Kaj Nyström

Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains

Abstract In this paper we prove the boundary Harnack inequality and Hölder continuity for ratios of p-harmonic functions vanishing on a portion of certain Reifenberg flat and Ahlfors regular NTA-domains. Applications are given to the p-Martin boundary problem for these domains.

Regularity and Free Boundary Regularity for the p-Laplace Operator in Reifenberg Flat and Ahlfors Regular Domains

Regularity and free boundary regularity for the p-Laplace operator in Reifenberg flat and Ahlfors regular domains

Regularity of flat free boundaries in two-phase problems for the p-Laplace operator

In this paper we continue the study in Lewis and Nystrom (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.

Integrability of Green potentials in fractal domains

We proveLq-inequalities for the gradient of the Green potential (Gf) in bounded, connected NTA-domains inRn,n≥2. These domains may have a highly non-rectifiable boundary and in the plane the set of all bounded simply connected NTA-domains coincides with the set of all quasidiscs. We get a restriction on the exponentq for which our inequalities are valid in terms of the validity of a reverse Hölder inequality for the Green function close to the boundary.

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 w ...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.

TUG‐OF‐WAR, MARKET MANIPULATION, AND OPTION PRICING

We develop an option pricing model based on a tug-of-war game. This two-player zero-sum stochastic differential game is formulated in the context of a multi-dimensional financial market. The issuer and the holder try to manipulate asset price processes in order to minimize and maximize the expected discounted reward. We prove that the game has a value and that the value function is the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the non-linear and completely degenerate infinity Laplace operator.

The weak-A∞ PROPERTY OF HARMONIC AND p-HARMONIC MEASURES IMPLIES UNIFORM RECTIFIABILITY

Let

Efficient filtering of financial time series and extreme value theory

E\subset \ree

On the Dimension of p-Harmonic Measure in Space

,

A boundary Harnack inequality for singular equations of p-parabolic type

n\ge 2