A second-order weak approximation of SDEs using a Markov chain without Lévy area simulation

2018 ◽  
Vol 24 (4) ◽  
pp. 289-308 ◽  
Author(s):  
Toshihiro Yamada ◽  
Kenta Yamamoto
Keyword(s):  
Markov Chain ◽  
Vector Fields ◽  
Correction Term ◽  
Random Variables ◽  
Simulation Method ◽  
Second Order ◽  
Weak Approximation ◽  
Lie Bracket ◽  
Discrete Moment ◽  
The Cost

Abstract This paper proposes a new Markov chain approach to second-order weak approximations of stochastic differential equations (SDEs) driven by d-dimensional Brownian motion. The scheme is explicitly constructed by polynomials of Brownian motions up to second order, and any discrete moment-matched random variables or the Lévy area simulation method are not used. The required number of random variables is still d in one-step simulation of the implementation of the scheme. In the Markov chain, a correction term with Lie bracket of vector fields associated with SDEs appears as the cost of not using moment-matched random variables.

scholarly journals Feedback synthesis for underactuated systems using sequential second-order needle variations

2018 ◽  
Vol 37 (13-14) ◽  
pp. 1826-1853 ◽  
Author(s):  
Giorgos Mamakoukas ◽  
Malcolm A. MacIver ◽  
Todd D. Murphey
Keyword(s):  
Vector Fields ◽  
Underwater Vehicle ◽  
Second Order ◽  
Three Dimensions ◽  
Control Synthesis ◽  
Nonlinear Feedback ◽  
First Order ◽  
Lie Bracket ◽  
Control Response ◽  
Needle Variations

This paper derives nonlinear feedback control synthesis for general control affine systems using second-order actions, the second-order needle variations of optimal control, as the basis for choosing each control response to the current state. A second result of this paper is that the method provably exploits the nonlinear controllability of a system by virtue of an explicit dependence of the second-order needle variation on the Lie bracket between vector fields. As a result, each control decision necessarily decreases the objective when the system is nonlinearly controllable using first-order Lie brackets. Simulation results using a differential drive cart, an underactuated kinematic vehicle in three dimensions, and an underactuated dynamic model of an underwater vehicle demonstrate that the method finds control solutions when the first-order analysis is singular. Finally, the underactuated dynamic underwater vehicle model demonstrates convergence even in the presence of a velocity field.

scholarly journals Uncertainty Cost Functions in Climate-Dependent Controllable Loads in Commercial Environments

Energies ◽  
2021 ◽  
Vol 14 (10) ◽  
pp. 2885
Author(s):  
Daniel Losada ◽  
Ameena Al-Sumaiti ◽  
Sergio Rivera
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Simulation Method ◽  
Electricity Sector ◽  
Cost Functions ◽  
Density Estimator ◽  
Monte Carlo Simulation Method ◽  
Commercial Building ◽  
Complementary Method ◽  
The Cost

This article presents the development, simulation and validation of the uncertainty cost functions for a commercial building with climate-dependent controllable loads, located in Florida, USA. For its development, statistical data on the energy consumption of the building in 2016 were used, along with the deployment of kernel density estimator to characterize its probabilistic behavior. For validation of the uncertainty cost functions, the Monte-Carlo simulation method was used to make comparisons between the analytical results and the results obtained by the method. The cost functions found differential errors of less than 1%, compared to the Monte-Carlo simulation method. With this, there is an analytical approach to the uncertainty costs of the building that can be used in the development of optimal energy dispatches, as well as a complementary method for the probabilistic characterization of the stochastic behavior of agents in the electricity sector.

scholarly journals A weak convergence approach to hybrid LQG problems with infinite control weights

2002 ◽  
Vol 15 (1) ◽  
pp. 1-21
Author(s):  
G. George Yin ◽  
Jiongmin Yong
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
Small Parameter ◽  
Limit Problem ◽  
Discrete Events ◽  
Linear Quadratic ◽  
Piecewise Constant ◽  
Constant Coefficients ◽  
The Cost ◽  
Random Times

This work is concerned with a class of hybrid LQG (linear quadratic Gaussian) regulator problems modulated by continuous-time Markov chains. In contrast to the traditional LQG models, the systems have both continuous dynamics and discrete events. In lieu of a model with constant coefficients, these coefficients vary with time and exhibit piecewise constant behavior. At any time t, the system follows a stochastic differential equation in which the coefficients take one of the m possible configurations where m is usually large. The system may jump to any of the possible configurations at random times. Further, the control weight in the cost functional is allowed to be indefinite. To reduce the complexity, the Markov chain is formulated as singularly perturbed with a small parameter. Our effort is devoted to solving the limit problem when the small parameter tends to zero via the framework of weak convergence. Although the limit system is still modulated by a Markov chain, it has a much smaller state space and thus, much reduced complexity.

On the Markov Chain Simulation Method for Uniform Combinatorial Distributions and Simulated Annealing

1987 ◽  
Vol 1 (1) ◽  
pp. 33-46 ◽  
Author(s):  
David Aldous
Keyword(s):  
Markov Chain ◽  
Simulated Annealing ◽  
Simulated Annealing Algorithm ◽  
Simulation Method ◽  
Careful Analysis ◽  
Eigenvalue Estimate ◽  
Uniform Distributions ◽  
Markov Chain Method ◽  
Annealing Algorithm ◽  
Second Largest Eigenvalue

Uniform distributions on complicated combinatorial sets can be simulated by the Markov chain method. A condition is given for the simulations to be accurate in polynomial time. Similar analysis of the simulated annealing algorithm remains an open problem. The argument relies on a recent eigenvalue estimate of Alon [4]; the only new mathematical ingredient is a careful analysis of how the accuracy of sample averages of a Markov chain is related to the second-largest eigenvalue.

On the limit distribution of a second-order semi-Markov chain in state and duration

2016 ◽  
Vol 46 (12) ◽  
pp. 5994-5999
Author(s):  
Guglielmo D’Amico ◽  
Filippo Petroni ◽  
Flavio Prattico
Keyword(s):  
Markov Chain ◽  
Limit Distribution ◽  
Second Order
Sign in / Sign up

Export Citation Format

Share Document