An integral equation for Root’s barrier and the generation of Brownian increments

Gassiat, Paul; Mijatović, Aleksandar; Oberhauser, Harald

doi:10.1214/14-aap1042

Cited by 13 publications

(28 citation statements)

References 30 publications

(65 reference statements)

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“…The boundary of Root's barrier is expressed as a function of space x → r(x) rather than a function of time. Under the assumption of atom-less target measures µ (as in our case) [15] prove that the boundary r( · ) solves a Volterra-type equation which is reminiscent of the ones obtained here. Uniqueness is provided via arguments based on viscosity theory in the special cases of laws µ's which produce continuous boundaries r( · ).…”

Section: Introductionsupporting

confidence: 59%

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Integral equations for Rost’s reversed barriers: Existence and uniqueness results

Angelis

Kitapbayev

2017

Stochastic Processes and their Applications

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We establish that the boundaries of the so-called Rost's reversed barrier are the unique couple of left-continuous monotonic functions solving a suitable system of nonlinear integral equations of Volterra type. Our result holds for atom-less target distributions µ of the related Skorokhod embedding problem.The integral equations we obtain here generalise the ones often arising in optimal stopping literature and our proof of the uniqueness of the solution goes beyond the existing results in the field.MSC2010: 60G40, 60J65, 60J55, 35R35, 45D05.

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Section: Introductionsupporting

confidence: 59%

“…In fact the numerical evaluation of the boundaries in [15] is restricted to the class of µ's that produce symmetric, continuous, monotonic on [0, ∞], maps x → r(x). Finally the existence results in our paper and in [15] are derived in completely different ways.…”

Section: Introductionmentioning

confidence: 99%

Integral equations for Rost’s reversed barriers: Existence and uniqueness results

Angelis

Kitapbayev

2017

Stochastic Processes and their Applications

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“…Other recent approaches to the Root and Rost embeddings can be found in [25,26,14,13]. These papers largely exploit PDE techniques, and as a consequence, are able to produce more explicit descriptions of the barriers, however the methods tend to be highly specific to the problem under consideration.…”

Section: 4mentioning

confidence: 99%

Optimal transport and Skorokhod embedding

2016

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The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to aThe first author was supported by the FWF-Grants P26736 and Y00782, the third author by the CRC 1060.

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“…In fact, to the best of our knowledge, even for the one-marginal problem in a standard Brownian setting, the only cases where an explicit barrier can be computed are measures supported on two points and Gaussian marginals. In some cases, the barrier can be characterised as the solution to an integral equation, see Gassiat et al [19]. As a result, numerical methods seem to be the only viable approach for explicit computation of Root-type barriers.…”

Section: Iterated Optimal Stopping and Multiple Marginal Barriersmentioning

confidence: 99%

The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

Cox

Obłój

Touzi

2018

Probab. Theory Relat. Fields

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We provide a complete characterisation of the Root solution to the Skorokhod embedding problem (SEP) by means of an optimal stopping formulation. Our methods are purely probabilistic and the analysis relies on a tailored time-reversal argument. This approach allows us to address the long-standing question of a multiple marginals extension of the Root solution of the SEP. Our main result establishes a complete solution to the n-marginal SEP using first hitting times of barrier sets by the time-space process. The barriers are characterised by means of a recursive sequence of optimal stopping problems. Moreover, we prove that our solution enjoys a global optimality property extending the one-marginal Root case. Our results hold for general, one-dimensional, martingale diffusions. Mathematics Subject Classification

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An integral equation for Root’s barrier and the generation of Brownian increments

Cited by 13 publications

References 30 publications

Integral equations for Rost’s reversed barriers: Existence and uniqueness results

Integral equations for Rost’s reversed barriers: Existence and uniqueness results

Optimal transport and Skorokhod embedding

The Root solution to the multi-marginal embedding problem: an optimal stopping and time-reversal approach

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