Set-valued Stochastic Differential Equations Research Articles

Symmetric Fuzzy Stochastic Differential Equations with Generalized Global Lipschitz Condition

The paper contains a discussion on solutions to symmetric type of fuzzy stochastic differential equations. The symmetric equations under study have drift and diffusion terms symmetrically on both sides of equations. We claim that such symmetric equations have unique solutions in the case that equations’ coefficients satisfy a certain generalized Lipschitz condition. To show this, we prove that an approximation sequence converges to the solution. Then, a study on stability of solution is given. Some inferences for symmetric set-valued stochastic differential equations end the paper.

Weak solutions of set-valued stochastic differential equations

The paper is devoted to the existence of weak solutions of set-valued stochastic differential equations. Such equations were considered in the authors paper [22] , where existence and uniqueness of strong solutions have been investigated. Now, apart from the existence of weak solutions, the weak compactness in distribution of the set of all weak solutions of set-valued stochastic differential equations, is considered.

一类泛函集值随机微分方程的解

一类泛函集值随机微分方程的解

Stochastic inclusions and set-valued stochastic equations driven by a two-parameter Wiener process

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations driven by a two-parameter Wiener process. We establish new connections between their solutions. We prove that attainable sets of solutions to such inclusions are subsets of values of multivalued solutions of associated set-valued stochastic equations. Next we show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. Additionally we establish other properties of such solutions. The results obtained in the paper extends results dealing with this topic known both in deterministic and stochastic cases.

On connections between stochastic differential inclusions and set-valued stochastic differential equations driven by semimartingales

In the paper we study properties of solutions to stochastic differential inclusions and set-valued stochastic differential equations with respect to semimartingale integrators. We present new connections between their solutions. In particular, we show that attainable sets of solutions to stochastic inclusions are subsets of values of multivalued solutions of certain set-valued stochastic equations. We also show that every solution to stochastic inclusion is a continuous selection of a multivalued solution of an associated set-valued stochastic equation. The results obtained in the paper generalize results dealing with this topic known both in deterministic and stochastic cases.

Properties of set-valued stochastic differential equations

The paper is devoted to properties of set-valued stochastic differential equations. The main result of the paper deals with existence and uniqueness of solutions. Furthermore, a connection between solutions of stochastic differential inclusions and solutions of set-valued stochastic differential equations are given. The result of the paper extends a lot of particular results dealing with such type equations.

Relationship between attainable sets of functional and set-valued functional stochastic inclusions

ABSTRACTIt is proved that attainable sets of stochastic differential inclusions, considered by Kisielewicz and M. Michta (J. Math. Anal. Appl. [submitted]), with a family of ΙF-non-anticipative matrix-valued stochastic processes for n ⩾ 1 such that ∑∞n = 1E|gnt|2 < ∞, are subsets of values of solutions of associated to them set-valued stochastic differential equations. Furthermore, the above type relationships are investigated for functional and set-valued functional stochastic inclusions. In particular, it is proved that for every 0 ⩽ t ⩽ T the closed convex hull of decomposable hull of attainable sets of functional stochastic inclusions covers with a set of all -measurable selectors of value Xt of any solution X of set-valued functional stochastic inclusions.

Fuzzy stochastic differential equations of decreasing fuzziness: Non-Lipschitz coefficients

We study fuzzy stochastic differential equations driven by multidimensional Brownian motion with solutions of decreasing fuzziness. The drift and diffusion coefficients are random. Under a non-Lipschitz condition, the existence and pathwise uniqueness of solutions to such the equations are proven. The solutions are considered to be fuzzy stochastic processes. The main result is obtained with a help of a sequence of approximate solutions that converge to a desired unique local solution with trajectories having decreasing fuzziness. A parallel assertion for solutions to fuzzy stochastic differential equations of increasing fuzziness is stated as well. We indicate that our considerations of fuzzy stochastic differential equations of decreasing fuzziness can be applied to examine non-Lipschitz set-valued stochastic differential equations with solutions being set-valued stochastic processes.

Bipartite Fuzzy Stochastic Differential Equations with Global Lipschitz Condition

We introduce and analyze a new type of fuzzy stochastic differential equations. We consider equations with drift and diffusion terms occurring at both sides of equations. Therefore we call them the bipartite fuzzy stochastic differential equations. Under the Lipschitz and boundedness conditions imposed on drifts and diffusions coefficients we prove existence of a unique solution. Then, insensitivity of the solution under small changes of data of equation is examined. Finally, we mention that all results can be repeated for solutions to bipartite set-valued stochastic differential equations.

Set-valued and fuzzy stochastic differential equations in M-type 2 Banach spaces

In this paper we study set-valued stochastic differential equations in M-type 2 Banach spaces. Their drift terms and diffusion terms are assumed to be set-valued and single-valued respectively. These coefficients are considered to be random which makes the equations to be truely nonautonomous. Firstly we define set-valued stochastic Lebesgue integral in a Banach space. This integral is a set-valued random variable. We state its properties such as additivity with respect to the interval of integration, continuity as a function of the upper limit of integration, integrable boundedness. The existence and uniqueness of solution to set-valued differential equations in M-type 2 Banach space is obtained by a method of successive approximations. We show that the approximations are uniformly bounded and converge to the unique solution. A distance between $n$th approximation and exact solution is estimated and a continuous dependence of solution with respect to the data of the equation is proved. Finally, we construct a fuzzy stochastic Lebesgue integral in a Banach space and examine fuzzy stochastic differential equations in M-type 2 Banach spaces. We investigate properties like those in set-valued cases. All the results are achieved without assumption on separability of underlying sigma-algebra.

Fuzzy stochastic differential equations of decreasing fuzziness: Approximate solutions

We consider recently introduced fuzzy stochastic differential equations with solutions of decreasing fuzziness. In general, such the equations do not have solutions that could be written in explicit, closed form. Therefore some methods of construction of approximate solutions are proposed in this p aper. In considered framework, approximate solutions are some measurable and adapted fuzzy stochastic processes. We analyze two kinds of sequences of approximate solutions. It is showed that each sequence of approximate solutions can be used to prove existence and uniqueness of solution to fuzzy stochastic differential equations of decreasing fuzziness. In fact, both the sequences converge to a unique solution. The rates of convergence related to both the sequences are investigated. All the results apply immediately to set-valued stochastic differential equations with solutions of decreasing diameter.

Fuzzy and Set-Valued Stochastic Differential Equations With Local Lipschitz Condition

We are concerned with the fuzzy stochastic differential equations driven by multidimensional Brownian motion viewed as a tool used to describe the behavior of dynamic systems operating in fuzzy environments with stochastic noises. Under the uniform Lipschitz condition, we prove the local uniqueness theorem for the solutions of fuzzy stochastic differential equations. Next we show, assuming the Lipschitz condition is satisfied only locally, that these equations have a unique solution. The fact that the solution is bounded is also proved. We conclude the paper with a number of corresponding results holding for the deterministic fuzzy differential equations and set-valued stochastic differential equations with local Lipschitz condition.

On solutions to set-valued and fuzzy stochastic differential equations

We study existence and uniqueness of solutions to nonlinear set-valued stochastic differential equations driven by multidimensional Brownian motion. The conditions imposed on the equation׳s coefficients are non-Lipschitz. The drift coefficient is set-valued and diffusion coefficient is single-valued, both coefficients are random. The approach used in this paper allows the solutions to be set-valued stochastic processes. The set-valued results are then extended for the parallel studies of nonlinear fuzzy stochastic differential equations with solutions being fuzzy stochastic processes.

The interrelation between stochastic differential inclusions and set-valued stochastic differential equations

In this paper we connect the well established theory of stochastic differential inclusions with a new theory of set-valued stochastic differential equations. Solutions to the latter equations are understood as continuous mappings taking on their values in the hyperspace of nonempty, bounded, convex and closed subsets of the space L2 consisting of square integrable random vectors. We show that for the solution X to a set-valued stochastic differential equation corresponding to a stochastic differential inclusion, there exists a solution x for this inclusion that is a ‖⋅‖L2-continuous selection of X. This result enables us to draw inferences about the reachable sets of solutions for stochastic differential inclusions, as well as to consider the viability problem for stochastic differential inclusions.

Some properties of strong solutions to stochastic fuzzy differential equations

We consider stochastic fuzzy differential equations driven by m-dimensional Brownian motion. Such equations can be useful in modeling of hybrid dynamic systems, where the phenomena are subjected to two kinds of uncertainties: randomness and fuzziness, simultaneously. Under a boundedness condition, which is weaker than linear growth condition, and the Lipschitz condition we obtain existence and uniqueness of solution to stochastic fuzzy differential equations. Solutions, which are fuzzy stochastic processes, and their uniqueness are considered to be in a strong sense. An estimation of error of the Picard approximate solution is established. We give a boundedness type result for the solution defined on finite time interval. Also the stabilities of solution on initial condition and coefficients of the equation are shown. The existence and uniqueness of a solution defined on infinite time interval is proven. Finally, some applications of fuzzy stochastic differential equations are considered. All the results presented in this paper apply to set-valued stochastic differential equations.

Existence Theorems for Solutions to Set-Valued Stochastic Differential Equations and Applications

In this paper, a class of new stochastic differential equations on semilinear Hausdorff space under Hukuhara derivative, called set-valued stochastic differential equations (SSDEs) driven by a Wiener process. Moreover, some corresponding properties of SSDEs are discussed such as existence, uniqueness of solution. Finaly, we give some applications to models of interval-valued stochastic differential equations such as stock prices model and the Langevin equation.

Strong solution of Itô type set-valued stochastic differential equation

In this paper, we shall firstly illustrate why we should introduce an Ito type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the Ito type set-valued stochastic differential equation.

Set-valued stochastic differential equation in M-type 2 Banach space

Let (›,F,{Ft},P) be a complete probability space with filtra- tion {Ft}, (X ,H,µ) an abstract Wiener space of M-type 2, and {Bt : t ‚ 0} an X -valued Brownian motion such that the distribution of the random func- tion ti1/2Bt : › ! X is µ for any t > 0. We consider the strong solutions to a set-valued stochastic dierential equation with a set-valued drift and a single valued diusion driven by dBt. Under some suitable conditions, the existence and uniqueness of strong solutions are obtained.

On the solutions of set-valued Stochastic differential equations in M-type 2 Banach spaces

In a certain Banach space called an M-type 2 Banach space (including Hilbert spaces), we consider a set-valued stochastic differential equation with a set-valued drift term and a single valued diffusion term, under the Lipschitz continuity conditions, and we prove the existence and uniqueness of strong solutions which are continuous in the Hausdorff distance.

On set-valued stochastic integrals in an M-type 2 Banach space

In a separable Banach space, for set-valued martingale, several equivalent conditions based on the measurable selections are discussed, and then, in an M-type 2 Banach space, at first we define single valued stochastic integral by the differential of a real valued Brownian motion, after that extend it to set-valued case. We prove that the set-valued stochastic integral becomes a set-valued submartingale, which is different from single valued case, and obtain the Castaing representation theorem for the set-valued stochastic integral, which is applicable for set-valued stochastic differential equations.