On explicit Milstein-type scheme for McKean–Vlasov stochastic differential equations with super-linear drift coefficient

This paper represents a generalization of the stability result on the Euler-Maruyama solution, which is established in the paper M. Milosevic, Almost sure exponential stability of solutions to highly nonlinear neutral stochastics differential equations with time-dependent delay and Euler-Maruyama approximation, Math. Comput. Model. 57 (2013) 887 - 899. The main aim of this paper is to reveal the sufficient conditions for the global almost sure asymptotic exponential stability of the ?-Euler-Maruyama solution (? ? [0, 1/2 ]), for a class of neutral stochastic differential equations with time-dependent delay. The existence and uniqueness of solution of the approximate equation is proved by employing the one-sided Lipschitz condition with respect to the both present state and delayed arguments of the drift coefficient of the equation. The technique used in proving the stability result required the assumption ? ?(0, 1/2], while the method is defined by employing the parameter ? with respect to the both drift coefficient and neutral term. Bearing in mind the difference between the technique which will be applied in the present paper and that used in the cited paper, the Euler-Maruyama case (? = 0) is considered separately. In both cases, the linear growth condition on the drift coefficient is applied, among other conditions. An example is provided to support the main result of the paper.

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On the Existence of Solutions of Stochastic Differential Equations with Integrable Drift Coefficient

Theory of Probability and Its Applications ◽

10.1137/1119061 ◽

1975 ◽

Vol 19 (3) ◽

pp. 552-557 ◽

Cited By ~ 2

Author(s):

N. I. Portenko

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Existence Of Solutions ◽

Drift Coefficient

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A note on the Euler–Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient

BIT Numerical Mathematics ◽

10.1007/s10543-008-0164-1 ◽

2008 ◽

Vol 48 (1) ◽

pp. 51-59 ◽

Cited By ~ 14

Author(s):

Nikolaos Halidias ◽

Peter E. Kloeden

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Drift Coefficient

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The Euler scheme for stochastic differential equations with discontinuous drift coefficient: a numerical study of the convergence rate

Advances in Difference Equations ◽

10.1186/s13662-019-2361-4 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 1

Author(s):

S. Göttlich ◽

K. Lux ◽

A. Neuenkirch

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Numerical Study ◽

Market Model ◽

Euler Scheme ◽

Convergence Properties ◽

Constant Diffusion ◽

Discontinuous Drift ◽

Long Time ◽

Drift Coefficient

Abstract The Euler scheme is one of the standard schemes to obtain numerical approximations of solutions of stochastic differential equations (SDEs). Its convergence properties are well known in the case of globally Lipschitz continuous coefficients. However, in many situations, relevant systems do not show a smooth behavior, which results in SDE models with discontinuous drift coefficient. In this work, we analyze the long time properties of the Euler scheme applied to SDEs with a piecewise constant drift and a constant diffusion coefficient and carry out intensive numerical tests for its convergence properties. We emphasize numerical convergence rates and analyze how they depend on the properties of the drift coefficient and the initial value. We also give theoretical interpretations of some of the arising phenomena. For application purposes, we study a rank-based stock market model describing the evolution of the capital distribution within the market and provide theoretical as well as numerical results on the long time ranking behavior.

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A note on strong solutions of stochastic differential equations with a discontinuous drift coefficient

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/jamsa/2006/73257 ◽

2006 ◽

Vol 2006 ◽

pp. 1-6 ◽

Cited By ~ 7

Author(s):

Nikolaos Halidias ◽

P. E. Kloeden

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Upper And Lower Solutions ◽

Strong Solutions ◽

Heaviside Function ◽

Lipschitz Continuous ◽

Discontinuous Drift ◽

Vector Valued ◽

Increasing Function ◽

Drift Coefficient

The existence of a mean-square continuous strong solution is established for vector-valued Itô stochastic differential equations with a discontinuous drift coefficient, which is an increasing function, and with a Lipschitz continuous diffusion coefficient. A scalar stochastic differential equation with the Heaviside function as its drift coefficient is considered as an example. Upper and lower solutions are used in the proof.

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Strong convergence of the tamed Euler method for stochastic differential equations with piecewise continuous arguments and Poisson jumps

Filomat ◽

10.2298/fil1712815y ◽

2017 ◽

Vol 31 (12) ◽

pp. 3815-3836

Author(s):

Huizi Yang ◽

Minghui Song ◽

Mingzhu Liu ◽

Hong Wang

Keyword(s):

Differential Equations ◽

Convergence Rate ◽

Stochastic Differential Equations ◽

Polynomial Growth ◽

Euler Method ◽

Poisson Jumps ◽

Lipschitz Continuous ◽

Piecewise Continuous ◽

Piecewise Continuous Arguments ◽

Drift Coefficient

In the present work, the tamed Euler method is proven to be strongly convergent for stochastic differential equations with piecewise continuous arguments and Poisson jumps, where the diffusion and jump coefficients are globally Lipschitz continuous, the drift coefficient is one-sided Lipschitz continuous, and its derivative demonstrates an at most polynomial growth. Moreover, the convergence rate is obtained.

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Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients

Symmetry ◽

10.3390/sym12040570 ◽

2020 ◽

Vol 12 (4) ◽

pp. 570

Author(s):

Haesung Lee ◽

Gerald Trutnau

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Borel Measure ◽

Measure Zero ◽

Discontinuous Coefficients ◽

Dispersion Matrix ◽

Well Posedness ◽

Uniqueness In Law ◽

Zero Time ◽

Drift Coefficient

We show uniqueness in law for a general class of stochastic differential equations in R d , d ≥ 2 , with possibly degenerate and/or fully discontinuous locally bounded coefficients among all weak solutions that spend zero time at the points of degeneracy of the dispersion matrix. Points of degeneracy have a d-dimensional Lebesgue–Borel measure zero. Weak existence is obtained for a more general, but not necessarily locally bounded drift coefficient.

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