Continuous stochastic processes in Riesz spaces: The Doob-Meyer decomposition. (English) Zbl 1216.46005
Riesz spaces provide an ideal framework for developing an abstract theory of stochastic processes. The general theory was considered by Kuo, Labuschagne, and Watson in a series of papers. These authors studied countable processes in the Riesz space setting.
The author of the present paper studies the aspects of continuous stochastic processes in Riesz spaces. He develops notions to prove the Doob-Meyer Decomposition Theorem for submartingales, which states that a submartingale satisfying a uniform integrability condition can be decomposed uniquely as the sum of a martingale and an increasing right continuous predictable process. This is nontrivial even in the classical case.
The author of the present paper studies the aspects of continuous stochastic processes in Riesz spaces. He develops notions to prove the Doob-Meyer Decomposition Theorem for submartingales, which states that a submartingale satisfying a uniform integrability condition can be decomposed uniquely as the sum of a martingale and an increasing right continuous predictable process. This is nontrivial even in the classical case.
Reviewer: Şafak Alpay (Ankara)
MSC:
| 46A40 | Ordered topological linear spaces, vector lattices |
| 60G44 | Martingales with continuous parameter |
| 60G07 | General theory of stochastic processes |
| 06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |
Keywords:
vector lattice; stochastic process with continuous parameter; Doob-Meyer decomposition; submartingaleReferences:
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