1997
DOI: 10.1017/s000186780002807x
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Integral Probability Metrics and Their Generating Classes of Functions

Abstract: We consider probability metrics of the following type: for a class of functions and probability measures P, Q we define A unified study of such integral probability metrics is given. We characterize the maximal class of functions that generates such a metric. Further, we show how some interesting properties of these probability metrics arise directly from conditions on the generating class of functions. The results are illustrated by several examples, including the Kolmogorov metric, the Dudley metric and th… Show more

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Cited by 228 publications
(291 citation statements)
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“…Of course, one could also assume that the decision rules of a trading strategy π also depend on past actions and past values of the portfolio process X ϕ . However, as already discussed in Remark 2.5(i), the corresponding historydependent trading strategies do not lead to an improved optimal value for the terminal wealth problem (28). ✸…”
Section: Embedding Into Mdm and Optimal Trading Strategiesmentioning
confidence: 77%
See 1 more Smart Citation
“…Of course, one could also assume that the decision rules of a trading strategy π also depend on past actions and past values of the portfolio process X ϕ . However, as already discussed in Remark 2.5(i), the corresponding historydependent trading strategies do not lead to an improved optimal value for the terminal wealth problem (28). ✸…”
Section: Embedding Into Mdm and Optimal Trading Strategiesmentioning
confidence: 77%
“…A strategy π P ∈ Π is called an optimal (self-financing) trading strategy w.r.t. P (and x 0 ) if it solves the maximization problem (28).…”
Section: Embedding Into Mdm and Optimal Trading Strategiesmentioning
confidence: 99%
“…The R-distance over policies for Markov decision processes is essentially an Integral Probability Metric (IPM) over stationary distributions (Müller, 1997). For different choices of R, we have various R-distances.…”
Section: Generalization Of Gailmentioning
confidence: 99%
“…Proof: The proof involves the estimation of R m (F; {X (1) i }) and R n (F; {X (2) i }) for F = F W , F β and F k , which is then used in (6). For details, see [18,Corollary 12].…”
Section: Theoremmentioning
confidence: 99%
“…The goal of this paper is to study the non-parametric estimation of another popular family (particularly in probability theory and mathematical statistics) of distance measures on probabilities, the integral probability metrics (IPM) [6], defined as…”
Section: Introductionmentioning
confidence: 99%