Path-regularity and martingale properties of set-valued stochastic integrals

Abstract

In this paper, we study the path-regularity and martingale properties of the set-valued stochastic integrals defined in our previous work [4]. Such integrals have some fundamental differences from the well-known Aumann-Itô stochastic 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-Itô integral, 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.

Keywords

Set-valued stochastic integral / Set-valued martingale / Martingale representation / Path-regularity / Random polytope / Normal fan

Acknowledgements

Çağın Ararat is supported by TÜBİTAK 2219 Program and by the Fulbright Scholar Program of the U.S. Department of State, sponsored by the Turkish Fulbright Commission. This work was partly completed while he was visiting the University of Southern California, whose hospitality is greatly appreciated. Jin Ma is supported in part by US NSF (Grant Nos. DMS#1908665 and #2205972).

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