Abstract
In this paper, we shall firstly illustrate why we should introduce an Itô type set-valued stochastic differential equation and why we should notice the almost everywhere problem. Secondly we shall give a clear definition of Aumann type Lebesgue integral and prove the measurability of the Lebesgue integral of set-valued stochastic processes with respect to time t. Then we shall present some new properties, especially prove an important inequality of set-valued Lebesgue integrals. Finally we shall prove the existence and the uniqueness of a strong solution to the Itô type set-valued stochastic differential equation.
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Supported by National Natural Science Foundation of China (Grant No. 10771010), PHR (IHLB), Research Fund of Beijing Educational Committee, China; and Grant-in-Aid for Scientific Research 19540140, Japan
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Li, J.G., Li, S.M. & Ogura, Y. Strong solution of Itô type set-valued stochastic differential equation. Acta. Math. Sin.-English Ser. 26, 1739–1748 (2010). https://doi.org/10.1007/s10114-010-8298-x
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DOI: https://doi.org/10.1007/s10114-010-8298-x
Keywords
- Set-valued stochastic process
- set-valued Lebesgue integral
- set-valued stochastic differential equation
- strong solution