non-Lipschitz Condition Research Articles

Averaging principle for stochastic Caputo fractional differential equations with non-Lipschitz condition

Averaging principle for stochastic Caputo fractional differential equations with non-Lipschitz condition

Finite-time stability and uniqueness theorem of solutions of nabla fractional $ (q, h) $-difference equations with non-Lipschitz and nonlinear conditions

<abstract><p>In this paper, the discrete $ (q, h) $-fractional Bihari inequality is generalized. On the grounds of inequality, the finite-time stability and uniqueness theorem of solutions of $ (q, h) $-fractional difference equations with non-Lipschitz and nonlinear conditions is concluded. In addition, the validity of our conclusion is illustrated by a nonlinear example with a non-Lipschitz condition.</p></abstract>

Consumption and portfolio optimization with generalized stochastic differential utility in incomplete markets

This paper examines continuous time intertemporal consumption and portfolio choice problems of an investor in a generalized stochastic differential utility preference of Epstein–Zin type with subjective beliefs and with ambiguity, respectively. In these two situations, the aggregators of the utility both contain the “variability” process. Based on the generalized verification theorem which formulates Hamilton–Jacobi-Bellman equations under a weak non-Lipschitz condition, we show the explicit closed-form optimal consumption and portfolio solutions with subjective beliefs and the numerical solutions with ambiguity for Heston model in incomplete market, respectively.

Stochastic averaging principle for McKean–Vlasov SDEs driven by Lévy noise

In this paper, we study McKean–Vlasov stochastic differential equations driven by Lévy processes. Firstly, under the non-Lipschitz condition which include classical Lipschitz conditions as special cases, we establish the existence and uniqueness for solutions of McKean–Vlasov stochastic differential equations using Carathéodory approximation. Then under certain averaging conditions, we establish a stochastic averaging principle for McKean–Vlasov stochastic differential equations driven by Lévy processes. We find that the solutions to stochastic systems concerned with Lévy noise can be approximated by solutions to averaged McKean–Vlasov stochastic differential equations driven by Lévy processes in the sense of convergence in [Formula: see text]th moment.

RETRACTED ARTICLE: Averaging principle for stochastic Caputo fractional differential equations with non-Lipschitz condition

RETRACTED ARTICLE: Averaging principle for stochastic Caputo fractional differential equations with non-Lipschitz condition

Effective dynamics for a class of stochastic weakly damped wave equation with a fast oscillation

The purpose of this paper is to consider the effective dynamic behavior of a class of stochastic weakly damped wave equations with a fast oscillation under the non-Lipschitz condition. We show that the slow component converges to the solution of the corresponding average equation. The result presented here extends the existing results from the Lipschitz to non-Lipschitz condition, which is a much weaker condition with a wider range of applications.

Stability analysis for a class of stochastic delay nonlinear systems driven by G-Lévy Process

The regularity and stability of the solution to a class of stochastic delay differential equation driven by G-Lévy processes are studied in this paper. Firstly, we introduce a new Burkholder–Davis–Gundy (BDG) inequality involving the jump measure. Secondly, we use the BDG inequality to establish the existence and uniqueness of the solution under non-Lipschitz condition. Thirdly, we establish the existence, uniqueness, quasi-sure exponential stability and pth moment exponential stability of the solution under local Lipschitz condition and one-sided polynomial growth condition.

On the averaging principle for stochastic differential equations driven by [formula omitted]-Lévy process

In this paper, we investigate the averaging principle for stochastic differential equation driven by G-Lévy process. By the BDG inequality for G-stochastic calculus with respect to G-Lévy process, we show that the solution of averaged stochastic differential equation driven by G-Lévy process converges to that of the standard one, under non-Lipschitz condition, in the mean square sense and also in capacity. An example is presented to illustrate the efficiency of the obtained results.

The Existence, Uniqueness, and Carathéodory’s Successive Approximation of Fractional Neutral Stochastic Differential Equation

The existence, uniqueness, and Carathe´odory’s successive approximation of the fractional neutral stochastic differential equation (FNSDE) in Hilbert space are considered in this paper. First, we give the Carathe´odory’s approximation solution for the FNSDE with variable time delays. We then establish the boundedness and continuity of the mild solution and Carathe´odory’s approximation solution, respectively. We prove that the mean-square error between the exact solution and the approximation solution depends on the supremum of time delay. Next, we give the Carathe´odory’s approximation solution for the general FNSDE without delay. Under uniform Lipschitz condition and linear growth condition, we show that the proof of the convergence of the Carathe´odory approximation represents an alternative to the procedure for establishing the existence and uniqueness of the solution. Furthermore, under the non-Lipschitz condition, which is weaker than Lipschitz one, we establish the existence and uniqueness theorem of the solution for the FNSDE based on the Carathe´odory’s successive approximation. Finally, a simulation is given to demonstrate the effectiveness of the proposed methods.

Global Adaptive Control for Uncertain Nonlinear Systems under Non-Lipschitz Condition with Quantized States

A novel adaptive control method is designed for the uncertain nonlinear systems with quantized states under the condition that system nonlinearities do not satisfy the Lipschitz condition. In this paper, the global control for an uncertain nonlinear system with quantized states in the case that the system nonlinearities do not satisfy the Lipschitz condition is first achieved. Separation theorem is used to model the uncertainties appropriately and then an adaptive term is designed to help attenuate the effects of system uncertainties. Furthermore, the global boundedness of all signals of the closed-loop system is proved based on the Lyapunov stability theorem. Finally, simulation results are given to demonstrate the effectiveness of the proposed methods.

Stability Analysis of Stochastic Differential Equation Driven by G-Brownian Motion under Non-Lipschitz Condition

This paper is devoted to studying the p-th moment exponential stability for a class of stochastic differential equation (SDE) driven by G-Brownian motion under non-Lipschitz condition. The delays considered in this paper are time-varying delays τ i t 1 ≤ i ≤ 3 . Since the coefficients are non-Lipschitz, the normal enlargement on the coefficients is not available and the Gronwall inequality is not suitable in this case. By Bihari inequality and Itô integral formula, it is pointed out that there exists a constant τ ∗ such that the p-th moment exponential stability holds if the time-varying delays are smaller than τ ∗ .

Ulam–Hyers Stability for a Class of Caputo-Type Fractional Stochastic System with Delays

In this study, we explore the Caputo-type fractional stochastic differential equations (FSDEs) with delays. Firstly, the existence and uniqueness of the solutions for the considered system are obtained under the non-Lipschitz condition by using the Carath e ´ odory approximation method with the aid of Fatous Lemma, It o ^ isometry, Cauchy–Schwarz inequality, and Gronwall’s inequality. Secondly, we study the Ulam–Hyers (U–H) stability of the addressed equations with the aid of the Cauchy–Schwarz inequality and Gronwall’s inequality. Finally, an example with numerical simulation is presented to support the validity of our theory.

A class of backward stochastic Bellman\u2013Bihari\u2019s inequality and its applications

In this paper, we propose and prove several different forms of backward stochastic Bellman–Bihari’s inequality. Then, as two applications, two different types of the comparison theorems for backward stochastic differential equation with stochastic non-Lipschitz condition are presented.

Weighted pseudo asymptotically antiperiodic sequential solutions to semilinear difference equations

In this paper, we firstly introduce the notion of weighted pseudo S-asymptotically ω-antiperiodic sequence and prove some fundamental properties of such sequences. Then we investigate some existence results for weighted pseudo S-asymptotically ω-antiperiodic solutions to a semilinear difference equation of convolution type. We establish some existence and uniqueness of weighted pseudo S-asymptotically ω-antiperiodic sequential solutions under the global Lipschitz growth condition on the second variable of the force term with different Lipschitz coefficients such as a constant, a summable function and a qth summable function, respectively. Particularly with an appropriate selection of a weight, one of our results shows that the strict contraction on the norm of Lipschitz coefficients is not necessarily required for the existence and uniqueness of weighted pseudo S-asymptotically ω-antiperiodic sequential solutions. We also prove some existence results for weighted pseudo S-asymptotically ω-antiperiodic sequential solutions under a local Lipschitz or a non-Lipschitz growth condition on the second variable of the force term, respectively.

Robust consumption portfolio optimization with stochastic differential utility

This paper examines a continuous time intertemporal consumption and portfolio choice problem with a stochastic differential utility preference of Epstein–Zin type for a robust investor, who worries about model misspecification and seeks robust decision rules. We provide a verification theorem which formulates the Hamilton–Jacobi–Bellman–Isaacs equation under a non-Lipschitz condition. Then, with the verification theorem, the explicit closed-form optimal robust consumption and portfolio solutions to a Heston model are given. Also we compare our robust solutions with the non-robust ones, and the comparisons shown in a few figures coincide with our common sense.

Ulam–Hyers Stability of Caputo-Type Fractional Stochastic Differential Equations with Time Delays

In this paper, we study a class of Caputo-type fractional stochastic differential equations (FSDEs) with time delays. Under some new criteria, we get the existence and uniqueness of solutions to FSDEs by Carath e ´ odory approximation. Furthermore, with the help of H o ¨ lder’s inequality, Jensen’s inequality, It o ^ isometry, and Gronwall’s inequality, the Ulam–Hyers stability of the considered system is investigated by using Lipschitz condition and non-Lipschitz condition, respectively. As an application, we give two representative examples to show the validity of our theories.

Stochastic partial differential equations with gradient driven by space-time fractional noises

We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Holder continuity of the solution to the SPDEs.

Existence results for second-order neutral stochastic equations driven by Rosenblatt process

In this paper we consider a class of second-order impulsive stochastic functional differential equations driven simultaneously by a Rosenblatt process and a standard Brownian motion in a Hilbert space. We prove an existence and uniqueness result under non-Lipschitz condition which is weaker than Lipschitz one and we establish some conditions ensuring the controllability for the mild solution by means of the Banach fixed point principle. At the end we provide a practical example in order to illustrate the viability of our result. <br><br> Розглянуто клас імпульсних стохастичних функціонально-диференціальних рівнянь другого порядку, які керуються процесом Розенблата і стандартним броунівський рухом у ґільбертовому просторі одночасно за умови, яка є слабкішою за умови Ліпшица. Також встановлено умови керованості для помірного розв'язку за допомоги принципу Банаха про нерухому точку. Наведено приклад з практики, що ілюструє отримані результати.

On finite-time stability and stabilization of nonlinear hybrid dynamical systems

<abstract> Finite time stability involving dynamical systems whose trajectories converge to a Lyapunov stable equilibrium state in finite time have been studied for both continuous-time and discrete-time systems. For continuous-time systems, finite time stability is defined for equilibria of continuous but non-Lipschitzian nonlinear dynamics, whereas discrete-time systems can exhibit finite time stability even when the system dynamics are linear, and hence, Lipschitz continuous. Alternatively, for impulsive dynamical systems it may be possible to reset the system states to an equilibrium state achieving finite time stability without requiring a non-Lipschitz condition for the continuous-time part of the hybrid system dynamics. In this paper, we develop sufficient Lyapunov conditions for finite time stability of impulsive dynamical systems using both a scalar differential Lyapunov inequality on the continuous-time dynamics as well as a scalar difference Lyapunov inequality on the discrete-time resetting dynamics. Furthermore, using our proposed finite time stability results, we design universal hybrid finite time stabilizing control laws for impulsive dynamical systems. Finally, we present several numerical examples for finite time stabilization of network impulsive dynamical systems. </abstract>

Existence and uniqueness of weak solution to a three-dimensional stochastic modified-Leray-alpha model of fluid turbulence

In this paper, we study the stochastic three-dimensional modified Leray-alpha model arising from the turbulent flows of fluids. We prove the existence of the probabilistic weak solution under the non-Lipschitz condition for the nonlinear forcing terms. We also discuss its uniqueness.