Jia-An Yan

Introduction to Infinite Dimensional Stochastic Analysis

Preface. I. Foundations of Infinite Dimensional Analysis. II. Malliavin Calculus. III. Stochastic Calculus of Variation for Wiener Functionals. IV. General Theory of White Noise Analysis. V. Linear Operators on Distribution Spaces. Appendices. Comments. References. Subject Index. Index of Symbols.

Optimal Insurance Design Under Rank‐Dependent Expected Utility

We consider an optimal insurance design problem for an individual whose preferences are dictated by the rank-dependent expected utility (RDEU) theory with a concave utility function and an inverse-S shaped probability distortion function. This type of RDEU is known to describe human behavior better than the classical expected utility. By applying the technique of quantile formulation, we solve the problem explicitly. We show that the optimal contract not only insures large losses above a deductible but also insures small losses fully. This is consistent, for instance, with the demand for warranties. Finally, we compare our results, analytically and numerically, both to those in the expected utility framework and to cases in which the distortion function is convex or concave.

Products and transforms of white-noise functionals (in general setting)

Some sharp results about Weiner and Wick products of whitenoise functionals are obtained. Using the inequality of Wick products we show to what extent scaling transformations, translations, and Sobolev differentiations can be performed on white-noise functionals.

On Wick product of generalized operators

Fundamental properties of Wick product of generalized operators are investigated. The annihilation and creation algebras are characterized from various points of view. Wick ordering widely used in quantum physics is interpreted as the Wick product of generalized operators.

Some Remarks on Arbitrage Pricing Theory

AbstractIn this note we report main results in a recent paper by the authors, in which we established a version of Kramkovs optional decomposition theorem in the setting of equivalent martingale measures and using this theorem we clarified some basic concepts and results in arbitrage pricing theory: superhedging, fair price, replicatable contingent claim, complete markets.

From Feynman-Kac formula to Feynman integrals via analytic continuation

By using a calculus based on Brownian bridge measures, it is shown that under mild assumptions on V (e.g. V is in the Kato class) the fundamental solution (FS) q (t,x,y) for the heat equation can be represented by the Feynman-Kac formula. Furthermore, it has an analytic continuation in t over +, where , and q([var epsilon] + it,x,y) can be expressed via Wiener path integrals. For small [var epsilon] > 0 it can be considered as an approximation of the FS for the Schrodinger equation . We also give an estimate of q(t,x,y) for t [set membership, variant] +.

A limit theorem for partial sums of random variables and its applications

In this paper, we establish a new limit theorem for partial sums of random variables. As corollaries, we generalize the extended Borel-Cantelli lemma, and obtain some strong laws of large numbers for Markov chains as well as a generalized strong ergodic theorem for irreducible and positive recurrent Markov chains.

Continuity of affine transformations of white noise test functionals and applications

Translations and scalings defined on the Schwartz space of tempered distributions induce continuous transformations on the space of white noise test functionals [25]. Continuity of the induced transformations with respect to their parameters is proved. As a consequence one obtains a direct simple proof of the fact that the space of white noise test functionals is infinitely differentiable in Frechet sense. Moreover, it is shown that the Wiener semigroup acts as a mollifier on the space of test functionals.

A New Space of White Noise Distributions and Applications to Spde’s

This paper deals with the so-called white noise calculus. Some of the shortcomings of the existing spaces of generalized functions are discussed and a new space of distributions is introduced. This new space of distributions is shown to be larger than the existing ones. We give a characterization of its elements in terms of a local S-transform. Finally an application to stochastic partial differential equations is given.

Inequalities for Products of White Noise Functionals

Some sharp inequalities for the Wiener and Wick products of white noise functionals in the new settings of white noise analysis are established.