In this paper,we study the path-regularity and martingale properties of the setvalued stochastic integrals defined in our previous work[4].Such integrals have some fundamental differences from the well-known Aumann-It...In this paper,we study the path-regularity and martingale properties of the setvalued stochastic integrals defined in our previous work[4].Such integrals have some fundamental differences from the well-known Aumann-Itôstochastic integrals,and are much better suitable for representing set-valued martingales,whence potentially useful in the study of set-valued backward stochastic differential equations.However,similar to the Aumann-Itôintegral,the new integral is only a set-valued submartingale in general,and there is very limited knowledge about the path regularity of the related indefinite integral,much less the sufficient conditions under which the integral is a true martingale.In this paper,we first establish the existence of right-and left-continuous modifications of set-valued submartingales in continuous time,and apply the results to set-valued stochastic integrals.Moreover,we show that a set-valued stochastic integral yields a martingale if and only if the set of terminal values of the stochastic integrals associated to the integrand is closed and decomposable.Finally,as a particular example,we study the set-valued martingale in the form of the conditional expectation of a set-valued random variable.We show that when the random variable is a convex random polytope,the conditional expectation of a vertex stays as a vertex of the set-valued conditional expectation if and only if the random polytope has a deterministic normal fan.展开更多
基金supported by TÜBİTAK 2219 Programthe Fulbright Scholar Program of the U.S.Department of State+1 种基金sponsored by the Turkish Fulbright Commissionsupported in part by US NSF(Grant Nos.DMS#1908665 and#2205972).
文摘In this paper,we study the path-regularity and martingale properties of the setvalued stochastic integrals defined in our previous work[4].Such integrals have some fundamental differences from the well-known Aumann-Itôstochastic integrals,and are much better suitable for representing set-valued martingales,whence potentially useful in the study of set-valued backward stochastic differential equations.However,similar to the Aumann-Itôintegral,the new integral is only a set-valued submartingale in general,and there is very limited knowledge about the path regularity of the related indefinite integral,much less the sufficient conditions under which the integral is a true martingale.In this paper,we first establish the existence of right-and left-continuous modifications of set-valued submartingales in continuous time,and apply the results to set-valued stochastic integrals.Moreover,we show that a set-valued stochastic integral yields a martingale if and only if the set of terminal values of the stochastic integrals associated to the integrand is closed and decomposable.Finally,as a particular example,we study the set-valued martingale in the form of the conditional expectation of a set-valued random variable.We show that when the random variable is a convex random polytope,the conditional expectation of a vertex stays as a vertex of the set-valued conditional expectation if and only if the random polytope has a deterministic normal fan.
