On Asymmetric Skew Bessel Process

Abstract

이 논문에서는 반비대칭 베셀 과정에 대해서 서술하겠습니다. 이를 위해 반 브라운 운동과 브라운 사이의 관계에 대해서 알아보고 반비대칭 베셀과정과 베셀과정 사이의 관계로 적용될 수 있는 것들을 알아보겠습니다. 속도측도와 비율함수로 부터 어떤 확률미분방정식의 해가되는 추계과정을 구하는 방법을 서술하겠습니다. 또한 반 비대칭 베셀과정의 일반화된 무한소생성장를 찾겠습니다. 또한 브라운 운동의 추이밀도함수로 부터 반 브라운 운동의 추이밀도함수를 구하는 과정을 통해서 반 비대칭 베셀과정의 추이밀도함수를 구할 수 있는 방법을 제시하겠습니다.

As shown by Decamps, Goovaerts and Schoutens[1] the asymmetric skew Bessel processes are very useful for the pricing of Asian Option . Because Asymmetric skew Bessel processes share many properties with the skew Brownian Motion(skew BM) and Bessel processes, many of its properties can be investigated through skew BM and Bessel Processes. The asymmetric skew Bessel processes are a good example of a skew reflected nonnegative valued process. \newline In this paper, the connection between skew BM and Brownian motion is reviewed and then we investigate the connection between asymmetric skew Bessel processes and Bessel processes. The construction of diffusions through their scale functions and speed measures is quite routine. Thus such a way of construction by Decamps, Goovaerts and Schoutens[1] is given with more details, especially in regard of an occuring symmetric local time. Infinitesimal generators of skew BM or asymmetric skew Bessel processes do not have a good form if they are applied on $C_K^2$, the generalized diffusion processes which allow local behavior that should be described by generalized function is presented. Asymmetric skew Bessel processes are shown to be generalized diffusion processes. The density of BM and the density of skew BM have a quite intuitive relation. Thus it is natural to expect a similar relation between Bessel processes of dimension greater than 2 and asymmetric skew Bessel processes. We also suggest an approach to find the distribution of asymmetric skew Bessel processes.