Set-valued Stochastic Processes Research Articles

On uniqueness and existence of solutions to stochastic set-valued differential equations with fractional Brownian motions

ABSTRACT This paper is concerned with a class of stochastic set differential equations (SSDEs) driven by a fractional Brownian motion (fBm) with the Lipschitzian condition. The solutions of SSDEs with an fBm are set-valued stochastic processes. We first provide some prior inequalities on set valued integrals including the Itô type set-valued one with respect to an fBm. Second, the existence and uniqueness theorem of solutions to SSDEs is proven under some assumptions. Furthermore, their continuous dependence on initial data is studied. Finally, a numerical example with the evolution of interest from the financial market is analysed. The Gronwall inequality is employed.

Approximation theorems for set-valued stochastic integrals

ABSTRACTThe article is devoted to new properties of Aumann, Lebesgue, and Itô set-valued stochastic integrals considered in papers [1,2]. In particular, it contains some approximation theorems for Aumann and Itô set-valued stochastic integrals. Hence, in particular, it follows that Aumann and Lebesgue set-valued stochastic integrals cover a.s., both for measurable and IF-nonanticipative integrably bounded set-valued stochastic processes.

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.

Set-Valued Stochastic Integrals with Respect to Finite Variation Processes

In a Euclidean space Rd, the Lebesgue-Stieltjes integral of set-valued stochastic processes with respect to real valued finite variation process is defined directly by employing all integrably bounded selections instead of taking the decomposable closure appearing in some existed references. We shall show that this kind of integral is measurable, continuous in t under the Hausdorff metric and L2-bounded.

Set-valued assessments of solutions to stochastic differential equations with random set parameters

We consider stochastic differential equations depending on parameters whose uncertainty is modeled by random compact sets. Several approaches are discussed how to construct set-valued stochastic processes from the solutions. The induced lower and upper probabilities are compared to a set-valued set function and a set of probability measures constructed from the distributions of the solutions and the selections of the random set.

Existence of solutions to stochastic set differential equations driven by local martingales

In this paper, a class of stochastic set differential equations (SSDEs) disturbed by a continuous local martingale with the Lipschitzian condition is studied. The solutions of SSDEs are set-valued stochastic processes. First, we give some prior inequalities on set valued integrals. Second, the existence and uniqueness theorem of solutions to SSDEs is proven. Moreover, their continuous dependence on initial conditions and a stability property are investigated. Finally, an illustrating example from the mathematical finance is discussed. The main mathematical tool is Gronwall’s inequality.

Characterizing joint distributions of random sets by multivariate capacities

By the Choquet theorem, distributions of random closed sets can be characterized by a certain class of set functions called capacity functionals. In this paper a generalization to the multivariate case is presented, that is, it is proved that the joint distribution of finitely many random sets can be characterized by a multivariate set function being completely alternating in each component, or alternatively, by a capacity functional defined on complements of cylindrical sets. For the special case of finite spaces a multivariate version of the Moebius inversion formula is derived. Furthermore, we use this result to formulate an existence theorem for set-valued stochastic processes.

Fuzzy set-valued stochastic Lebesgue integral

This paper studies Lebesgue integral of a fuzzy closed set-valued stochastic process with respect to the time t. Firstly, a progressively measurable fuzzy closed set-valued stochastic process is discussed and an almost everywhere problem in the former Aumann type Lebesgue integral of the level-set process is pointed out. Secondly, a new definition of the Lebesgue integral by decomposable closure is given, focusing on Aumann representation theorem, representation theorem and property of convexity. It is proved that the fuzzy closed set-valued stochastic Lebesgue integral is a fuzzy closed set-valued stochastic process which is widely used in the fuzzy world with randomness. Finally, the fuzzy closed set-valued stochastic Lebesgue integral in Lp-space is studied, especially on an almost everywhere problem.

On solutions of stochastic differential equations with parameters modeled by random sets

We consider ordinary stochastic differential equations whose coefficients depend on parameters. After giving conditions under which the solution processes continuously depend on the parameters random compact sets are used to model the parameter uncertainty. This leads to continuous set-valued stochastic processes whose properties are investigated. Furthermore, we define analogues of first entrance times for set-valued processes called first entrance and inclusion times. The theoretical concept is applied to a simple example from mechanics.

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.

Aumann Type Set-valued Lebesgue Integral and Representation Theorem

In this paper, we shall firstly illustrate why we should discuss the Aumann type set-valued Lebesgue integral of a set-valued stochastic process with respect to time t under the condition that the set-valued stochastic process takes nonempty compact subset of d-dimensional Euclidean space. After recalling some basic results about set-valued stochastic processes, we shall secondly prove that the Aumann type set-valued Lebesgue integral of a set-valued stochastic process above is a set-valued stochastic process. Finally we shall give the representation theorem, and prove an important inequality of the Aumann type set-valued Lebesgue integrals of set-valued stochastic processes with respect to t, which are useful to study set-valued stochastic differential inclusions with applications in finance.

Statistical aspects of fuzzy monotone set-valued stochastic processes. Application to birth-and-growth processes

The paper considers a particular family of fuzzy monotone set-valued stochastic processes. The proposed setting allows us to investigate suitable α -level sets of such processes, modeling birth-and-growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set-valued estimators are studied for growth process. Moreover, the nucleation process is studied via the hitting function, and a consistent estimator of the nucleation hitting function is derived.

Set-Valued Stochastic Lebesgue Integral And Representation Theorems

In this paper, we shall firstly illustrate why we should introduce set-valued stochastic integrals, and then we shall discuss some properties of set-valued stochastic processes and the relation between a set-valued stochastic process and its selection set. After recalling the Aumann type definition of stochastic integral, we shall introduce a new definition of Lebesgue integral of a set-valued stochastic process with respect to the time t. Finally we shall prove the presentation theorem of set-valued stochastic integral and discuss further properties that will be useful to study set-valued stochastic differential equations with their applications.

Martingale selection problem and asset pricing in finite discrete time

Given a set-valued stochastic process $(V_t)_{t=0}^T$, we say that the martingale selection problem is solvable if there exists an adapted sequence of selectors $\xi_t\in V_t$, admitting an equivalent martingale measure. The aim of this note is to underline the connection between this problem and the problems of asset pricing in general discrete-time market models with portfolio constraints and transaction costs. For the case of relatively open convex sets $V_t(\omega)$ we present effective necessary and sufficient conditions for the solvability of a suitably generalized martingale selection problem. We show that this result allows to obtain computationally feasible formulas for the price bounds of contingent claims. For the case of currency markets we also sketch a new proof of the first fundamental theorem of asset pricing.

Representation theorems, set-valued and fuzzy set-valued Ito integral

In this paper, we shall present representation theorems of set-valued martingales and set-valued processes of finite variation with continuous time. We shall also obtain a representation theorem of a predictable set-valued stochastic process. We shall give a new definition of Ito integral of a set-valued stochastic process with respect to a Brownian motion based on the work [E.J. Jung, J.H. Kim, On set-valued stochastic integrals, Stochastic Anal. Appl. 21(2) (2003) 401–418.]. We shall also discuss some properties of set-valued Ito integral, especially the presentation theorem of set-valued Ito integral. Finally, we extend some of above results to the fuzzy set-valued case.

Right continuous fuzzy set-valued stochastic processes with left limitation

The concept of (d) quasi-left continuous fuzzy set-valued stochastic process is proposed. A (d) cadlag. fuzzy set-valued stochastic process, i.e. (d) right continuous fuzzy set-valued stochastic processes with left limitation, is (d) quasi-left continuous if and only if for each positive predictable time this process is (d) left continuous almost surely on the set on which the predictable time is finite. A (d) cadlag. fuzzy set-valued stochastic process only has accessible jump if and only if for each totally inaccessible time this process is (d) left continuous almost surely on the set on which the totally inaccessible time is finite.

Stochastic Functional Inclusion Driven by Semimartingale

We define a set-valued stochastic process and an integral of a such process with respect to a semimartingale. Next we consider a stochastic integral inclusion. Using fixed point methods we prove some results on the existence of its solutions

Selections of set‐valued stochastic processes

We show that ℱt‐adapted, set‐valued stochastic processes satisfying mild continuity conditions admit, ℱt‐adapted, stochastically continuous selections.