Shadow couplings
Mathias Beiglböck, Nicolas Juillet
Abstract
A classical result of Strassen asserts that given probabilities <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="mu comma nu"> <mml:semantics> <mml:mrow> <mml:mi> μ </mml:mi> <mml:mo>,</mml:mo> <mml:mi> ν </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mu , \nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the real line which are in convex order, there exists a <italic>martingale coupling</italic> with these marginals, i.e. a random vector <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper X 1 comma upper X 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(X_1,X_2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X 1 tilde mu comma upper X 2 tilde nu"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo> ∼ </mml:mo> <mml:mi> μ </mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo> ∼ </mml:mo> <mml:mi> ν </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X_1\sim \mu , X_2\sim \nu</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper E left-bracket upper X 2 vertical-bar upper X 1 right-bracket equals upper X 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">E</mml:mi> </mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>X</mml:mi> <mml:mn>1</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathbb {E}[X_2|X_1]=X_1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Remarkably, it is a non-trivial problem to construct particular solutions to this problem. Based on the concept of <italic>shadow</italic> for measures in convex order, we introduce a family of such martingale couplings, each of which admits several characterizations in terms of optimality properties/geometry of the support set/representation through a Skorokhod embedding. As a particular element of this family we recover the (left-)curtain martingale transport, which has recently been studied (see Beiglböck, Henry-Labordère, and Touzi [Stochastic Process. Appl. 127 (2017), pp. 3005–3013]; Beiglböck and Juillet [Ann. Probab. 44 (2016), pp. 42–106]; Campi, Laachir, and Martini [Finance Stoch. 21 (2017), pp. 471–486; Henry-Labordère and Touzi [Finance Stoch. 20 (2016), pp. 635–668]) and which can be viewed as a martingale analogue of the classical monotone rearrangement. As another canonical element of this family we identify a martingale coupling that resembles the usual <italic>product coupling</italic> and appears as an optimizer in the general transport problem recently introduced by Gozlan et al. In addition, this coupling provides an explicit example of a Lipschitz kernel, shedding new light on Kellerer’s proof of the existence of Markov martingales with specified marginals.