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Kaj Nyström

Publications and source records attributed to Kaj Nyström.

42 records · Page 3Linked to original sources
arXiv2014

A new characterization of chord-arc domains

We show that if $Ω\subset \mathbb{R}^{n+1}$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $Ω$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $Ω$ is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew conditions, and an interior Harnack Chain condition. We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain free boundary problems.

math.CA↗
arXiv2014

Optimal Switching Problems under Partial Information

In this paper we formulate and study an optimal switching problem under partial information. In our model the agent/manager/investor attempts to maximize the expected reward by switching between different states/investments. However, he is not fully aware of his environment and only an observation process, which contains partial information about the environment/underlying, is accessible. It is based on the partial information carried by this observation process that all decisions must be made. We propose a probabilistic numerical algorithm based on dynamic programming, regression Monte Carlo methods, and stochastic filtering theory to compute the value function. In this paper, the approximation of the value function and the corresponding convergence result are obtained when the underlying and observation processes satisfy the linear Kalman-Bucy setting. A numerical example is included to show some specific features of partial information.

math.OC↗
arXiv2014

A framework for the modeling of order book dynamics based on event sizes

We propose a modeling framework for the dynamics of a reduced form order book in event time and based on event sizes. Our framework for the order book is influenced by [9], but compared to [9] we allow the best bid ask spread to be larger than one tick. Based on the modeling assumption that the best bid as well as the best ask price can only move by at most one tick (up or down), when an event occurs, we show that the dynamics of this simplified order book is completely described by a non-linear transformation of two processes (X, Y ). A key challenge in the modeling is the empirical fact that the high frequency order flow is strongly autocorrelated, a fact we have to deal with in the modeling of (X, Y ). The core of our framework is a semi linear regression type model for (X, Y ), influence by more classical ARMA and AR models, and one key degree of freedom is the potentially non-linear basis functions used in the regression. We use results from the theory of random iterative function systems to understand issues concerning stationarity and ergodicity in our models. We show how to rapidly calibrate the model by inverting block Toeplitz matrices in an efficient way. All components are worked through and explain in an application and the predictability of the model for order flows and price moves are analyzed in the context of a high frequency dataset.

math.PR↗
arXiv2010

The Skorohod oblique reflection problem in time-dependent domains

The deterministic Skorohod problem plays an important role in the construction and analysis of diffusion processes with reflection. In the form studied here, the multidimensional Skorohod problem was introduced, in time-independent domains, by H. Tanaka [61] and further investigated by P.-L. Lions and A.-S. Sznitman [42] in their celebrated article. Subsequent results of several researchers have resulted in a large literature on the Skorohod problem in time-independent domains. In this article we conduct a thorough study of the multidimensional Skorohod problem in time-dependent domains. In particular, we prove the existence of càdlàg solutions $(x,λ)$ to the Skorohod problem, with oblique reflection, for $(D,Γ,w)$ assuming, in particular, that $D$ is a time-dependent domain (Theorem 1.2). In addition, we prove that if $w$ is continuous, then $x$ is continuous as well (Theorem 1.3). Subsequently, we use the established existence results to construct solutions to stochastic differential equations with oblique reflection (Theorem 1.9) in time-dependent domains. In the process of proving these results we establish a number of estimates for solutions to the Skorohod problem with bounded jumps and, in addition, several results concerning the convergence of sequences of solutions to Skorohod problems in the setting of time-dependent domains.

math.PR↗
arXiv2009

p Harmonic Measure in Simply Connected Domains

We extend to all planar simply connected domains Makarov-type results about the Hausdorff dimension of $p$-harmonic measure pioneered by Lewis and Bennewitz in the context of quasidisks. The key to our analysis is a gradient estimate using the distance to the boundary and constants that only depend on $p$. This is achieved by studying the conformal map from the unit disk to the simply connected domain to construct good quasicurves from a point in the domain to the boundary.

math.CV↗