Geometry of distribution-constrained optimal stopping problems

Beiglböck, Mathias; Eder, Manu; Elgert, Christiane; Schmock, Uwe

doi:10.1007/s00440-017-0805-x

Cited by 22 publications

(17 citation statements)

References 36 publications

(64 reference statements)

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“…Recent developments and aims of this article. More recently, variants of this 'monotonicity priniciple' have been applied in transport problems for finitely or infinitely many marginals [38,19,27,8,45], the martingale version of the optimal transport problem [9,36,11], stochastic portfolio theory [37], the Skorokhod embedding problem [5,28], the distribution constrained optimal stopping problem [6,10] and the weak transport problem [26,3,4].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

A non-linear monotonicity principle and applications to Schrödinger type problems

Backhoff-Veraguas¹,

Beiglböck²,

Conforti³

2021

Preprint

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A basic idea in optimal transport is that optimizers can be characterized through a geometric property of their support sets called cyclical monotonicity. In recent years, similar monotonicity principles have found applications in other fields where infinite dimensional linear optimization problems play an important role.In this note, we observe how this approach can be transferred to non-linear optimization problems. Specifically we establish a monotonicity principle that is applicable to the Schrödinger problem and use it to characterize the structure of optimizers for target functionals beyond relative entropy. In contrast to classical convex duality approaches, a main novelty is that the monotonicity principle allows to deal also with non-convex functionals.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Putting the two together, we find that ϕ is Borel. 6 Proof of Theorem 1.5. The start of the proof is the same as the one of Theorem 1.4: Let Γ Q * , Γ P be as in Lemma 1.3.…”

Section: Lemma 22 ([7 Proposition 21])mentioning

confidence: 96%

A non-linear monotonicity principle and applications to Schrödinger type problems

Backhoff-Veraguas¹,

Beiglböck²,

Conforti³

2021

Preprint

Self Cite

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“…Optimal stopping under expectation constraints is studied in [3,9] while stochastic control under expectation constraints is studied in [44]. Distributionconstrained optimal stopping is studied in [8,10]. It should also be mentioned that meanvariance problems are sometimes formulated as constrained optimization problems.…”

Section: Background and Related Literaturementioning

confidence: 99%

Moment constrained optimal dividends: precommitment \& consistent planning

Christensen,

Lindensjö

2019

Preprint

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A moment constraint that limits the number of dividends in the optimal dividend problem is suggested. This leads to a new type of time-inconsistent stochastic impulse control problem. First, the optimal solution in the precommitment sense is derived. Second, the problem is formulated as an intrapersonal sequential dynamic game in line with Strotz' consistent planning. In particular, the notions of pure dividend strategies and a (strong) subgame perfect Nash equilibrium are adapted. An equilibrium is derived using a smooth fit condition. The equilibrium is shown to be strong. The uncontrolled state process is a fairly general diffusion.

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“…The existence of optimal ψ has also been established for the case when L has quadratic growth in u and the target distribution is smooth with ν > 0 for a version of the problem including a mean field cost posed on the torus [27,28], which makes use of the variational structure and energy estimates. On the other hand, stopping uncontrolled processes with distribution constraints has a vast literature, in particular pertaining to applications in finance; for some of the approaches related to (1.1) see [1,2,3,17].…”

Section: Introductionmentioning

confidence: 99%

Stochastic optimal transport with free end time

Dweik¹,

Ghoussoub²,

Kim³

et al. 2019

Preprint

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We consider a stochastic transportation problem between two prescribed probability distributions (a source and a target) over processes with general drift dependence and with free end times. First, and in order to establish a dual principle, we associate two equivalent formulations of the primal problem in order to guarantee its convexity and lower semi-continuity with respect to the source and target distributions. We exhibit an equivalent Eulerian formulation, whose dual variational principle is given by Hamilton-Jacobi-Bellman type variational inequalities. In the case where the drift is bounded, regularity results on the minimizers of the Eulerian problem then enable us to prove attainment in the corresponding dual problem. We also address attainment when the drift component of the cost defining Lagrangian L is superlinear L ≈ |u| p with 1 < p < 2, in which case the setting is reminiscent of our approach -in a previous work-on deterministic controlled transport problems with free end time. We finally address criteria under which the optimal drift and stopping time are unique, namely strict convexity in the drift component and monotonicity in time of the Lagrangian.

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Geometry of distribution-constrained optimal stopping problems

Cited by 22 publications

References 36 publications

A non-linear monotonicity principle and applications to Schrödinger type problems

A non-linear monotonicity principle and applications to Schrödinger type problems

Moment constrained optimal dividends: precommitment \& consistent planning

Stochastic optimal transport with free end time

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