Some Integration-by-Parts Formulas Involving 2-Copulas
Chapter
Abstract
Abstract We note examples of probabilistic interpretations of integrals involving 2-copulas. We then use the theory of strong convergence of copulas to justify an integration-by-parts formula involving 2-copulas, where A and B are arbitrary 2-copulas and f is continuously differentiable.
$$ \int_{{I^2}} {f\left( A \right)dB = \int_0^1 {f\left( t \right)dt - \int_{{I^2}} {f'\left( A \right){D_1}A\,{D_2}B = \int_0^1 {f\left( t \right)dt - \int_{{I^2}} {f'\left( A \right){D_2}A\,{D_1}B.} } } } } $$
Keywords
Integration-by-parts Strong convergence IdentityPreview
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