On the existence of solutions with smooth density of stochastic differential equations in plane

1992 ◽  
Vol 8 (4) ◽  
pp. 432-446 ◽  
Author(s):  
Zhou Xianyin
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence Of Solutions ◽  
Smooth Density

scholarly journals Comparison Theorems for the Multidimensional BDSDEs and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Bo Zhu ◽  
Baoyan Han
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Comparison Theorem ◽  
Existence Of Solutions ◽  
Minimal Solution ◽  
Comparison Theorems ◽  
Doubly Stochastic

A class of backward doubly stochastic differential equations (BDSDEs) are studied. We obtain a comparison theorem of these multidimensional BDSDEs. As its applications, we derive the existence of solutions for this multidimensional BDSDEs with continuous coefficients. We can also prove that this solution is the minimal solution of the BDSDE.

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.

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