Stochastic Differential Equations with Discontinuous Drift

2011 ◽  
pp. 185-200
Author(s):  
Leszek Gawarecki ◽  
Vidyadhar Mandrekar
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Discontinuous Drift

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 On the existence of solutions for stochastic differential equations driven by fractional Brownian motion

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1695-1700
Author(s):  
Zhi Li
Keyword(s):  
Brownian Motion ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Fractional Brownian Motion ◽  
Growth Condition ◽  
Linear Growth ◽  
Existence Of Solutions ◽  
Hurst Parameter ◽  
Linear Growth Condition ◽  
Discontinuous Drift

In this paper, we are concerned with a class of stochastic differential equations driven by fractional Brownian motion with Hurst parameter 1/2 < H < 1, and a discontinuous drift. By approximation arguments and a comparison theorem, we prove the existence of solutions to this kind of equations under the linear growth condition.

scholarly journals A study on stochastic differential equation

2021 ◽  
Vol 5 (2) ◽  
pp. 68-75
Author(s):  
Govindaraju P ◽  
Senthil Kumar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Density Function ◽  
Discontinuous Drift ◽  
Point Of Departure

In this paper we study solutions to stochastic differential equations (SDEs) with discontinuous drift. In this paper we discussed The Euler-Maruyama method and this shows that a candidate density function based on the Euler-Maruyama method. The point of departure for this work is a particular SDE with discontinuous drift.

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 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.

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