Set-Valued Stochastic Equation with Set-Valued Square Integrable Martingale

Jun-Gang Li,Shi-Qing Zheng,L Long,H Yang,Y Li,X Li,Y Dai

doi:10.1051/itmconf/20171203002

Jun-Gang Li, Shi-Qing Zheng + Show 5 more

Open Access

https://doi.org/10.1051/itmconf/20171203002

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Abstract

In this paper, we shall introduce the stochastic integral of a stochastic process with respect to set-valued square integrable martingale. Then we shall give the Aumann integral measurable theorem, and give the set-valued stochastic Lebesgue integral and set-valued square integrable martingale integral equation. The existence and uniqueness of solution to set-valued stochastic integral equation are proved. The discussion will be useful in optimal control and mathematical finance in psychological factors.

Highlights

Set-valued theory is used in optimal control(cf.[1]), mathematical finance(cf. [2]), fixed point theory(cf. [3])
Set-valued and fuzzy set-valued theory can be used to account for psychological factors(cf.[4,5]).In [6], stochastic control problems are discussed by stochastic integral with respect to set-valued square integral martingales
Throughout this paper, assume that (:, ࣛ, P) is a complete probability space, the V -field filtration { ࣛ t : t I} satisfies the usual conditions (i.e. Containing all nullsets, non-decreasing and right continuous), I [0,T ] with T > 0, R is the set of all real numbers, N is the set of all natural numbers, Rd is the d - dimensional Euclidean space with usual norm || ×||

Summary

Introduction

Set-valued theory is used in optimal control(cf.[1]), mathematical finance(cf. [2]), fixed point theory(cf. [3]). M. Malinowski et al discussed the setvalued stochastic integral driven by semi-martingale and the set-valued stochastic differential equations driven by semi-martingale in [7]. J. Li et al discussed the space of fuzzy set-valued square integrable martingales in [12] and fuzzy set-valued stochastic Lebesgue integral in [13]. We shall give the set-valued stochastic integral equation : the first integral is set-valued stochastic Lebesgue integral (see [9,10]), the second integral is set-valued square integrable martingale integral (see [6]).

Preliminary on Set-Valued Random Variables and Martingales

Set-Valued Stochastic Integral Equation