Existence theorems for solutions of stochastic differential equations with discontinuous right-hand sides

2000 ◽  
Vol 36 (1) ◽  
pp. 56-63 ◽  
Author(s):  
A. A. Levakov
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence Theorems ◽  
Right Hand

scholarly journals CONTINUOUS DEPENDENCE ON THE INITIAL DATA OF THE SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS WITH FRACTIONAL BROWNIAN MOTIONS

2018 ◽  
Vol 54 (2) ◽  
pp. 193-209
Author(s):  
I. V. Kachan
Keyword(s):  
Initial Data ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Continuous Dependence ◽  
Initial Conditions ◽  
Hölder Continuous ◽  
Brownian Motions ◽  
Fractional Brownian Motions ◽  
Right Hand ◽  
The Right

In the present acticle we consider finite-dimensional stochastic differential equations with fractional Brownian motions having different Hurst indices larger than 1/3 and a drift. These heterogeneous components of the equations are combined into a single process. The solutions of the equations are understood in the integral sense, and the integrals in turn are Gubinelli’s rough path integrals [1] realizing the well-known approach of the rough paths theory [2]. The existence and uniqueness conditions of the solutions of these stochastic differential equations are specified. Such conditions are sufficient to obtain the results related the continuous dependence on the initial data. In this article, we have first proved a continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations under consideration for almost all their trajectories. The result obtained does not depend on the probabilistic properties of fractional Brownian motions, and therefore it can be easily generalized to the case of arbitrary Holder-continuous processes with an exponent greater than 1/3. In this case, the constant arising in the estimates appears to be exponentially dependent on the norms of fractional Brownian motions. Taking into account the last fact and the proved result, an expected logarithmic continuous dependence on the initial conditions and the right-hand sides of the solutions of the stochastic differential equations con - si dered is subsequently derived. This is the major result of this article.

scholarly journals Existence Theorems for Solutions to Set-Valued Stochastic Differential Equations and Applications

2012 ◽  
Vol 1 ◽  
pp. 1-15
Author(s):  
Vu Ho ◽  
Van Hoa Ngo
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Wiener Process ◽  
Langevin Equation ◽  
Stock Prices ◽  
Hausdorff Space ◽  
Existence Theorems ◽  
Uniqueness Of Solution ◽  
Hukuhara Derivative ◽  
Interval Valued

In this paper, a class of new stochastic differential equations on semilinear Hausdorff space under Hukuhara derivative, called set-valued stochastic differential equations (SSDEs) driven by a Wiener process. Moreover, some corresponding properties of SSDEs are discussed such as existence, uniqueness of solution. Finaly, we give some applications to models of interval-valued stochastic differential equations such as stock prices model and the Langevin equation.

On the regularity of stochastic difference equations in hyperfinite-dimensional vector spaces and applications to -valued stochastic differential equations

1994 ◽  
Vol 124 (6) ◽  
pp. 1089-1117 ◽  
Author(s):  
Jiang-Lun Wu
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Difference Equations ◽  
Nonstandard Analysis ◽  
Existence Theorems ◽  
Vector Spaces ◽  
Stochastic Integrals ◽  
Dimensional Vector ◽  
Stochastic Difference Equations ◽  
Infinite Dimensional

Nonstandard analysis is used, in this paper, to give a construction of a Wiener -process Wt, t ∈ [0, ∞). From this, a hyperfinite representation of stochastic integrals for operatorvalued processes with respect to Wt is derived, and existence theorems in the spirit of Keisler are proved for (infinite-dimensional) stochastic differential equations of Itô's type one and a certain kind of Itô's type two, via regularity of hyperfinite stochastic difference equations.

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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