Marginal Density Expansions for Diffusions and Stochastic Volatility II: Applications

In Part I (Comm. Pure Appl. Math., 67 (2014), no. 1, 40–82) we discussed density expansions for multidimensional diffusions (X¹,…,X^d), at fixed time T and projected to their first l coordinates in the small‐noise regime. Global conditions were found that replace the well‐known “not‐in‐cut‐locus” condition known from heat kernel asymptotics. In the present paper we discuss financial applications; these include tail and implied volatility asymptotics in some correlated stochastic volatility models. In particular, we solve a problem left open by A. Gulisashvili and E. M. Stein. © 2013 Wiley Periodicals, Inc.     

Entropic repulsion of the lattice free field

Consider the massless free field on thed-dimensional lattice ^d ,d3; that is the centered Gaussian field on with covariances given by the Green function of the simple random walk on ^d . We show that the probability, that all the spins are positive in a box of volumeN ^d , decays exponentially at a rate of orderN ^d–2 logN and compute explicitly the corresponding constant in terms of the capacity of the unit cube. The result is extended to a class of transient random walks with transition functions in the domain of the normal and -stable law.This research was partially supported by the foundation for promotion of research at the Technion.     

Entropic repulsion of the lattice free field,II. The 0-boundary case

This paper is a continuation of [5]. We consider the Euclidean massless free field on a boxV _N of volumeN ^d with O-boundary condition; that is the centered Gaussian field with covariances given by the Green function of the simple random walk on ^d ,d3, killed as it exitsV _N . We show that the probability, that all the spins are positive in the boxV _N decays exponentially at a surface rateN ^d–1 . This is in contrast with the rateN ^d–2 logN for the infinite field of [5].     

Aging for interacting diffusion processes

We study the aging phenomenon for a class of interacting diffusion processes {Xt(i),i∈Zd}. In this framework we see the effect of the lattice dimension d on aging, as well as that of the class of test functions f(Xt) considered. We further note the sensitivity of aging to specific details, when degenerate diffusions (such as super random walk, or parabolic Anderson model), are considered. We complement our study of systems on the infinite lattice, with that of their restriction to finite boxes. In the latter setting we consider different regimes in terms of box size scaling with time, as well as the effect that the choice of boundary conditions has on aging. The key tool for our analysis is the random walk representation for such diffusions.     

Non-Gaussian surface pinned by a weak potential

We consider a model of a two-dimensional interface of the (continuous) SOS type, with finite-range, strictly convex interactions. We prove that, under an arbitrarily weak pinning potential, the interface is localized. We consider the cases of both square well and δ potentials. Our results extend and generalize previous results for the case of nearest-neighbours Gaussian interactions in [7] and [11]. We also obtain the tail behaviour of the height distribution, which is not Gaussian. Received: 3 November 1998 / Revised version: 14 June 1999