Stochastic Inclusions Research Articles

ON STOCHASTIC EQUATIONS AND INCLUSIONS WITH BACKWARD MEAN DERIVATIVES

ON STOCHASTIC EQUATIONS AND INCLUSIONS WITH BACKWARD MEAN DERIVATIVES

Solution Sets for Young Differential Inclusions

The paper deals with some properties of solutions of differential inclusions driven by set-valued integrals of a Young type. The existence of solutions, boundedness, closedness of the set of solutions and continuous dependence type results are considered. These inclusions contain as a particular case set-valued stochastic inclusions with respect to a fractional Brownian motion (fBm), and therefore, their properties are crucial for investigation the properties of solutions of fBm stochastic differential inclusions.

On the computation of equilibria in monotone and potential stochastic hierarchical games

We consider a class of noncooperative hierarchical $${\textbf{N}}$$ -player games where the ith player solves a parametrized stochastic mathematical program with equilibrium constraints (MPEC) with the caveat that the implicit form of the ith player’s in MPEC is convex in player strategy, given rival decisions. Few, if any, general purpose schemes exist for computing equilibria, motivating the development of computational schemes in two regimes: (a) Monotone regimes. When player-specific implicit problems are convex, then the necessary and sufficient equilibrium conditions are given by a stochastic inclusion. Under a monotonicity assumption on the operator, we develop a variance-reduced stochastic proximal-point scheme that achieves deterministic rates of convergence in terms of solving proximal-point problems in monotone/strongly monotone regimes with optimal or near-optimal sample-complexity guarantees. Finally, the generated sequences are shown to converge to an equilibrium in an almost-sure sense in both monotone and strongly monotone regimes; (b) Potentiality. When the implicit form of the game admits a potential function, we develop an asynchronous relaxed inexact smoothed proximal best-response framework, requiring the efficient computation of an approximate solution of an MPEC with a strongly convex implicit objective. To this end, we consider an $$\eta $$ -smoothed counterpart of this game where each player’s problem is smoothed via randomized smoothing. In fact, a Nash equilibrium of the smoothed counterpart is an $$\eta $$ -approximate Nash equilibrium of the original game. Our proposed scheme produces a sequence and a relaxed variant that converges almost surely to an $$\eta $$ -approximate Nash equilibrium. This scheme is reliant on resolving the proximal problem, a stochastic MPEC whose implicit form has a strongly convex objective, with increasing accuracy in finite-time. The smoothing framework allows for developing a variance-reduced zeroth-order scheme for such problems that admits a fast rate of convergence. Numerical studies on a class of multi-leader multi-follower games suggest that variance-reduced proximal schemes provide significantly better accuracy with far lower run-times. The relaxed best-response scheme scales well with problem size and generally displays more stability than its unrelaxed counterpart.

Stochastic inclusions and set-valued stochastic equations with mixed integrals in the plane

The paper refers to the study of properties of solutions to two-parameter stochastic inclusions and set-valued stochastic equations with set-valued mixed integrals driven by finite variation processes and martingales. We present new types of such inclusions and equations that generalize those studied earlier. Apart from existence results to such inclusions and equations also topological properties of their solutions are studied. Additionally some connections between their solutions are established. The results obtained in the paper present a set-valued counterpart dealing with this topic known both in deterministic and stochastic cases.

Approximate controllability for Hilfer fractional stochastic evolution inclusion with nonlocal conditions

In this paper, the approximate controllability of Hilfer fractional stochastic evolution inclusion with nonlocal conditions is investigated. By using fractional calculus, semigroups theory, stochastic analysis and the fixed point theorem for multi-valued maps, a new sufficient condition for the existence of mild solution of the Hilfer fractional stochastic system in the space of weighted continuous functions is derived. Furthermore, a novel sufficient condition for the approximate controllability of the Hilfer fractional stochastic evolution inclusion with nonlocal conditions is established. Finally, an example is presented to illustrate the main results.

Existence of solutions for non-autonomous second-order stochastic inclusions with Clarke’s subdifferential and non instantaneous impulses

This manuscript explores a new class of non-autonomous second-order stochastic inclusions of Clarke?s subdifferential form with non-instantaneous impulses (NIIs), unbounded delay, and the Rosenblatt process in Hilbert spaces. The existence of a solution is deduced by employing a fixed point strategy for a set-valued map together with the evolution operator and stochastic analysis approach. An example is analyzed for theoretical developments.

Solutions of semi-linear stochastic evolution integro-differential inclusions with Poisson jumps and non-local initial conditions

In this paper, we mainly investigate properties on solution sets of a semi-linear stochastic evolution integro-differential inclusion with Poisson jumps and non-local initial conditions. We firstly establish the existence of solutions to the addressed equation via weak topology technique. Through establishing the estimate on noncompactness measure of integrals for Poisson jumps, we further show that the solution set is a compact -set when the resolvent operator is compact or non-compact, respectively.

Weak compactness of weak solutions sets to stochastic differential inclusions

The main result of the paper deals with weak compactness in distribution of weak solutions sets to stochastic differential inclusions defined by set-valued stochastic integrals presented in the paper [Kisielewicz, M., Michta, M. (2017). Integrably bounded set-valued stochastic integrals. J. Math. Anal. Appl. 449(2):1892–1910]. Up to now only stochastic functional inclusions (see [Kisielewicz, M. (2013). Stochastic Differential and Inclusions Applications. New York: Springer]) have been investigated. The introductory parts of the paper deal with some properties of set-valued stochastic integrals and selected notions dealing with weak convergence in distribution of sequences of continuous stochastic processes.

Stochastic Inclusions with Forward Mean Derivatives Having Decomposable Right-Hand Sides

A.V. Makarova,N.E. Zhukovsky and Y.A. Gagarin Air Force Academy, Voronezh,Russian Federation, allagm@mail.ru Алла Викторовна Макарова, кандидат физико-математических наук, преподаватель, Военный учебно-научный центр ВВС≪Академия имени профессора Н.Е. Жуковского и Ю.А. Гагарина≫(г. Воронеж, Российская Федерация), allagm@mail.ru.

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.

Stochastic Inclusions with Current Velocities Having Decomposable Right-Hand Sides

An existence of solution theorem is obtained for stochastic differential inclusions given in terms of the so-called current velocities (symmetric mean derivatives, a direct analogs of ordinary velocity of deterministic systems) and quadratic mean derivatives (giving information on the diffusion coefficient) on the flat $n$-dimensional torus. Right-hand sides in both the current velocity part and the quadratic part are set-valued, lower semi-continuous but not necessarily have convex images. Instead we suppose that they are decomposable.

Upper separated multifunctions in deterministic and stochastic optimal control

Abstract Let X be a Banach space while (Y,⪯) a Banach lattice. We consider the class of “upper separated” set-valued functions F : X → 2 Y and investigate the problem of the existence of order-convex selections of F. First, we present results on the existence of the Carathéodory-convex type selections of upper separated multifunctions and apply them to investigation of the existence of solutions of differential and stochastic inclusions. We will discuss the applicability of obtained selection results to some deterministic and stochastic optimal control problems.

Topological properties of solution sets for stochastic evolution inclusions

ABSTRACTIn this paper, we investigate the topological structure of solution sets for stochastic evolution inclusions in Hilbert spaces when the semigroup is compact as well as noncompact. It is shown that the solution set is nonempty, compact, and an Rδ-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As applications of the obtained results, an example is given.

Detecting stochastic inclusions in electrical impedance tomography

This work considers the inclusion detection problem of electrical impedance tomography with stochastic conductivities. It is shown that a conductivity anomaly with a random conductivity can be identified by applying the factorization method or the monotonicity method to the mean value of the corresponding Neumann-to-Dirichlet map provided that the anomaly has high enough contrast in the sense of expectation. The theoretical results are complemented by numerical examples in two spatial dimensions.

Existence of the Mild Solution for an Impulsive Nonlocal Neutral Stochastic Fractional Differential Inclusions with Infinite Delay

This paper studies an impulsive nonlocal neutral stochastic fractional integro-differential inclusions with infinite delays in an arbitrary separable Hilbert space X. The sufficient conditions proving the existence of mild solution of the nonlocal stochastic inclusion problem with impulsive conditions are derived by using resolvent operator and fixed point theorem for multi-valued operators due to O’Regan. An example is also presented to illustrate the obtained theory.

Topological properties of solution sets of fractional stochastic evolution inclusions

In this paper, we investigate the topological structure for the solution set of Caputo type neutral fractional stochastic evolution inclusions in Hilbert spaces. We introduce the concept of mild solutions for fractional neutral stochastic inclusions and show that the solution set is nonempty compact and R_{delta}-set, which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point in the sense that it has the same homology group as one-point space. Finally, we illustrate the obtained theory with the aid of an example.

Approximate controllability of fractional nonlinear neutral stochastic differential inclusion with nonlocal conditions and infinite delay

In this paper we consider a class of fractional nonlinear neutral stochastic evolution inclusions with nonlocal initial conditions in Hilbert space. Using fractional calculus, stochastic analysis theory, operator semigroups and Bohnenblust–Karlin’s fixed point theorem, a new set of sufficient conditions are formulated and proved for the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusions under the assumption that the associated linear part of the system is approximately controllable. An example is provided to illustrate the theory.

Approximate Controllability of a Neutral Stochastic Fractional Integro-Differential Inclusion with Nonlocal Conditions

This paper discusses the approximate controllability of a neutral functional integro-differential inclusion involving Caputo fractional derivative in a Hilbert space under the assumptions that the corresponding linear system is approximately controllable. A new set of sufficient conditions for approximate controllability of neutral fractional stochastic functional integro-differential inclusions are formulated and established by utilizing stochastic analysis theory, fractional calculus and the technique of fixed point theorem with analytic compact resolvent operator. An example is also considered for illustrating the discussed theory.

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.

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.