Amarts: A class of asymptotic martingales B. Continuous parameter

Gerald A. Edgar and Louis Sucheston

Journal of Multivariate Analysis, 1976, vol. 6, issue 4, 572-591

Abstract: A continuous-parameter ascending amart is a stochastic process (Xt)t[set membership, variant]+ such that E[X[tau]n] converges for every ascending sequence ([tau]n) of optional times taking finitely many values. A descending amart is a process (Xt)t[set membership, variant]+ such that E[X[tau]n] converges for every descending sequence ([tau]n), and an amart is a process which is both an ascending amart and a descending amart. Amarts include martingales and quasimartingales. The theory of continuous-parameter amarts parallels the theory of continuous-parameter martingales. For example, an amart has a modification every trajectory of which has right and left limits (in the ascending case, if it satisfies a mild boundedness condition). If an amart is right continuous in probability, then it has a modification every trajectory of which is right continuous. The Riesz and Doob-Meyer decomposition theorems are proved by applying the corresponding discrete-parameter decompositions. The Doob-Meyer decomposition theorem applies to general processes and generalizes the known Doob decompositions for continuous-parameter quasimartingales, submartingales, and supermartingales. A hyperamart is a process (Xt) such that E[X[tau]n] converges for any monotone sequence ([tau]n) of bounded optional times, possibly not having finitely many values. Stronger limit theorems are available for hyperamarts. For example: A hyperamart (which satisfies mild regularity and boundedness conditions) is indistinguishable from a process all of whose trajectories have right and left limits.

Keywords: Amart; Martingale; Quasimartingale; Right; and; left; limits; Continuity; of; trajectories; Riesz; decomposition; Doob; decomposition; Hyperamart (search for similar items in EconPapers)
Date: 1976
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