non-Lipschitz Coefficients Research Articles

Backward Doubly Stochastic Differential Equations with Stochastic Non-Lipschitz Coefficients

Backward Doubly Stochastic Differential Equations with Stochastic Non-Lipschitz Coefficients

Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions

Regularity preservation in Kolmogorov equations for non-Lipschitz coefficients under Lyapunov conditions

Stochastic averaging principle for neutral stochastic functional differential equations driven by G-Lévy process

In this paper, we aim to establish an averaging principle for neutral stochastic functional differential equations driven by G-Lévy processes with non-Lipschitz coefficients. Utilizing the theory of sub-linear expectations, we show that the solutions of the considered stochastic systems can be approximated by the solutions of averaged neutral stochastic functional differential equations driven by G-Lévy processes in the mean square convergence and convergence in the sense of capacity. Finally, we give an example to support our main results.

Existence of weak solutions to stochastic heat equations driven by truncated α-stable white noises with non-Lipschitz coefficients

We consider a class of stochastic heat equations driven by truncated α-stable white noises for α∈(1,2) with noise coefficients that are continuous but not necessarily Lipschitz continuous. We prove the existence of weak solution in probabilistic sense, taking values in two different forms under different conditions, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. More precisely, for α∈(1,2) there exists a measure-valued weak solution. However, for α∈(1,5/3) there exists a function-valued weak solution, and in this case we further show that for p∈(α,5/3) the uniform p-th moment in Lp-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in Lp sense.

SIR epidemic model with non-Lipschitz stochastic perturbations

This paper investigates a stochastic SIR epidemic model with non-Lipschitz coefficients in the diffusion term. Firstly, we study the existence and uniqueness of the global positive solution of this model. Then, we prove that the stochastic process has a stationary distribution, and we also establish an exact expression of the density function of the stationary distribution under certain parameter constraints. Finally, we discuss the extinction of the epidemic and the asymptotic properties around the equilibrium points of the deterministic SIR model.

Multi-dimensional mean-reflected BSDEs driven by [formula omitted]-Brownian motion with time-varying non-Lipschitz coefficients

In this paper, we focus on the well-posedness problem of the multi-dimensional mean-reflected BSDEs driven by G-Brownian motion (G-BSDEs) with time-varying non-Lipschitz coefficients. The existence and uniqueness of the solution are gotten by systematically using nonlinear stochastic analysis and Picard iteration argument only for the Y component. Moreover, the backward Bihari’s inequality and some existing results of time-varying Lipschitz G-BSDEs play a key role in the proof.

SPDEs with non-Lipschitz coefficients and nonhomogeneous boundary conditions

SPDEs with non-Lipschitz coefficients and nonhomogeneous boundary conditions

Delay BSDEs driven by fractional Brownian motion

Abstract This paper deals with a class of delay backward stochastic differential equations driven by fractional Brownian motion (with Hurst parameter H greater than 1 2 {\frac{1}{2}} ). In this type of equation, a generator at time t can depend not only on the present but also on the past solutions. We essentially establish existence and uniqueness of a solution in the case of Lipschitz coefficients and non-Lipschitz coefficients. The stochastic integral used throughout the paper is a divergence-type integral.

Correction: Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Correction: Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Mean-square convergence rates of implicit Milstein type methods for SDEs with non-Lipschitz coefficients

Well-posedness of a system of SDEs driven by jump random measures

We establish well-posedness for a class of systems of SDEs with non-Lipschitz coefficients in the diffusion and jump terms and with two sources of interdependence: a monotone function of all the components in the drift of each SDE and the correlation between the driving Brownian motions and jump random measures. Pathwise uniqueness is derived by employing some standard techniques. Then, we use a comparison theorem along with our uniqueness result to construct non-negative, [Formula: see text]-integrable càdlàg solutions as monotone limits of solutions to approximating SDEs, allowing for time-inhomogeneous drift terms to be included. Our approach allows also for a comparison property to be established for the solutions to the systems we investigate. The applicability of certain systems in financial modeling is also discussed.

Fractional Itô–Doob Stochastic Differential Equations Driven by Countably Many Brownian Motions

This article is devoted to showing the existence and uniqueness (EU) of a solution with non-Lipschitz coefficients (NLC) of fractional Itô-Doob stochastic differential equations driven by countably many Brownian motions (FIDSDECBMs) of order ϰ∈(0,1) by using the Picard iteration technique (PIT) and the semimartingale local time (SLT).

Stochastic Integral Evolution Equations with Locally Monotone and Non-Lipschitz Coefficients

In this work the existence and uniqueness of strong solutions are established for a class of stochastic integral evolution equations with locally monotone and non-Lipschitz coefficients. In particular, our results can be applied to a large class of stochastic partial differential equation models with hereditary or memory effects such as quasilinear SPDEs like stochastic porous medium equations and semilinear SPDEs like stochastic 2D Navier—Stokes equations with hereditary or memory terms.

On the Discrete-Time Simulation of the Rough Heston Model

We study Euler-type discrete-time schemes for the rough Heston model, which can be described by a stochastic Volterra equation (with non-Lipschitz coefficient functions) or by an equivalent integrated variance formulation. Using weak convergence techniques, we prove that the limits of the discrete-time schemes are a solution to some modified Volterra equations. Such modified equations are then proved to share the same unique solution as the initial equations, which implies the convergence of the discrete-time schemes. Numerical examples are also provided in order to evaluate different derivative options prices under the rough Heston model.

Mean Square Finite-Approximate Controllability of Semilinear Stochastic Differential Equations with Non-Lipschitz Coefficients

In this paper, we present a study on mean square approximate controllability and finite-dimensional mean exact controllability for the system governed by linear/semilinear infinite-dimensional stochastic evolution equations. We introduce a stochastic resolvent-like operator and, using this operator, we formulate a criterion for mean square finite-approximate controllability of linear stochastic evolution systems. A control is also found that provides finite-dimensional mean exact controllability in addition to the requirement of approximate mean square controllability. Under the assumption of approximate mean square controllability of the associated linear stochastic system, we obtain sufficient conditions for the mean square finite-approximate controllability of a semilinear stochastic systems with non-Lipschitz drift and diffusion coefficients using the Picard-type iterations. An application to stochastic heat conduction equations is considered.

Well-posedness of mean reflected BSDEs with non-Lipschitz coefficients

This paper aims at solving a new type of BSDE with mean reflection under weaker assumptions on the coefficients. We establish the well-posedness of mean reflected BSDEs whenever the generator is non-Lipschitz in the y argument.

Continuity and approximation properties of solutions to fractional neutral stochastic functional differential equations with non-Lipschitz coefficients

ABSTRACT This paper aims to investigate a fractional neutral stochastic functional differential equation (FNSFDE) with non-Lipschitz coefficients. Under the assumptions, we first establish the continuity of the solution in the fractional order of the equation. Furthermore, an Euler-Maruyama (EM) approximation is constructed and then we obtain the strong convergence of the numerical scheme. Specially, if the non-Lipschitz conditions are replaced with the Lipschitz conditions, we shall get a definite convergence rate, which is related to the fractional order of the equation. Finally, we consider the averaging principle for the fractional neutral stochastic equation, which provides us with an easy way to study the properties of the equation.

A Probabilistic Method for a Class of Non-Lipschitz BSDEs with Application to Fund Management

The present work is devoted to a study of the solvability of a class of non-Lipschitz and noncanonical backward stochastic differential equations (BSDEs) that naturally arises from an intertemporal mutual fund management problem; to this end, we propose a method of combining the techniques of Malliavin calculus and a discussion on the Jacobian flow of the BSDE. Specifically, based on the intimate relationship between $Y_t$ and $Z_t$ of the BSDE via the Malliavin derivative of the former, namely, $D_tY_t=Z_t$, we construct an iterative Picard converging scheme for approximating the underlying solution pair by first obtaining $Z_t$ from the derived BSDE with respect to the Malliavin derivative and then recovering $Y_t$ from the underlying BSDE. A local unique existence result is first warranted over a short time horizon with carefully examined a priori estimates; indeed, each term in the iterative sequence is related to different Girsanov transforms for change of measure, and comparing them demands a delicate analysis. The use of Jacobian flow further enables us to properly control the lower and upper bounds for a certain product of the forward process and $Z_t$, which enables us to extend the solution globally by an inductive argument. Our proposed method is fundamentally different from other probabilistic methods that also involve estimating or bounding Malliavin traces such as [E. Pardoux and S. Peng, Some Backward SDEs with Non-Lipschitz Coefficients, Technical note, Université de Provence, Aix-en-Provence, France, 1996] and [M. C. Zedouri, Equations Différentielles Stochastiques Rétrogrades avec Générateurs Lipschitiziens Stochastiques, Master's Thesis, Université Mohammed Seddik Ben Yahia-Jijel, Jijel, Algeria, 2010]. We believe that our new approach proposed here can be potentially applied to resolve many other general non-Lipschitz forward-backward stochastic differential equations (FBSDEs) encountered in economics and finance, especially in the presence of generic utility functions.

Stochastic functional differential equations with infinite delay under non-Lipschitz coefficients: Existence and uniqueness, Markov property, ergodicity, and asymptotic log-Harnack inequality

This paper focuses on a class of stochastic functional differential equations with infinite delay and non-Lipschitz coefficients. Under one-sided super-linear growth and non-Lipschitz conditions, this paper establishes the existence and uniqueness of strong solutions and strong Markov properties of the segment processes. Under additional assumption on non-degeneracy of the diffusion coefficient, exponential ergodicity for the segment process is derived by using asymptotic coupling method. In addition, the asymptotic log-Harnack inequality is established for the associated Markovian semigroup by using coupling and change of measures, which implies the asymptotically strong Feller property. Finally, an example is given to demonstrate these results.

Anticipated Backward Doubly Stochastic Differential Equations with Non-Lipschitz Coefficients

The work presented in this paper focuses on a type of differential equations called anticipated backward doubly stochastic differential equations (ABDSDEs) whose generators not only depend on the anticipated terms of the solution (Y·,Z·) but also satisfy one kind of non-Lipschitz assumption. Firstly, we give the existence and uniqueness theorem. Further, two comparison theorems for the solutions of these equations are obtained after finding a new comparison theorem for backward doubly stochastic differential equations (BDSDEs) with non-Lipschitz coefficients.