scholarly journals Bias behaviour and antithetic sampling in mean-field particle approximations of SDEs nonlinear in the sense of McKean

2019 ◽  
Vol 65 ◽  
pp. 219-235
Author(s):  
O. Bencheikh ◽  
B. Jourdain
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Mean Field ◽  
Sampling Technique ◽  
Interacting Particles ◽  
Time Step ◽  
Field Interaction ◽  
Weak Error ◽  
Antithetic Sampling

In this paper, we prove that the weak error between a stochastic differential equation with nonlinearity in the sense of McKean given by moments and its approximation by the Euler discretization with time-step h of a system of N interacting particles is 𝒪(N-1+h). We provide numerical experiments confirming this behaviour and showing that it extends to more general mean-field interaction and study the efficiency of the antithetic sampling technique on the same examples.

Diffusion Approximation of State-Dependent G-Networks Under Heavy Traffic

2008 ◽  
Vol 45 (2) ◽  
pp. 347-362 ◽  
Author(s):  
Saul C. Leite ◽  
Marcelo D. Fragoso
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Diffusion Approximation ◽  
Numerical Experiments ◽  
Heavy Traffic ◽  
Weak Sense ◽  
Feedforward Network ◽  
State Dependent ◽  
Number Of Customers

This paper is concerned with the characterization of weak-sense limits of state-dependent G-networks under heavy traffic. It is shown that, for a certain class of networks (which includes a two-layer feedforward network and two queues in tandem), it is possible to approximate the number of customers in the queue by a reflected stochastic differential equation. The benefits of such an approach are that it describes the transient evolution of these queues and allows the introduction of controls, inter alia. We illustrate the application of the results with numerical experiments.

The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field

1991 ◽  
Vol 23 (2) ◽  
pp. 303-316 ◽  
Author(s):  
Kiyomasa Narita
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Diffusion Process ◽  
Langevin Equation ◽  
Mean Field ◽  
Two Dimensional ◽  
Large Parameter ◽  
Rigorous Evaluation ◽  
One Dimensional ◽  
Limit Diffusion

The oscillator of the Liénard type with mean-field containing a large parameter α < 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

On the theory and application of one-step numerical schemes for solving quantum stochastic differential equation (QSDE)

2014 ◽  
Vol 07 (03) ◽  
pp. 1450037
Author(s):  
T. O. Akinwumi ◽  
B. J. Adegboyegun
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Numerical Schemes ◽  
Quantum Stochastic Differential Equation ◽  
One Step ◽  
Definition Of ◽  
Convergent Sequences ◽  
Solution Techniques

This paper presents one-step numerical schemes for solving quantum stochastic differential equation (QSDE). The algorithms are developed based on the definition of QSDE and the solution techniques yield rapidly convergent sequences which are readily computable. As well as developing the schemes, we perform some numerical experiments and the solutions obtained compete favorably with exact solutions. The solution techniques presented in this work can handle all class of QSDEs most especially when the exact solution does not exist.

scholarly journals Existence of solution for Mean-field Reflected Discontinuous Backward Doubly Stochastic Differential Equation

2020 ◽  
Vol 5 (2) ◽  
pp. 205-216
Author(s):  
Mostapha Abdelouahab Saouli
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Mean Field ◽  
Existence Of Solution ◽  
Maximal Solution ◽  
Existence Of A Solution ◽  
Doubly Stochastic

AbstractIn this paper we prove the existence of a solution for mean-field reflected backward doubly stochastic differential equations (MF-RBDSDEs) with one continuous barrier and discontinuous generator (left-continuous). By a comparison theorem establish here for MF-RBDSDEs, we provide a minimal or a maximal solution to MF-RBDSDEs.

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

2015 ◽  
Vol 5 (2) ◽  
pp. 192-208 ◽  
Author(s):  
Ning Li ◽  
Bo Meng ◽  
Xinlong Feng ◽  
Dongwei Gui
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Finite Element ◽  
Finite Difference ◽  
Stochastic Differential Equation ◽  
Numerical Experiments ◽  
Polynomial Chaos ◽  
Numerical Solutions ◽  
Numerical Comparison ◽  
Fe Method

AbstractA numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field

1991 ◽  
Vol 23 (02) ◽  
pp. 303-316 ◽  
Author(s):  
Kiyomasa Narita
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Diffusion Process ◽  
Langevin Equation ◽  
Mean Field ◽  
Two Dimensional ◽  
Large Parameter ◽  
Rigorous Evaluation ◽  
One Dimensional ◽  
Limit Diffusion

The oscillator of the Liénard type with mean-field containing a large parameter α &lt; 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

A weak approximation with Malliavin weights for local stochastic volatility model

2017 ◽  
Vol 04 (01) ◽  
pp. 1750002
Author(s):  
Toshihiro Yamada
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Stochastic Differential Equation ◽  
Stochastic Volatility ◽  
Numerical Experiments ◽  
Option Price ◽  
Stochastic Volatility Model ◽  
Weak Approximation ◽  
Volatility Model ◽  
Sabr Model

This paper introduces a new efficient and practical weak approximation for option price under local stochastic volatility model as marginal expectation of stochastic differential equation, using iterative asymptotic expansion with Malliavin weights. The explicit Malliavin weights for SABR model are shown. Numerical experiments confirm the validity of our discretization with a few time steps.

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