Elton P. Hsu

Asymptotics of Implied Volatility in Local Volatility Models

Using an expansion of the transition density function of a 1-dimensional time inhomogeneous diffusion, we obtain the first and second order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first order and second order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusions using probabilistic methods, i.e. the so called principle of not feeling the boundary.

Estimates of derivatives of the heat kernel on a compact Riemannian manifold

We give global estimates on the covariant derivatives of the heat kernel on a compact Riemannian manifold on a fixed finite time interval. The proof is based on analyzing the behavior of the heat kernel along Riemannian Brownian bridge.

Volume growth and escape rate of Brownian motion on a complete Riemannian manifold

We give an effective upper escape rate function for Brownian motion on a complete Riemannian manifold in terms of the volume growth of the manifold. An important step in the work is estimating the small tail probability of the crossing time between two concentric geodesic spheres by reflecting Brownian motions on the larger geodesic ball.

Volume Growth and Escape Rate of Brownian Motion on a Cartan-Hadamard Manifold

We prove an upper bound for the escape rate of Brownian motion on a Cartan-Hadamard manifold in terms of the volume growth function. One of the ingredients of the proof is the Sobolev inequality on such manifolds.

Stochastic Local Gauss-Bonnet-Chern Theorem

The Gauss-Bonnet-Chern theorem for compact Riemannian manifold (without boundary) is discussed here to exhibit in a clear manner the role Riemannian Brownian motion plays in various probabilistic approaches to index theorems. The method with some modifications works also for the index theorem for the Dirac operator on the bundle of spinors, see Hsu.(7)

GRADIENT ESTIMATES FOR HARMONIC FUNCTIONS ON MANIFOLDS WITH LIPSCHITZ METRICS

We introduce a distributional Ricci curvature on complete smooth man- ifolds with Lipschitz continuous metrics. Under an assumption on the volume growth of geodesics balls, we obtain a gradient estimate for weakly harmonic functions if the distributional Ricci curvature is bounded below. 1. Introduction. The study of harmonic functions has a long history. On Rieman- nian manifolds, Yau (12) proved a fundamental gradient estimate for harmonic functions in terms of the lower bound of Ricci curvature. As one of the many applications of Yaus gradient estimate, we have a generalization of Liouvilles theorem to the effect that a positive harmonic function on a complete Riemannian manifold of nonnegative Ricci curvature is constant. Recently there is an increasing interest in the study of harmonic functions (and harmonic mappings) on nonsmooth spaces. In this paper, we study the behavior of weakly harmonic functions on smooth manifolds with Lipschitz Riemannian metrics. We introduce the concept of distributional Ricci curvature and, under a mild volume growth condition, we derive a gradient estimate for positive (weakly) harmonic functions in terms of the lower bound of distributional Ricci curvature. Liouvilles theorem also follows from our gradient estimate. The main result of our work can be stated as follows. THEOREM 1.1. Let M be a complete differentiable manifold with a Lipschitz Rie- mannian metric such that volumes of geodesic balls satisfies sub-quadratic exponential growth condition, i.e., Vol B ( R ) e o ( R 2 )

A domain monotonicity property of the Neumann heat kernel

Physically pa{ty x, y) represents the temperautre distribution in Ω at time t and point y if a heat source of total capacity one is present at point x at time 0 with the assumption that the boundary 9Ω is impervious to heat conduction (adiabatic boundary). From this interpretation of the Neumann heat kernel it was conjectured (see Chavel [2] and Kendall [7]) that if Ω is a smooth convex domain and D is another smooth domain containing Ω, then for all (ΐy xy y) e (0, oo)χΩχΩ

Selected works of Kai Lai Chung

This unique volume presents a collection of the extensive journal publications written by Kai Lai Chung over a span of 70-odd years. It was produced to celebrate his 90th birthday. The selection is only a subset of the many contributions that he made throughout his prolific career. Another volume, Chance and Choice, published by World Scientific in 2004, contains yet another subset, with four articles in common with this volume. Kai Lai Chungs research contributions have had a major influence on several areas in probability. Among his most significant works are those related to sums of independent random variables, Markov chains, time reversal of Markov processes, probabilistic potential theory, Brownian excursions, and gauge theorems for the Schrodinger equation.As Kai Lai Chungs contributions spawned critical new developments, this volume also contains retrospective and perspective views provided by collaborators and other authors who themselves advanced the areas of probability and mathematics.