Shadow martingales – a stochastic mass transport approach to the peacock problem

Abstract

Given a family of real probability measures (µt) t≥0 increasing in convex order (a peacock) we describe a systematic method to create a martingale exactly fitting the marginals at any time. The key object for our approach is the obstructed shadow of a measure in a peacock, a generalization of the (obstructed) shadow introduced in [13,46]. As input data we take an increasing family of measures (ν α ) α∈[0,1] with ν α (R) = α that are submeasures of µ0, called a parametrization of µ0. Then, for any α we define an evolution (η α t ) t≥0 of the measure ν α = η α 0 across our peacock by setting η α t equal to the obstructed shadow of ν α in (µs) s∈[0,t] . We identify conditions on the parametrization (ν α ) α∈[0,1] such that this construction leads to a unique martingale measure π, the shadow martingale, without any assumptions on the peacock. In the case of the left-curtain parametrization (ν α lc ) α∈[0,1] we identify the shadow martingale as the unique solution to a continuous-time version of the martingale optimal transport problem.Furthermore, our method enriches the knowledge on the Predictable Representation Property (PRP) since any shadow martingale comes with a canonical Choquet representation in extremal Markov martingales.

Supermartingale Brenier's Theorem with full-marginals constraint

“…, X i−1 ] ≤ X i−1 P-a.s., for all i = 1, ..., T . (For the martingale version with general cost functions c : R T +1 → R, see Nutz et al [58], while a corresponding continuous-time extension is given by Juillet et al [20].) Then, if each c i (or −c i ) is supermartingale Spence-Mirrlees, the optimal supermartingale coupling (with fixed T + 1 marginals) can be obtained by a Markovian iteration of the increasing (or decreasing) one-period supermartingale transport plans.…”

“…The search of a model that produces the highest no-arbitrage price of an exotic claim, among all calibrated models, then naturally corresponds to the MOT problem. For more recent developments of MOT problems, see for example Acciaio et al [1], Backhoff-Veraguas et al [2,3], Beiglböck et al [6,16,7,9,10,11], Beiglböck and Juillet [12,13], Brückerhoff et al [20], Campi et al [21], De Marco and Henry-Labordère [30], Dolinsky and Soner [31], Fahim nad Huang [34], Gaoyue et al [39], Hobson and Neuberger [46], Hobson and Klimmek [45], Hobson and Norgilas [47], Nutz et al [58], Wiesel [63].…”