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Nikolay Lysenko

Publications and source records attributed to Nikolay Lysenko.

2 recordsLinked to original sources
arXiv2015

Maximization of Functionals Depending on the Terminal Value and the Running Maximum of a Martingale: A Mass Transport Approach

It is known that the Azema-Yor solution to the Skorokhod embedding problem maximizes the law of the running maximum of an uniformly integrable martingale with given terminal value distribution. Recently this optimality property has been generalized to expectations of certain bivariate cost functions depending on the terminal value and the running maximum. In this paper we give an extension of this result to another class of functions. In particular, we study a class of cost functions with the property that the corresponding optimal embeddings are not Azema-Yor. The suggested approach is quite straightforward modulo basic facts of the Monge-Kantorovich mass transportation theory. Loosely speaking, the joint distribution of the running maximum and the terminal value in the Azema-Yor embedding is concentrated on the graph of a monotone function, and we show that this fact follows from the cyclical monotonicity criterion for solutions to the Monge-Kantorovich problem. Keywords: Skorokhod problem, Azema-Yor embedding, Monge-Kantorovich problem, optimal transport, supermodular functions, running maximum and the terminal value of a martingale.

math.PR↗
arXiv2015

Remarks on mass transportation minimizing expectation of a minimum of affine functions

We study the Monge--Kantorovich problem with one-dimensional marginals $\mu$ and $\nu$ and the cost function $c = \min\{l_1, \ldots, l_n\}$ that equals the minimum of a finite number $n$ of affine functions $l_i$ satisfying certain non-degeneracy assumptions. We prove that the problem is equivalent to a finite-dimensional extremal problem. More precisely, it is shown that the solution is concentrated on the union of $n$ products $I_i \times J_i$, where $\{I_i\}$ and $\{J_i\}$ are partitions of the real line into unions of disjoint connected sets. The families of sets $\{I_i\}$ and $\{J_i\}$ have the following properties: 1) $c=l_i$ on $I_i \times J_i$, 2) $\{I_i\}, \{J_i\}$ is a couple of partitions solving an auxiliary $n$-dimensional extremal problem. The result is partially generalized to the case of more than two marginals.

math.PR↗