The Existence of Strong Solutions for a Class of Stochastic Differential Equations

International Journal of Differential Equations ◽

10.1155/2018/2059694 ◽

2018 ◽

Vol 2018 ◽

pp. 1-5

Author(s):

Junfei Zhang

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Strong Solution ◽

Strong Solutions ◽

Lipschitz Continuous ◽

Discontinuous Drift ◽

Increasing Function

In this paper, we will consider the existence of a strong solution for stochastic differential equations with discontinuous drift coefficients. More precisely, we study a class of stochastic differential equations when the drift coefficients are an increasing function instead of Lipschitz continuous or continuous. The main tools of this paper are the lower solutions and upper solutions of stochastic differential equations.

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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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McKean–Vlasov type stochastic differential equations arising from the random vortex method

SN Partial Differential Equations and Applications ◽

10.1007/s42985-021-00146-z ◽

2021 ◽

Vol 3 (1) ◽

Author(s):

Zhongmin Qian ◽

Yuhan Yao

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Vortex Dynamics ◽

Existence And Uniqueness ◽

Strong Solutions ◽

Vortex Method ◽

Wasserstein Distance ◽

New Approach ◽

Lipschitz Continuous ◽

Space Of Distributions

AbstractWe study a class of McKean–Vlasov type stochastic differential equations (SDEs) which arise from the random vortex dynamics and other physics models. By introducing a new approach we resolve the existence and uniqueness of both the weak and strong solutions for the McKean–Vlasov stochastic differential equations whose coefficients are defined in terms of singular integral kernels such as the Biot–Savart kernel. These SDEs which involve the distributions of solutions are in general not Lipschitz continuous with respect to the usual distances on the space of distributions such as the Wasserstein distance. Therefore there is an obstacle in adapting the ordinary SDE method for the study of this class of SDEs, and the conventional methods seem not appropriate for dealing with such distributional SDEs which appear in applications such as fluid mechanics.

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