scholarly journals The numerical solution of stochastic differential equations

1977 ◽  
Vol 20 (1) ◽  
pp. 8-12 ◽  
Author(s):  
P. E. Kloeden ◽  
R. A. Pearson
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Numerical Solution ◽  
Stochastic Differential Equations ◽  
Iterative Scheme ◽  
Correction Term ◽  
Second Order ◽  
Runge Kutta ◽  
The Given

AbstractA method is proposed for the numerical solution of Itô stochastic differential equations by means of a second-order Runge–Kutta iterative scheme rather than the less efficient Euler iterative scheme. It requires the Runge–Kutta iterative scheme to be applied to a different stochastic differential equation obtained by subtraction of a correction term from the given one.

scholarly journals Stability and Boundedness of Solutions to a Certain Second-Order Nonautonomous Stochastic Differential Equation

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
A. T. Ademola ◽  
S. Moyo ◽  
B. S. Ogundare ◽  
M. O. Ogundiran ◽  
O. A. Adesina
Keyword(s):  
Differential Equation ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Relevant Literature ◽  
Second Order ◽  
Boundedness Of Solutions ◽  
Nonlinear Functions ◽  
Lyapunov’S Second Method

This paper focuses on stability and boundedness of certain nonlinear nonautonomous second-order stochastic differential equations. Lyapunov’s second method is employed by constructing a suitable complete Lyapunov function and is used to obtain criteria, on the nonlinear functions, that guarantee stability and boundedness of solutions. Our results are new; in fact, according to our observations from the relevant literature, this is the first attempt on stability and boundedness of solutions of second-order nonlinear nonautonomous stochastic differential equations. Finally, examples together with their numerical simulations are given to authenticate and affirm the correctness of the obtained results.

scholarly journals TOPOLOGICAL EQUIVALENCE FOR DISCONTINUOUS RANDOM DYNAMICAL SYSTEMS AND APPLICATIONS

2013 ◽  
Vol 14 (01) ◽  
pp. 1350007 ◽  
Author(s):  
HUIJIE QIAO ◽  
JINQIAO DUAN
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Lévy Processes ◽  
Levy Processes ◽  
Topological Equivalence ◽  
Random Differential Equation ◽  
Jump Discontinuities ◽  
Non Gaussian

After defining non-Gaussian Lévy processes for two-sided time, stochastic differential equations with such Lévy processes are considered. Solution paths for these stochastic differential equations have countable jump discontinuities in time. Topological equivalence (or conjugacy) for such an Itô stochastic differential equation and its transformed random differential equation is established. Consequently, a stochastic Hartman–Grobman theorem is proved for the linearization of the Itô stochastic differential equation. Furthermore, for Marcus stochastic differential equations, this topological equivalence is used to prove the existence of global random attractors.

scholarly journals Mean reflected stochastic differential equations with jumps

2020 ◽  
Vol 52 (2) ◽  
pp. 523-562
Author(s):  
Phillippe Briand ◽  
Abir Ghannoum ◽  
Céline Labart
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Numerical Scheme ◽  
The Law

AbstractIn this paper, a reflected stochastic differential equation (SDE) with jumps is studied for the case where the constraint acts on the law of the solution rather than on its paths. These reflected SDEs have been approximated by Briand et al. (2016) using a numerical scheme based on particles systems, when no jumps occur. The main contribution of this paper is to prove the existence and the uniqueness of the solutions to this kind of reflected SDE with jumps and to generalize the results obtained by Briand et al. (2016) to this context.

scholarly journals Lie-Point Symmetries and Backward Stochastic Differential Equations

Symmetry ◽  
2019 ◽  
Vol 11 (9) ◽  
pp. 1153
Author(s):  
Na Zhang ◽  
Guangyan Jia
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Stochastic Differential Equations ◽  
Backward Stochastic Differential Equation ◽  
Backward Stochastic Differential Equations ◽  
Point Symmetry ◽  
Lie Point Symmetries ◽  
Lie Point Symmetry ◽  
Symmetry Method

In this paper, we introduce the Lie-point symmetry method into backward stochastic differential equation and forward–backward stochastic differential equations, and get the corresponding deterministic equations.

scholarly journals An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications

2020 ◽  
Vol 28 (1) ◽  
pp. 1-18
Author(s):  
Dahbia Hafayed ◽  
Adel Chala
Keyword(s):  
Differential Equation ◽  
Optimal Control ◽  
Maximum Principle ◽  
Differential Equations ◽  
Stochastic Differential Equation ◽  
Control Problem ◽  
Stochastic Differential Equations ◽  
Doubly Stochastic ◽  
Risk Sensitive ◽  
Performance Functional

AbstractIn this paper, we are concerned with an optimal control problem where the system is driven by a backward doubly stochastic differential equation with risk-sensitive performance functional. We generalized the result of Chala [A. Chala, Pontryagin’s risk-sensitive stochastic maximum principle for backward stochastic differential equations with application, Bull. Braz. Math. Soc. (N. S.) 48 2017, 3, 399–411] to a backward doubly stochastic differential equation by using the same contribution of Djehiche, Tembine and Tempone in [B. Djehiche, H. Tembine and R. Tempone, A stochastic maximum principle for risk-sensitive mean-field type control, IEEE Trans. Automat. Control 60 2015, 10, 2640–2649]. We use the risk-neutral model for which an optimal solution exists as a preliminary step. This is an extension of an initial control system in this type of problem, where an admissible controls set is convex. We establish necessary as well as sufficient optimality conditions for the risk-sensitive performance functional control problem. We illustrate the paper by giving two different examples for a linear quadratic system, and a numerical application as second example.

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