The Markov-quantile process attached to a family of marginals

Boubel, Charles; Juillet, Nicolas

doi:10.5802/jep.177

Cited by 2 publications

(2 citation statements)

References 38 publications

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“…Richard, Tan, and Touzi [52] give a continuous time version of the Root solution to the Skorokhod embedding problem. In a slightly different but related direction, Boubel and Juillet [12] consider a continuum of marginals on the real line that do not satisfy an order condition and construct a canonical Markov-process matching these marginals. We also refer to the book [28] of Hirsch, Profeta, Roynette, and Yor that collects a variety of related constructions.…”

Section: Overview Of Related Results In the Literaturementioning

confidence: 99%

From Bachelier to Dupire via Optimal Transport

Beiglböck¹,

Pammer²,

Schachermayer³

2021

Preprint

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Famously mathematical finance was started by Bachelier in his 1900 PhD thesis whereamong many other achievements -he also provides a formal derivation of the Kolmogorov forward equation. This forms also the basis for Dupire's (again formal) solution to the problem of finding an arbitrage free model calibrated to the volatility surface. The later result has rigorous counterparts in the theorems of Kellerer and Lowther. In this survey article we revisit these hallmarks of stochastic finance, highlighting the role played by some optimal transport results in this context.

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Section: Overview Of Related Results In the Literaturementioning

confidence: 99%

From Bachelier to Dupire via Optimal Transport

Beiglböck¹,

Pammer²,

Schachermayer³

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Similarly, there are no results about uniquely constructing martingales by solely describing how parts of the initial distribution evolve. In fact, the only approach in this direction that we are aware of is [16] but in a non-martingale setup.…”

Section: Related Literaturementioning

confidence: 99%

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

Brückerhoff

Huesmann

Juillet

2022

Electron. J. Probab.

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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.

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The Markov-quantile process attached to a family of marginals

Cited by 2 publications

References 38 publications

From Bachelier to Dupire via Optimal Transport

From Bachelier to Dupire via Optimal Transport

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

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