Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.

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Existence and Uniqueness Theorem for Stochastic Differential Equations with Self-Exciting Switching

Discrete Dynamics in Nature and Society ◽

10.1155/2011/549651 ◽

2011 ◽

Vol 2011 ◽

pp. 1-12

Author(s):

Guixin Hu ◽

Ke Wang

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Uniqueness Theorem ◽

Existence And Uniqueness ◽

Sufficient Condition ◽

Existence And Uniqueness Theorem ◽

Uniqueness Of The Solution

We introduce a new kind of equation, stochastic differential equations with self-exciting switching. Firstly, we give some preliminaries for this kind of equation, and then, we get the main results of our paper; that is, we gave the sufficient condition which can guarantee the existence and uniqueness of the solution.

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Lp Solutions of BSDEs with Stochastic Lipschitz Condition

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/2007/78196 ◽

2007 ◽

Vol 2007 ◽

pp. 1-14 ◽

Cited By ~ 8

Author(s):

Jiajie Wang ◽

Qikang Ran ◽

Qihong Chen

Keyword(s):

Brownian Motion ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Lipschitz Condition ◽

Existence And Uniqueness ◽

Special Class ◽

Lipschitz Continuity ◽

Backward Stochastic Differential Equations ◽

Uniqueness Of The Solution ◽

Lp Solutions

We are concerned with the solutions of a special class of backward stochastic differential equations which are driven by a Brownian motion, where the uniform Lipschitz continuity is replaced by a stochastic one. We prove the existence and uniqueness of the solution in Lp with p>1.

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Some Existence Results for Systems of Impulsive Stochastic Differential Equations

Annales Mathematicae Silesianae ◽

10.2478/amsil-2020-0028 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Sliman Mekki ◽

Tayeb Blouhi ◽

Juan J. Nieto ◽

Abdelghani Ouahab

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Banach Spaces ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Sufficient Conditions ◽

Existence Results ◽

Impulsive Systems ◽

Uniqueness Of Solutions

Abstract In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.

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Periodic Solutions for Stochastic Differential Equations Driven by General Counting Processes: Application to Malaria

Journal of Mathematics Research ◽

10.5539/jmr.v13n4p1 ◽

2021 ◽

Vol 13 (4) ◽

pp. 1

Author(s):

KOUAME Yao Simplice ◽

NZI Modeste

Keyword(s):

Periodic Solution ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Markov Processes ◽

Existence And Uniqueness ◽

Sufficient Conditions ◽

Uniqueness Result ◽

Poisson Processes ◽

Counting Processes ◽

Periodic Environment

In this paper, a class of periodic stochastic differential equations driven by general counting processes (SDEsGp) is studied. First, an existence-uniqueness result for the solution of general SDEsGp based on Poisson processes with т-periodic stochastic intensity of time t has been given, for some  т> 0. Then, using the properties of periodic Markov processes, sufficient conditions for the existence and uniqueness of a periodic solution of the considered equations are obtained. We will then apply the obtained results to the propagation of malaria in a periodic environment.

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Bounded solutions of functional differential equations of the neutral type with infinite delays

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500031632 ◽

1982 ◽

Vol 93 (1-2) ◽

pp. 33-39 ◽

Cited By ~ 1

Author(s):

V. G. Angelov ◽

D. D. Bainov

Keyword(s):

Differential Equations ◽

Initial Value Problem ◽

Existence And Uniqueness ◽

Functional Differential Equations ◽

Neutral Type ◽

Sufficient Conditions ◽

Bounded Solutions ◽

Initial Value ◽

Infinite Delays ◽

Functional Differential

SynopsisIn this paper the authors obtain sufficient conditions for the existence and uniqueness of the initial value problem of functional differential equations of neutral type with infinite delays, making use of some earlier results of the present authors.

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REFLECTED BSDES WITH STOCHASTIC MONOTONE GENERATOR AND APPLICATION TO VALUING AMERICAN OPTIONS

International Journal of Theoretical and Applied Finance ◽

10.1142/s021902492050034x ◽

2020 ◽

Vol 23 (05) ◽

pp. 2050034

Author(s):

MOHAMED MARZOUGUE

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Existence And Uniqueness ◽

American Options ◽

Growth Conditions ◽

Backward Stochastic Differential Equations ◽

Stochastic Monotonicity ◽

Uniqueness Of The Solution ◽

Fair Valuation ◽

Reflected Bsdes

In this paper, we prove the existence and uniqueness of the solution to backward stochastic differential equations with lower reflecting barrier in a Brownian setting under stochastic monotonicity and general increasing growth conditions. As an application, we study the fair valuation of American options.

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Existence and Uniqueness of Mild Solutions for Nonlinear Stochastic Impulsive Differential Equation

Abstract and Applied Analysis ◽

10.1155/2011/439724 ◽

2011 ◽

Vol 2011 ◽

pp. 1-10

Author(s):

L. J. Shen ◽

J. T. Sun

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Existence And Uniqueness ◽

Stochastic Analysis ◽

Impulsive Differential Equation ◽

Mild Solutions ◽

Sufficient Conditions ◽

Impulsive Differential Equations ◽

Analysis Technique

This paper investigates the existence and uniqueness of mild solutions to the general nonlinear stochastic impulsive differential equations. By using Schaefer's fixed theorem and stochastic analysis technique, we propose sufficient conditions on existence and uniqueness of solution for stochastic differential equations with impulses. An example is also discussed to illustrate the effectiveness of the obtained results.

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Reflected backward stochastic differential equations under monotonicity and general increasing growth conditions

Advances in Applied Probability ◽

10.1239/aap/1113402403 ◽

2005 ◽

Vol 37 (1) ◽

pp. 134-159 ◽

Cited By ~ 9

Author(s):

J.-P. Lepeltier ◽

A. Matoussi ◽

M. Xu

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Unique Solution ◽

Existence And Uniqueness ◽

Growth Conditions ◽

Backward Stochastic Differential Equations ◽

Contingent Claim ◽

Terminal Time ◽

Uniqueness Of The Solution ◽

American Contingent Claim

We prove the existence and uniqueness of the solution to certain reflected backward stochastic differential equations (RBSDEs) with one continuous barrier and deterministic terminal time, under monotonicity, and general increasing growth conditions on the associated coefficient. As an application, we obtain, in some constraint cases, the price of an American contingent claim as the unique solution of such an RBSDE.

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Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

Advances in Difference Equations ◽

10.1186/s13662-021-03313-z ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Muhammad Bahar Ali Khan ◽

Thabet Abdeljawad ◽

Kamal Shah ◽

Gohar Ali ◽

Hasib Khan ◽

...

Keyword(s):

Differential Equations ◽

Delay Differential Equations ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Research Work ◽

Sufficient Conditions ◽

Point Of View ◽

Uniqueness Of The Solution ◽

Delay Differential ◽

Ulam Type Stability

AbstractIn this research work, a class of multi-term fractional pantograph differential equations (FODEs) subject to antiperiodic boundary conditions (APBCs) is considered. The ensuing problem involves proportional type delay terms and constitutes a subclass of delay differential equations known as pantograph. On using fixed point theorems due to Banach and Schaefer, some sufficient conditions are developed for the existence and uniqueness of the solution to the problem under investigation. Furthermore, due to the significance of stability analysis from a numerical and optimization point of view Ulam type stability and its various forms are studied. Here we mention different forms of stability: Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam Rassias (HUR) and generalized Hyers–Ulam–Rassias (GHUR). After the demonstration of our results, some pertinent examples are given.

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On the Existence and Uniqueness of Solutions for Multidimensional Fractional Stochastic Differential Equations with Variable Order

Mathematics ◽

10.3390/math9172106 ◽

2021 ◽

Vol 9 (17) ◽

pp. 2106

Author(s):

Seyfeddine Moualkia ◽

Yong Xu

Keyword(s):

Differential Equations ◽

Stochastic Differential Equations ◽

Fractional Order ◽

Existence And Uniqueness ◽

Sufficient Conditions ◽

Variable Order ◽

Uniqueness Of Solutions ◽

Fractional Stochastic Differential Equations ◽

Picard Iterations

Fractional stochastic differential equations are still in their infancy. Based on some existing results, the main difficulties here are how to deal with those equations if the fractional order is varying with time and how to confirm the existence of their solutions in this case. This paper is about the existence and uniqueness of solutions to the fractional stochastic differential equations with variable order. We prove the existence by using the Picard iterations and propose new sufficient conditions for the uniqueness.

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