Bilateral set-valued stochastic integral equations

Abstract

We investigate bilateral set-valued stochastic integral equations and these equations combine widening and narrrowing set-valued stochastic integral equations studied in literature. An existence and uniqueness theorem is established using approximate solutions. In addition stability of the solution with respect to small changes of the initial state and coefficients is established, also we provide a result on boundedness of the solution, and an estimate on a distance between the exact solution and the approximate solution is given. Finally some implications for deterministic set-valued integral equations are presented.

On a Set-Valued Young Integral with Applications to Differential Inclusions

“…In the case of multifunctions with convex compact values, other approaches such as Hukuhara's [15] or Debreu's [10] are restricted to multifunctions with compact convex values and use the cone structure of the space of convex compact sets. The concept of Aumann integral has been applied in several papers to integration of multivalued stochastic processes using classical stochastic calculus and a definition of the form (1), where w is a Brownian motion, or more generally a semimartingale, e.g., [18,22,23]. Despite this similarity, the setting of stochastic calculus is quite different from ours, since the stochastic integral needs a probability space to make sense.…”