Strong comparison result for a class of reflected stochastic differential equations with non-Lipschitzian coefficients

In this paper, we are concerned with a class of reflected stochastic differential equations (reflected SDEs) with non-Lipschitzian coefficients. Under the same coefficients assumptions as Fang and Zhang [Probab. Theory Relat. Fields, 2005, 132(3): 356–390] for a class of SDEs, we establish the pathwise uniqueness for the reflected SDEs. Furthermore, a strong comparison theorem is proved for the reflected SDEs in a one-dimensional case.      

An Approximation Scheme for Reflected Stochastic Differential Equations with Non-Lipschitzian Coefficients

Journal of Theoretical Probability - In this paper, we study a numerical approximation scheme for reflected stochastic differential equations (SDEs) with non-Lipschitzian coefficients in a bounded...     

Strong Convergence of Wong-Zakai Approximations of Reflected SDEs in a Multidimensional General Domain

In this paper, we obtain the strong convergence of Wong-Zakai approximations of reflected SDEs in a general multidimensional domain giving an affirmative answer to the question posed by Evans and Stroock (Stoch. Process. Appl. 121, 1464–1491, 2011).     

A class of degenerate stochastic differential equations with non-Lipschitz coefficients

We obtain sufficient condition for SDEs to evolve in the positive orthant. We use arguments based on comparison theorems for SDEs to achieve this. As an application we prove the existence of a unique strong solution for a class of multidimensional degenerate SDEs with non-Lipschitz diffusion coefficients.     

Reflected forward–backward stochastic differential equations with continuous monotone coefficients

In this work, we prove that there exists at least one solution for the reflected forward–backward stochastic differential equations satisfying the obstacle constraint with continuous monotone coefficients. The distinct character of our result is that the coefficient of the forward SDEs contains the solution variable of the reflected BSDEs.     

RELATIONS BETWEEN SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SEMIMARTINGALE WITH NON-LIPSCHITZ COEFFICIENTS

In this paper, a class of stochastic differential equations (SDEs) driven by semi-martingale with non-Lipschitz coefficients is studied. We investigate the dependence of solutions to SDEs on the initial value. To obtain a continuous version, we impose the conditions on the local characteristic of semimartingale. In this case, it gives rise to a flow of homeomorphisms if the local characteristic is compactly supported.     

A class of orthogonal integrators for stochastic differential equations

The purpose of this paper is to construct a class of orthogonal integrators for stochastic differential equations (SDEs). The family of SDEs with orthogonal solutions is univocally characterized. For this, a class of orthogonal integrators is introduced by imposing constraints to Runge–Kutta (RK) matrices and weights of the standard stochastic RK schemes.The performance of the method is illustrated by means of numerical simulations.     

Mean square polynomial stability of numerical solutions to a class of stochastic differential equations

The exponential stability of numerical methods to stochastic differential equations (SDEs) has been widely studied. In contrast, there are relatively few works on polynomial stability of numerical methods. In this letter, we address the question of reproducing the polynomial decay of a class of SDEs using the Euler–Maruyama method and the backward Euler–Maruyama method. The key technical contribution is based on various estimates involving the gamma function.     

First order strong approximations of scalar SDEs defined in a domain

We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the SDEs and is strongly convergent with rate one. Moreover, we show that this general result can be applied to many SDEs we encounter in mathematical finance and bio-mathematics. We will demonstrate flexibility of our approach by analyzing classical examples of SDEs with sublinear coefficients (CIR, CEV models and Wright–Fisher diffusion) and also with superlinear coefficients (3/2-volatility, Aït-Sahalia model). Our goal is to justify an efficient Multilevel Monte Carlo method for a rich family of SDEs, which relies on good strong convergence properties.     

Characterizing Gaussian Flows Arising from Itô’s Stochastic Differential Equations

In order to identify which of the strong solutions of Itô’s stochastic differential equations (SDEs) are Gaussian, we introduce a class of diffusions which ‘depend deterministically on the initial condition’ and then characterize the class. This characterization allows us to show, using the Monotonicity inequality, that the transpose of the flows generated by the SDEs, for an extended class of initial conditions, are the unique solutions of the class of stochastic partial differential equations introduced in Rajeev and Thangavelu (Potential Anal. 28(2), 139–162 2008), ‘Probabilistic Representations of Solutions of the Forward Equations’.     

High strong order stochastic Runge-Kutta methods for Stratonovich stochastic differential equations with scalar noise

This paper concerns the stochastic Runge-Kutta (SRK) methods with high strong order for solving the Stratonovich stochastic differential equations (SDEs) with scalar noise. Firstly, the new SRK methods with strong order 1.5 or 2.0 for the Stratonovich SDEs with scalar noise are constructed by applying colored rooted tree analysis and the theorem of order conditions for SRK methods proposed by Rößler (SIAM J. Numer. Anal. 48(3), 922–952, 2010). Secondly, a specific SRK method with strong order 2.0 for the Stratonovich SDEs whose drift term vanishes is proposed. And another specific SRK method with strong order 1.5 for the Stratonovich SDEs whose drift and diffusion terms satisfy the commutativity condition is proposed. The two specific SRK methods need only to use one random variable and do not need to simulate the multiple Stratonovich stochastic integrals. Finally, the numerical results show that performance of our methods is better than those of well-known SRK methods with strong order 1.0 or 1.5.     

非Lipschitz条件下半鞅随机微分方程解的唯一性(英文)

本文研究了非Lipschitz条件下半鞅随机微分方程.利用It(o)分析和Gronwall不等式,探讨了随机微分方程无爆炸解,并证明了随机微分方程解的唯一性.     

B-convergence of split-step one-leg theta methods for stochastic differential equations

For stochastic differential equations (SDEs) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient, the explicit schemes fail to converge strongly to the exact solution (see, Hutzenthaler, Jentzen and Kloeden in Proc. R. Soc. A, rspa.2010.0348v1?Crspa.2010.0348, 2010). In this article a class of implicit methods, called split-step one-leg theta methods (SSOLTM), are introduced and are shown to be mean-square convergent for such SDEs if the method parameter satisfies $\frac{1}{2}\leq\theta \leq1$ . This result gives an extension of B-convergence from the theta method for deterministic ordinary differential equations (ODEs) to SSOLTM for SDEs. Furthermore, the optimal rate of convergence can be recovered if the drift coefficient behaves like a polynomial. Finally, numerical experiments are included to support our assertions.     

BSDEs and SDEs with time-advanced and -delayed coefficients

ABSTRACT

This paper introduces a class of backward stochastic differential equations (BSDEs), whose coefficients not only depend on the value of its solutions of the present but also the past and the future. For a sufficiently small time delay or a sufficiently small Lipschitz constant, the existence and uniqueness of such BSDEs is obtained. As an adjoint process, a class of stochastic differential equations (SDEs) is introduced, whose coefficients also depend on the present, the past and the future of its solutions. The existence and uniqueness of such SDEs is proved for a sufficiently small time advance or a sufficiently small Lipschitz constant. A duality between such BSDEs and SDEs is established.     

SDELab: A package for solving stochastic differential equations in MATLAB

We introduce SDELab, a package for solving stochastic differential equations (SDEs) within MATLAB. SDELab features explicit and implicit integrators for a general class of Itô and Stratonovich SDEs, including Milstein's method, sophisticated algorithms for iterated stochastic integrals, and flexible plotting facilities.     

Integration by parts formulas concerning maxima of some SDEs with applications to study on density functions

In this article, we prove integration by parts (IBP) formulas concerning maxima of solutions to some stochastic differential equations (SDEs). We will deal with three types of maxima. First, we consider discrete time maximum, and then continuous time maximum in the case of one-dimensional SDEs. Finally, we deal with the maximum of the components of a solution to multi-dimensional SDEs. Applications to study their probability density functions by means of the IBP formulas are also discussed.     

Almost sure exponential stability of backward Euler-Maruyama discretizations for hybrid stochastic differential equations

This is a continuation of the first author’s earlier paper [1] jointly with Pang and Deng, in which the authors established some sufficient conditions under which the Euler-Maruyama (EM) method can reproduce the almost sure exponential stability of the test hybrid SDEs. The key condition imposed in [1] is the global Lipschitz condition. However, we will show in this paper that without this global Lipschitz condition the EM method may not preserve the almost sure exponential stability. We will then show that the backward EM method can capture the almost sure exponential stability for a certain class of highly nonlinear hybrid SDEs.     

SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations

It is known that the transition probabilities of a solution to a classical It? stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Time-fractional Kolmogorov-type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Lévy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman–Kac formula.     

SOLVXBILITY OF FORWARD－BACKWARD SDES AND THE NODAL SET OF HAMILTON－JACOBI－BELLMAN EQUATIONS

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SOLVABILITY OF FORWARD-BACKWARD SDES AND THE NODAL SET OF HAMILTON-JACOBI-BELLMAN EQUATIONS

The solvability of a class of forward-backward stochastic differential equations (SDEs for short) over an arbitrarily prescribed time duration is studied. The authors design a stochastic relaxed control problem, with both drift and difftusion all being controlled, so that the solvability problem is converted to a problem of finding the nodal set of the viscosity solution to a certain Hamilton-Jacobi-Bellman equation. This method overcomes the fatal difficulty encountered in the traditional contraction mapping approach to the existence theorem of such SDEs.