scholarly journals Fixed Points and Stability in Neutral Stochastic Differential Equations with Variable Delays

2008 ◽  
Vol 2008 ◽  
pp. 1-12 ◽  
Author(s):  
Meng Wu ◽  
Nan-jing Huang ◽  
Chang-Wen Zhao
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Stochastic Differential Equations ◽  
Variable Delays

scholarly journals A Compensated Numerical Method for Solving Stochastic Differential Equations with Variable Delays and Random Jump Magnitudes

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Ying Du ◽  
Changlin Mei
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Mean Square ◽  
Variable Delays ◽  
Mean Square Stability ◽  
The Mean ◽  
Random Jump

Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-stepθmethod is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability.

Stochastic formulations of the parametrix method

2018 ◽  
Vol 22 ◽  
pp. 178-209
Author(s):  
Arturo Kohatsu-Higa ◽  
Gô Yûki
Keyword(s):  
Differential Equations ◽  
Fixed Points ◽  
Stochastic Differential Equations ◽  
Probability Density ◽  
Lower Bounds ◽  
Probability Density Functions ◽  
Upper And Lower Bounds ◽  
Density Functions ◽  
Holder Continuity ◽  
Gaussian Type

In this manuscript, we consider stochastic expressions of the parametrix method for solutions of d-dimensional stochastic differential equations (SDEs) with drift coefficients which belong to Lp(Rd), p > d. We prove the existence and Hölder continuity of probability density functions for distributions of solutions at fixed points and obtain an explicit expansion via (stochastic) parametrix methods. We also obtain Gaussian type upper and lower bounds for these probability density functions.

scholarly journals FIXED POINTS AND STABILITY OF A CLASS OF NONLINEAR DELAY INTEGRO-DIFFERENTIAL EQUATIONS WITH VARIABLE DELAYS

2017 ◽  
pp. 031
Author(s):  
Hocine Gabsi ◽  
Abdelouaheb Ardjouni ◽  
Ahcene Djoudi
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Fixed Points ◽  
Fixed Point Theory ◽  
Zero Solution ◽  
Point Theory ◽  
Second Order ◽  
Asymptotically Stable ◽  
Variable Delays ◽  
Nonlinear Delay

In this work we study a class of second order nonlinear neutral integro-differential equations    x(t)+f(t,x(t),x(t))x(t)+∑_{j=1}^{N}∫_{t-τ_{j}(t)}^{t}a_{j}(t,s)g_{j}(s,x(s))ds    +∑_{j=1}^{N}b_{j}(t)x′(t-τ_{j}(t))=0,with variable delays and give some new conditions ensuring that the zero solution is asymptotically stable by means of the fixed point theory. Our work extends and improves previous results in the literature such as, D. Pi <cite>pi2,pi3</cite> and T. A. Burton <cite>b12</cite>. An example is given to illustrate our claim.

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