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

2006 ◽  
Vol 2006 ◽  
pp. 1-6 ◽  
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.

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

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.

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

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.

Existence of Solutions for Stochastic Differential Equations under G-Brownian Motion with Discontinuous Coefficients

2012 ◽  
Vol 67 (12) ◽  
pp. 692-698 ◽  
Author(s):  
Faiz Faizullah
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence Of Solutions ◽  
Upper And Lower Solutions ◽  
Discontinuous Coefficients ◽  
Existence Theory ◽  
Discontinuous Functions ◽  
Vector Valued

The existence theory for the vector valued stochastic differential equations under G-Brownian motion (G-SDEs) of the type Xt = X0+ ∫to(v;Xv)dv+ ∫t0 g(v;Xv)d(B)v+ ∫t0 h(v;Xv)dBv; t ∊ [0;T]; with first two discontinuous coefficients is established. It is shown that the G-SDEs have more than one solution if the coefficient g or the coefficients f and g simultaneously, are discontinuous functions. The upper and lower solutions method is used and examples are given to explain the theory and its importance.

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

2019 ◽  
Vol 2019 (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.

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

Filomat ◽  
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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