The Method of Upper and Lower Solutions of Stochastic Differential Equations and Applications

2007 ◽  
Vol 26 (1) ◽  
pp. 16-28 ◽  
Author(s):  
Nikolaos Halidias ◽  
Mariusz Michta
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Upper And Lower Solutions

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 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.

Stochastic Differential Equations for Power Law Behaviors

2012 ◽  
Author(s):  
Bo Jiang ◽  
Roger Brockett ◽  
Weibo Gong ◽  
Don Towsley
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Power Law

scholarly journals Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations

2020 ◽  
Vol 53 (2) ◽  
pp. 2220-2224
Author(s):  
William M. McEneaney ◽  
Hidehiro Kaise ◽  
Peter M. Dower ◽  
Ruobing Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Strong Solution ◽  
Solution Existence
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