Distributional compatibility for change of measures

Jie Shen (), Yi Shen (), Bin Wang () and Ruodu Wang ()
Additional contact information
Jie Shen: University of Waterloo
Yi Shen: University of Waterloo
Bin Wang: Chinese Academy of Sciences
Ruodu Wang: University of Waterloo

Finance and Stochastics, 2019, vol. 23, issue 3, No 9, 794 pages

Abstract: Abstract In this paper, we characterise compatibility of distributions and probability measures on a measurable space. For a set of indices J $\mathcal{J}$ , we say that the tuples of probability measures ( Q i ) i ∈ J $(Q_{i})_{i\in \mathcal{J}} $ and distributions ( F i ) i ∈ J $(F_{i})_{i\in \mathcal{J}} $ are compatible if there exists a random variable having distribution F i $F_{i}$ under Q i $Q_{i}$ for each i ∈ J $i\in \mathcal{J}$ . We first establish an equivalent condition using conditional expectations for general (possibly uncountable) J $\mathcal{J}$ . For a finite n $n$ , it turns out that compatibility of ( Q 1 , … , Q n ) $(Q_{1},\dots ,Q_{n})$ and ( F 1 , … , F n ) $(F_{1},\dots ,F _{n})$ depends on the heterogeneity among Q 1 , … , Q n $Q_{1},\dots ,Q_{n}$ compared with that among F 1 , … , F n $F_{1},\dots ,F_{n}$ . We show that under an assumption that the measurable space is rich enough, ( Q 1 , … , Q n ) $(Q_{1},\dots ,Q_{n})$ and ( F 1 , … , F n ) $(F_{1},\dots ,F_{n})$ are compatible if and only if ( Q 1 , … , Q n ) $(Q_{1},\dots ,Q _{n})$ dominates ( F 1 , … , F n ) $(F_{1},\dots ,F_{n})$ in a notion of heterogeneity order, defined via the multivariate convex order between the Radon–Nikodým derivatives of ( Q 1 , … , Q n ) $(Q_{1},\dots ,Q_{n})$ and ( F 1 , … , F n ) $(F_{1},\dots ,F_{n})$ with respect to some reference measures. We then proceed to generalise our results to stochastic processes, and conclude the paper with an application to portfolio selection problems under multiple constraints.

Keywords: Change of measure; Compatibility; Heterogeneity order; Optimisation; 60E05; 60E15 (search for similar items in EconPapers)
JEL-codes: C18 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (7)

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DOI: 10.1007/s00780-019-00393-4

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