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June, 1968 A Potential Theory for Supermartingales
E. Count Curtis
Ann. Math. Statist. 39(3): 802-814 (June, 1968). DOI: 10.1214/aoms/1177698312


A potential theory for supermartingales is presented below. It is much like the classical Newtonian potential theory and is a generalization of the potential theory for transient Markov chains. While dealing with stochastic processes with more general dependence relations the new theory retains what we believe to be the important features of the transient Markov chain theory. Briefly, a pure potential is a non-negative supermartingale $\{Z_n, F_n\}$ which satisfies the condition $E\lbrack Z_{n+k}\mid F_n\rbrack \rightarrow 0 \text{a.e. as} k \rightarrow \infty$ for every $n.$ The potential principles of domination, Riesz decomposition, lower envelope, balayage, equilibrium and minimum are proved for these potentials. It is shown how the corresponding results of the transient Markov chain theory can be derived from the new theory. Also, some applications to standard martingale theory are given.


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E. Count Curtis. "A Potential Theory for Supermartingales." Ann. Math. Statist. 39 (3) 802 - 814, June, 1968.


Published: June, 1968
First available in Project Euclid: 27 April 2007

zbMATH: 0162.48702
MathSciNet: MR228055
Digital Object Identifier: 10.1214/aoms/1177698312

Rights: Copyright © 1968 Institute of Mathematical Statistics

Vol.39 • No. 3 • June, 1968
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