  
pp. 2050015 ◽  
Author(s):  
Hua Zhang
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Moderate Deviation ◽  
Reflected Brownian Motion ◽  
Moderate Deviation Principle ◽  
Deviation Principle ◽  
Weak Convergence Approach

In this paper, we prove a moderate deviation principle for the multivalued stochastic differential equations whose proof are based on recently well-developed weak convergence approach. As an application, we obtain the moderate deviation principle for reflected Brownian motion.

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