Absolute continuity/singularity and relative entropy properties for probability measures induced by diffusions on infinite time intervals

This paper develops a new divergence that generalizes relative entropy and can be used to compare probability measures without a requirement of absolute continuity. We establish properties of the divergence, and in particular derive and exploit a representation as an infimum convolution of optimal transport cost and relative entropy. Also included are examples of computation and approximation of the divergence, and the demonstration of properties that are useful when one quantifies model uncertainty.

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Trace inequalities for mixtures of Markov chains

Advances in Applied Probability ◽

10.1017/s0001867800029165 ◽

1977 ◽

Vol 9 (04) ◽

pp. 747-764

Author(s):

Burton Singer ◽

Seymour Spilerman

Keyword(s):

Markov Chains ◽

Special Kind ◽

Probability Measures ◽

Stochastic Matrices ◽

Transition Matrices ◽

Time Intervals ◽

Panel Studies ◽

Image Position ◽

Trace Inequalities

In a wide variety of multi-wave panel studies in economics and sociology, comparisons between the observed transition matrices and predictions of them based on time-homogeneous Markov chains have revealed a special kind of discrepancy: the trace of the observed matrices tends to be larger than the trace of the predicted matrices. A common explanation for this discrepancy has been via mixtures of Markov chains. Specializing to mixtures of Markov semi-groups of the form we exhibit classes of stochastic matrices M, probability measures µ and time intervals Δ such that for k = 2, 3 and 4. These examples contradict the substantial literature which suggests — implicitly — that the above inequality should be reversed for general mixtures of Markov semi-groups.

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Weak convergence of stochastic processes defined on semi-infinite time intervals

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1963-0153046-2 ◽

1963 ◽

Vol 14 (5) ◽

pp. 694-694 ◽

Cited By ~ 104

Author(s):

Charles Stone

Keyword(s):

Weak Convergence ◽

Stochastic Processes ◽

Infinite Time ◽

Time Intervals

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On the equivalence of mixed and behavior strategies in finitely additive decision problems

Journal of Applied Probability ◽

10.1017/jpr.2019.47 ◽

2019 ◽

Vol 56 (3) ◽

pp. 810-829

Author(s):

János Flesch ◽

Dries Vermeulen ◽

Anna Zseleva

Keyword(s):

Mixed Strategy ◽

Action Space ◽

Decision Problems ◽

Probability Measures ◽

Infinite Time ◽

Behavior Strategy ◽

Behavior Strategies ◽

And Behavior ◽

Arbitrary Action ◽

Action Spaces

AbstractWe consider decision problems with arbitrary action spaces, deterministic transitions, and infinite time horizon. In the usual setup when probability measures are countably additive, a general version of Kuhn’s theorem implies under fairly general conditions that for every mixed strategy of the decision maker there exists an equivalent behavior strategy. We examine to what extent this remains valid when probability measures are only assumed to be finitely additive. Under the classical approach of Dubins and Savage (2014), we prove the following statements: (1) If the action space is finite, every mixed strategy has an equivalent behavior strategy. (2) Even if the action space is infinite, at least one optimal mixed strategy has an equivalent behavior strategy. The approach by Dubins and Savage turns out to be essentially maximal: these two statements are no longer valid if we take any extension of their approach that considers all singleton plays.

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