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Register a new userOn the Root solution to the Skorokhod embedding problem given full marginals
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This paper examines the Root solution of the Skorohod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on finitely-many marginals Root solution of Cox, Obloj and Touzi. Our main result provides a characterization of the corresponding potential function by means of a convenient parabolic PDE.
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We consider the optimal Skorokhod embedding problem (SEP) given full marginals over the time interval $[0,1]$. The problem is related to the study of extremal martingales associated with a peacock (process increasing in convex order, by Hirsch, Profeta, Roynette and Yor). A general duality result is obtained by convergence techniques. We then study the case where the reward function depends on the maximum of the embedding process, which is the limit of the martingale transport problem studied in Henry-Labordere, Obloj, Spoida and Touzi. Under technical conditions, some explicit characteristics of the solutions to the optimal SEP as well as to its dual problem are obtained. We also discuss the associated martingale inequality.
All spacecraft attitude estimation methods are based on Wahbas optimization problem. This problem can be reduced to finding the largest eigenvalue and the corresponding eigenvector for Davenports $K$-matrix. Several iterative algorithms, such as QUEST and FOMA, were proposed, aiming at reducing the computational cost. But their computational time is unpredictable because the iteration number is not fixed and the solution is not accurate in theory. Recently, an analytical solution, ESOQ was suggested. The advantages of analytical solutions are that their computational time is fixed and the solution should be accurate in theory if there is no numerical error. In this paper, we propose a different analytical solution to the Wahbas problem. We use simple and easy to be verified examples to show that this method is numerically more stable than ESOQ, potentially faster than QUEST and FOMA. We also use extensive simulation test to support this claim.
This paper characterizes the solution to a finite horizon min-max optimal control problem where the system is linear and discrete-time with control and state constraints, and the cost quadratic; the disturbance is negatively costed, as in the standard H-infinity problem, and is constrained. The cost is minimized over control policies and maximized over disturbance sequences so that the solution yields a feedback control. It is shown that the value function is piecewise quadratic and the optimal control policy piecewise affine, being quadratic and affine, respectively, in polytopes that partition the domain of the value function.
In a discrete-time financial market, a generalized duality is established for model-free superhedging, given marginal distributions of the underlying asset. Contrary to prior studies, we do not require contingent claims to be upper semicontinuous, allowing for upper semi-analytic ones. The generalized duality stipulates an extended version of risk-neutral pricing. To compute the model-free superhedging price, one needs to find the supremum of expected values of a contingent claim, evaluated not directly under martingale (risk-neutral) measures, but along sequences of measures that converge, in an appropriate sense, to martingale ones. To derive the main result, we first establish a portfolio-constrained duality for upper semi-analytic contingent claims, relying on Choquets capacitability theorem. As we gradually fade out the portfolio constraint, the generalized duality emerges through delicate probabilistic estimations.
We consider the Skorokhod problem in a time-varying interval. We prove existence and uniqueness for the solution. We also express the solution in terms of an explicit formula. Moving boundaries may generate singularities when they touch. We establish two sets of sufficient conditions on the moving boundaries that guarantee that the variation of the local time of the associated reflected Brownian motion is, respectively, finite and infinite. We also apply these results to study the semimartingale property of a class of two-dimensional reflected Brownian motions.
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