A Dynamic Programming Principle for Distribution-Constrained Optimal Stopping

Källblad, Sigrid

doi:10.48550/arxiv.1703.08534

Cited by 2 publications

(8 citation statements)

References 23 publications

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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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Section: Introductionmentioning

confidence: 99%

Stochastic optimal transport with free end time

Dweik¹,

Ghoussoub²,

Kim³

et al. 2019

Preprint

View full text Add to dashboard Cite

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“…Under stronger conditions, we are also able to prove comparison for the HJB equation, allowing characterisation of the value function as the unique solution to this equation. The probability measure-valued evolution we wish to study as our underlying state variable is the class of measure-valued martingales, or MVMs, introduced in Cox and Källblad (2017). A process (ξ t ) t≥0 , taking values in the space of probability measures on R d is an MVM if ξ t (ϕ) := R d ϕ(x)ξ t (dx) is a martingale for every bounded continuous function ϕ ∈ C b (R d ).…”

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confidence: 99%

“…In Cox and Källblad (2017), MVMs were introduced in the context of modelindependent pricing and hedging of financial derivatives. In this application, the measure µ has an interpretation as the implied distribution of the asset price S T given the information at time t, ξ t (A) = Q(S T ∈ A|F t ), where Q is the risk-neutral measure.…”

mentioning

confidence: 99%

“…While increasing the complexity of the optimisation problem by making the state variable infinite dimensional, this avoids the tricky distributional constraint on the terminal law of the process. In Cox and Källblad (2017) and Bayraktar et al (2018), this connection was used to characterise the model-independent bounds of Asian and American-type options. See also e.g.…”

mentioning

confidence: 99%

“…See also e.g. Källblad (2017) for the use of MVMs to address distribution-constrained optimal stopping problems.…”

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confidence: 99%

See 2 more Smart Citations

Controlled Measure-Valued Martingales: a Viscosity Solution Approach

Cox¹,

Källblad²,

Larsson³

et al. 2021

Preprint

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We consider a class of stochastic control problems where the state process is a probability measure-valued process satisfying an additional martingale condition on its dynamics, called measure-valued martingales (MVMs). We establish the 'classical' results of stochastic control for these problems: specifically, we prove that the value function for the problem can be characterised as the unique solution to the Hamilton-Jacobi-Bellman equation in the sense of viscosity solutions. In order to prove this result, we exploit structural properties of the MVM processes. Our results also include an appropriate version of Itô's lemma for controlled MVMs.We also show how problems of this type arise in a number of applications, including model-independent derivatives pricing, the optimal Skorokhod embedding problem, and two player games with asymmetric information.

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A Dynamic Programming Principle for Distribution-Constrained Optimal Stopping

Cited by 2 publications

References 23 publications

Stochastic optimal transport with free end time

Stochastic optimal transport with free end time

Controlled Measure-Valued Martingales: a Viscosity Solution Approach

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