Open AccessExistence and Uniqueness of Solutions for Some Basic Stochastic Differential Equations
Author(s) -
Putong Qi
Publication year - 2021
Publication title -
journal of physics. conference series
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.21
H-Index - 85
eISSN - 1742-6596
pISSN - 1742-6588
DOI - 10.1088/1742-6596/1802/4/042094
Subject(s) - lipschitz continuity , uniqueness , mathematics , stochastic differential equation , ordinary differential equation , martingale (probability theory) , differential equation , mathematical analysis
Stochastic differential equation (sde) has plenty of applications on economic and physics. However, how to find the explicit solution usually remains a serious trouble. In this paper, we aim to study the uniqueness and existence of the stochastic differential equations. To this end, we used the Ito formula and the Lipschitz condition. As a result, we found that the Lipschitz condition is a sufficient condition for a stochastic differential equation to have a unique solution. If the Lipschitz condition can be proved by calculating the index, then it is possible to find the numerical solution. Research on the stochastic differential equation has gained much attention after the finding of the Ito formula. The Ito formula illustrates the chain rule of a semi-martingale. Due to the difficulty in obtaining the explicit solution, one primary problem is whether there exists a unique solution. One approach to solve this problem is the use of Lipschitz condition, which is used in the ordinary differential equations.