Existence, uniqueness, and stability of uncertain delay differential equations with V-jump

Zhifu Jia,Cunlin Li,Xinsheng Liu

doi:10.1186/s13662-020-02895-4

Abstract

No previous study has involved uncertain delay differential equations with jump. In this paper, we consider the uncertain delay differential equations with V-jump, which is driven by both an uncertain V-jump process and an uncertain canonical process. First of all, we give the equivalent integral equation. Next, we establish an existence and uniqueness theorem of solution to the differential equations we proposed in the finite domain and the infinite domain, respectively. Once more, the concept of stability for uncertain delay differential equations with V-jump is proposed. In addition, the sufficient condition for stability theorem is derived. To judge existence, uniqueness, and stability briefly, we provide some examples in the end.

Highlights

More than half a century ago, when the Itô’s [1] landmark work “On stochastic differential equations” (Itô, 1951) came out, the stochastic differential equations (SDEs), as a new branch of mathematics, aroused great interest in academic circles
SDEs have accumulated many results, which played an important role in financial [2], control theory [3], biomathematics [4], game theory [5], and other models hidden in the observed data
Sometimes we have a lot of available sample data, the frequency obtained by sample data is, not close enough to the distribution function obtained in some practical problems, and we need to invite some domain experts to evaluate the belief degree that each event may happen in these situations

Summary

Introduction

More than half a century ago, when the Itô’s [1] landmark work “On stochastic differential equations” (Itô, 1951) came out, the stochastic differential equations (SDEs), as a new branch of mathematics, aroused great interest in academic circles. 2, we prove an existence, uniqueness, and stability theorem of the solution to uncertain delay differential equations with V -jump and give some examples. The following Theorem 1 will give the result of existence and uniqueness of uncertain delay integral equation with V -jump (8). According to Theorem 1, the uncertain delay integral equation with V -jump (8) has a unique solution on the local interval [k0, k0 + α]. Proof Define ρ = {k | uncertain delay integral equation with V -jump (7) has a unique continuous solution on [0, k)}, and ρ = sup. Theorem 1 means that there exists a positive number α such that uncertain delay with V -jump integral equation (13) has a unique continuous solution Zk(γ ) on the interval [ρ, ρ + α]. We can obtain the explicit solution of uncertain delay differential equation with V -jump (15)

Stability of the solution

Conclusions