Decomposition of supermartingales indexed by a linearly ordered set

Stochastic Processes and their Applications

2010

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We prove results on the existence of Doléans-Dade measures and of the Doob-Meyer decomposition for supermartingales indexed by a general index set.We prove, in Theorem 1, a necessary and sufficient condition, the class D 0 property, for the existence of a Doléans-Dade measure associated to a supermartingale. Based on this, we establish, in Theorem 2, a sufficient condition, the class D * property, for the existence of a Doob Meyer decomposition. In Corollary 1 we consider supermartingales of uniformly integrable variation. The two key properties involved in our results are the possibility of extending the original supermartingale to a larger filtration and some form of the optional sampling theorem. From a mathematical perspective we exploit extensively results from the theory of finitely additive measures.

Section: Thus Example 2 Does Not Applymentioning

confidence: 99%

Supermartingale decomposition with a general index set

Stochastic Processes and their Applications

2010

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“…of a countable set. In [3] it was shown that the class D property could likewise be restricted to consider only stopping times with countably many values.…”

Section: Decompositionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…Moreover, we allow the index set to be just a linearly ordered set. The setting is in fact the same as that proposed in [3], where a version of the Doob Meyer decomposition was obtained. A suitable example of a linearly ordered index set would be the whole real line.In our general model the main source of difficulty (and of interest) lies in the need to forsake the stopping machinery, a most useful tool in stochastic analysis.…”

mentioning

confidence: 99%

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Quasi-martingales with a linearly ordered index set

Statistics & Probability Letters

2010

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“…The class D property may thus be replaced by a corresponding property, the class D σ , in which the stopping times are restricted to have countable range, see [9].…”

Section: Then ξ Admits a Doob Meyer Decomposition If And Only Ifmentioning

confidence: 99%

Finitely Additive Supermartingales

2006

SSRN Journal

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Abstract. The concept of finitely additive supermartingales, originally due to Bochner, is revived and developed. We exploit it to study measure decompositions over filtered probability spaces and the properties of the associated Doléans-Dade measure. We obtain versions of the Doob Meyer decomposition and, as an application, we establish a version of the Bichteler and Dellacherie theorem with no exogenous probability measure.