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In this article, we study dynamic risk measures by means of backward doubly stochastic Volterra integral equations (BDSVIEs, for short) with jumps. We establish the well-posedness of BDSVIEs with jumps in the sense of M-solution and prove a comparison theorem of BDSVIEs with jumps. Finally, we study properties of dynamic risk measures induced by BDSVIEs with jumps. Our results extend the well-posedness and the comparison theorem of BDSVIEs without jumps to the setting with jumps, and extend dynamic risk measures induced by BSDEs, BDSDEs, and BSVIEs to the case of BDSVIEs with jumps. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in solving backward doubly stochastic differential equations in Lp, for any p ∈ (1 , 2) , when the coefficients are continuous with stochastic linear growth. Via approximation and comparison theorem, the existence of L p − minimal and L p − maximal solutions are obtained. [ABSTRACT FROM AUTHOR]

In this article, we study the viability for the multi-dimensional stochastic system described by Itô stochastic differential equations under the non-Lipschitzian assumption on the coefficients. We establish a necessary and sufficient condition for viability property with respect to a given nonempty closed convex set. And with this result, we explore the multi-dimensional comparison theorems about stochastic differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate an existence of minimal (resp. maximal) solutions for backward doubly stochastic differential equations (BDSDEs, in short) when the coefficient with respect to the forward integral is left (resp. right)-continuous in y and continuous in z with stochastic linear growth in (y, z). Also, the associated comparison theorem is obtained. [ABSTRACT FROM AUTHOR]

This article is interested in Lp ðp > 1Þ solutions of general time interval backward stochastic differential equations (BSDEs) with generator g satisfying a p-order weak stochastic-monotonicity condition and a general growth condition in y. An existence and uniqueness result of the Lp solution is proved for the multidimensional BSDEs where the generator g is stochastic-Lipschitz in z. And, a general comparison theorem and an existence and uniqueness theorem of the Lp solution is established for the one-dimensional BSDEs under a stochastic uniform-continuity condition of the generator g in z. These results strengthen some known works. The stochastic Gronwall-type inequality and the stochastic Bihari-type inequality play an important role in the whole article. [ABSTRACT FROM AUTHOR]

In this paper, we focus on a class of stochastic differential equations driven by the standard Brownian motion and fractional Brownian motion with Hurst parameter 1 / 2 < H < 1 under a weaker conditions than Lipschitz one. In the sense of the pathwise Riemann–Stieltjes integral, we prove the convergence of solutions for the considered equations. By making use of some transformation techniques and approximation means, we obtain some sufficient conditions on the viability for the stochastic systems under investigation. Subsequently, by using some distance functions, we establish some necessary conditions and some sufficient conditions for the viability property with respect to a given non-empty closed set K, but K can be not smooth. As applications of the results, we explore the comparison theorems about the mixed stochastic differential equations driven by fractional Brownian motion under some non-Lipschitz conditions. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the problem of viability property for the multi-dimensional stochastic differential equations, where the coefficients of the stochastic differential equations may be random and discontinuous in the time variable. First, we establish a necessary and sufficient condition for viability property with respect to a given non-empty closed convex set. And furthermore, with the help of this result, we also study the multi-dimensional comparison theorems about stochastic differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we investigate mean-field backward stochastic differential equations driven by G-Brownian motion with uniformly continuous coefficients. The existence and uniqueness theorem are obtained via a monotone convergence argument and a linearization method. Moreover, we establish the corresponding comparison theorem. [ABSTRACT FROM AUTHOR]

Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M-matrix in the Gauss–Seidel double SOR method to obtain a comparison result. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with the zero-sum stochastic differential game problems with the cost functional depending not only on g (X T) but also on { g (X t) , t ∈ [ T , T + θ ] }. The main approach is the anticipated backward stochastic differential equations under weaker conditions. For these equations, we prove the existence and uniqueness theorem and the comparison theorems. Then, by using these techniques, we get the saddle-point strategy for the zero-sum stochastic differential games when the Isaacs' condition holds. [ABSTRACT FROM AUTHOR]

In this paper, we study SDEs with respect to optional semimartingales or optional SDEs. Our leading idea is to explore the concept and technique of local time of optional processes to extend several results on comparison and pathwise uniqueness of solutions of such stochastic equations. We also obtain a comparison result for optional stochastic equations with different jump-diffusions. Moreover, we apply our comparison theorem to calculate option price bounds in mathematical finance. Our findings are supported by numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we study a kind of reflected backward stochastic differential equations (BSDEs) whose generators are of quadratic growth in z and linear growth in y. We first give an estimate of solutions to such reflected BSDEs. Then under the condition that the generators are convex with respect to z, we can obtain a comparison theorem, which implies the uniqueness of solutions for this kind of reflected BSDEs. Besides, the assumption of convexity also leads to a stability property in the spirit of above estimate. We further establish the nonlinear Feynman-Kac formula of the related obstacle problems for partial differential equations (PDEs) in our framework. At last, a numerical example is given to illustrate the applications of our theoretical results, as well as its connection with an optimal stopping time problem. [ABSTRACT FROM AUTHOR]

This paper is devoted to the question of existence and uniqueness of strong solutions of stochastic differential equations with respect to optional semimartingales. Optional semimartingales have right and left limits (ladlag) but are not necessarily continuous, therefore, defined on un usual probability spaces. Integration theory with respect to optional semimartingales is well developed. However, not much attention is given to stochastic equations of optional semimartingales. An existence and uniqueness result for strong solutions of a very general class of optional stochastic equations with non-Lipschitz coefficients is given. Moreover, potential applications to physics and finance are given. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we start with establishing the existence of a minimal (maximal) Lp (1 < p ⩽ 2) solution to a one-dimensional backward stochastic differential equation (BSDE), where the generator g satisfies a p-order weak monotonicity condition together with a general growth condition in y and a linear growth condition in z. Then, we propose and prove a comparison theorem of Lp (1 < p ⩽ 2) solutions to one-dimensional BSDEs with q-order (1 ⩽ q < p) weak monotonicity and uniform continuity generators. As a consequence, an existence and uniqueness result of Lp (1 < p ⩽ 2) solutions is also given for BSDEs whose generator g is q-order (1 ⩽ q < p) weakly monotonic with a general growth in y and uniformly continuous in z. [ABSTRACT FROM AUTHOR]

We first give the existence and uniqueness results for infinite horizon backward stochastic differential equations with Markov chains, taking advantage of the martingale representation theorem and fixed point principle. Then we prove the well-posedness results for infinite horizon reflected backward stochastic differential equations with Markov chains, by virtue of the Snell envelope theory and contraction mapping method. Comparison theorems for the above two kinds of equations are also obtained, via the linearization approach or properties of reflected backward stochastic differential equations, respectively. [ABSTRACT FROM AUTHOR]

This paper studies Backward Stochastic Volterra Integral Equations (BSVIEs) driven by finite state, continuous time Markov chains. First, the existence and uniqueness of the solutions to two types of BSVIEs are established. Second, some scalar and vector comparison theorems are given. Finally, the applications of BSVIEs to a linear-quadratic optimal control problem and time-inconsistent coherent risk measures are presented. [ABSTRACT FROM AUTHOR]

This article mainly considers a new class of anticipated BSDEs driven by Brownian motion and continuous increasing process, which are called generalized anticipated backward stochastic differential equations(GABSDEs). We first give the form of GABSDEs. Then, existence and uniqueness result for GABSDEs is established as well as a comparison theorem is obtained under the certain assumptions. At last, we give an application about the duality between SDDEs and GABSDEs. [ABSTRACT FROM AUTHOR]

This article mainly considers a new class of anticipated BSDEs driven by Brownian motion and continuous increasing process, which are called generalized anticipated backward stochastic differential equations(GABSDEs). We first give the form of GABSDEs. Then, existence and uniqueness result for GABSDEs is established as well as a comparison theorem is obtained under the certain assumptions. At last, we give an application about the duality between SDDEs and GABSDEs. [ABSTRACT FROM AUTHOR]

In this paper, we discuss a class of anticipated backward stochastic differential equations related to a finite continuous time single jump process. We prove the existence and uniqueness of the adapted solution. Moreover, a comparison theorem for the solutions is also established. [ABSTRACT FROM PUBLISHER]

Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to prove an existence and uniqueness result for solutions of a multidimensional backward stochastic differential equation (BSDE) with a general time interval (including the deterministic and stochastic cases), where the generatorgof the BSDE is weakly monotonic and of general growth iny, and Lipschitz continuous inz, both non-uniformly with respect tot. And, the corresponding comparison theorem for the solutions of one-dimensional BSDEs is provided. As applications, we establish a nonlinear Doob-Meyer’s decomposition theorem for general continuousg-supermartingales under an additional assumption of the generatorg. Some new problems in our setting arise naturally and are well overcome. These results generalize and improve some known works. [ABSTRACT FROM PUBLISHER]

Different classes of matrix splittings of semi-monotone matrices lead to many comparison results, which are useful tools in examining the convergence rate of iterative methods for solving rectangular systems of linear equations in a faster way. In this context, the theory of alternating iterations for rectangular matrices was introduced recently, in order to arrive at the desired solution of accuracy or at the exact solution. In this article, we expand convergence theory of such alternating iterations and obtain comparison results for such iterations. [ABSTRACT FROM AUTHOR]

By applying the comparison theorem and the Lyapunov method of the impulsive differential equations, this paper gives some new sufficient conditions for the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution in a class of two-species Lotka–Volterra cooperative systems with impulses. The method used in this paper provides a possible method to study the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the models with impulsive perturbations in biological populations. Finally, an example and numerical simulations are given to illustrate that our results are feasible. [ABSTRACT FROM AUTHOR]

In 2013, Lu and Ren considered anticipated backward stochastic differential equations driven by finite state, continuous time Markov chain noise and established the existence and uniqueness of the solutions of these equations and a scalar comparison theorem. In this article, we provide an estimate for their solutions and study the duality between these equations and stochastic differential delayed equations with Markov chain noise. Finally, we derive another comparison theorem for these solutions depending only on the two drivers. [ABSTRACT FROM PUBLISHER]

In this paper, we deal with a class of anticipated backward stochastic differential equations (anticipated BSDEs) under weak assumptions on the coefficients. We prove the existence and uniqueness result forsolutions of anticipated BSDEs under monotonicity and general increasing conditions on, with Lipschitz on. Furthermore, we give several comparison theorems with respect to 1-dimensional anticipated BSDEs. [ABSTRACT FROM PUBLISHER]

Conditional moment estimates on the cumulative sum of conditionally independent random variables are derived, conditional prophet inequalities for conditionally independent random variables are established, a comparison theorem on conditional moment inequalities between conditionally independent and conditionally negatively associated random variables is established. As applications of these results, a conditional Rosenthal type inequality and two conditional Kolmogorov exponential inequalities for conditionally negatively associated random variables are obtained. [ABSTRACT FROM PUBLISHER]

This paper is concerned with backward stochastic differential equations with Markov switching driven by Brownian motion and Poisson random measure. The motivation is a constrained stochastic Riccati equation derived from a stochastic linear quadratic optimal control problem with both Poisson and Markovian jumps. The existence and uniqueness of an adapted solution under global Lipschitz condition on the generator is obtained. The continuous dependence of the solution on parameters is proved. Two comparison theorems are also derived by a generalized Girsanov transformation theorem. [ABSTRACT FROM PUBLISHER]

In this article, we prove a comparison theorem for multidimensional stochastic differential equations driven byG-Brownian motion (G-SDEs, for short). Moreover, we obtain respectively the sufficient conditions and necessary conditions of the monotonicity and order-preservation for two multidimensionalG-diffusion processes. Finally, we give some applications. [ABSTRACT FROM AUTHOR]

This article studies the theory of discrete-time backward stochastic differential equations (also called BSDEs) with a random terminal time, which is not a stopping time. We follow Cohen and Elliott [2] and consider a reference filtration generated by a general discrete-time finite-state process. The martingale representation theorem for essentially bounded martingales under progressively enlarged filtration is established. Then we prove the existence and uniqueness theorem of BSDEs under enlarged filtration using some weak assumptions of the driver. We also present conditions for a comparison theorem. Applications to nonlinear expectations and optimal design of dynamic default risk are explored. [ABSTRACT FROM PUBLISHER]

In this article, we discuss the existence of multiple solutions to a one-dimensional stochastic differential delay equation with continuous drift coefficients and derive a related comparison theorem. [ABSTRACT FROM AUTHOR]

In this article, a convergence theorem and comparison theorems for Brou-splittings and Bran-splittings (introduced in [D. Mishra, K.C. Sivakumar, Generalizations of matrix monotonicity and their relationships with certain subclasses of proper splittings, Linear Algebra Appl. (accepted)]) are presented. We then conclude with a new characterization of range monotone matrices. [ABSTRACT FROM AUTHOR]

In this paper, we establish global existence of smooth solutions for Cauchy problem of the critical nonlinear wave equation with time dependent variable coefficients in three space dimensions where (g ij (t, x)) is a regular function valued in the spacetime of 3 × 3 positive definite matrix and (g ij (t, x)) its inverse matrix. Here and in the sequence, a repeated sum on an index in lower and upper position is never indicated. In the constant coefficients case, the result of global existence is due to Grillakis [5]; and in the time-independent variable coefficients case, the result of global existence and regularity is due to Ibrahim and Majdoub [7]. The key point of our proofs is to show that the energy cannot concentrate at any point. For that purpose, following Christodoulou and Klainerman [3], we use a null frame associated to an optical function to construct a geometric multiplier similar to the well-known Morawetz multiplier. Then we use comparison theorem originated from Riemannian Geometry to estimate the error terms. Finally, using Strichartz inequality due to [15] as Ibrahim and Majdoub [7], we obtain global existence. [ABSTRACT FROM AUTHOR]

We consider backward stochastic differential equations (BSDEs) related to a finite continuous time single jump process. We prove the existence and uniqueness of solutions when the coefficients satisfy Lipschitz continuity conditions. A comparison theorem for these solutions is also given. Applications to the theory of nonlinear expectations are then investigated. [ABSTRACT FROM AUTHOR]

We study the equation with Hamiltonian H ( x , p ) measurable with respect to the state variable, and convex and coercive in p . We introduce a notion of solution based on viscosity test functions, appropriate averages in measure-theoretic sense of the Hamiltonian, and t -partial-sup-convolutions. We get existence results, comparison principles, and stability properties. We show that our solutions are uniform limits of viscosity solutions (in the sense of Crandall and Lions) of approximated continuous equations. [ABSTRACT FROM AUTHOR]

A class of backward doubly stochastic differential equations (BDSDEs for short) are studied. We obtain a comparison theorem of these BDSDEs. As one of its applications, we derive the existence of solutions for BDSDEs with continuous coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin-Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples. [ABSTRACT FROM AUTHOR]

We deal with reflected backward stochastic differential equations with right continuous and left limited barrier. We show the existence and uniqueness of the solution and we give a comparison theorem. As an application, we study the link between such an equations with stochastic mixed control problems. [ABSTRACT FROM AUTHOR]

The authors consider m -th order nonlinear difference equations of the form D m p x n + δh j ( n , x s j ( n ) )=0, j =1,2,( E j ) where m ≥1, n ∈ N 0 ={0,1,2,…}, D 0 p x n = x n , D i p x n = p n i Δ( D i -1 p x n ), i =1,2,…, m , Δ x n = x n +1 - x n , { p n 1 },…,{ p n m } are real sequences, p n i >0, and p n m ≡1. In Eq. ( E 1 ) , p = a and p n i = a n i , and in Eq. ( E 2 ) , p = A and p n i = A n i , i =1,2,…, m . Here, { s j ( n )} are sequences of nonnegative integers with s j ( n )→∞ as n →∞, and h j : N 0 × R → R is continuous with uh j ( n , u )>0 for u ≠0. They prove a comparison result on the oscillation of solutions and the asymptotic behavior of nonoscillatory solutions of Eq. ( E j ) for j =1,2. Examples illustrating the results are also included. [ABSTRACT FROM AUTHOR]

Given a d -dimensional Wiener process W , with its natural filtration { F t }, a F T -measurable random variable ξ in R , a bounded measure ν on R , and an adapted process ( s , y , z )→ h ( s , y , z ), we consider the following BSDE: Y t = ξ + ∫ t T h ( s , Y s , Z s ) d s +∫ R ( L T a ( Y )- L t a ( Y )) ν (d a )- ∫ t T Z s d W s for 0≤ t ≤ T . Here L t a ( Y ) stands for the local time of Y at level a . For h =0, we establish the existence and the uniqueness of the processes ( Y , Z ), and if h is continuous with linear growth we establish the existence of a solution. We prove limit theorems for solutions of backward stochastic differential equations of the above form. Those limit theorems permit us to deduce that any solution of that equation is the limit, in a strong sense, of a sequence of semi-martingales, which are solutions of ordinary BSDEs of the form Y t = ξ + ∫ t T f ( Y s ) Z s 2 d s - ∫ t T Z s d W s . A comparison theorem for BSDEs involving measures is discussed. As an application we obtain, with the help of the connection between BSDE and PDE, some corresponding limit theorems for a class of singular non-linear PDEs and a new probabilistic proof of the comparison theorem for PDEs. [ABSTRACT FROM AUTHOR]

In this paper, we establish comparison results (maximum principles) which allow us to use the monotone method and the method of upper and lower solutions in order to build convergent sequences to the solutions of difference equations of the type Δ u k = f k , u k +1 , max l ∈{ k - h +1,…, k +1} u l , k ∈ I , u 0 = u T , with Δ u k = u k +1 - u k , I ={0,1,…, T -1} and f ∈ C ( I × R × R , R ). [ABSTRACT FROM AUTHOR]