Three dimensional distribution of Brownian motion extrema

Abstract

This paper describes the joint distributions of minima, maxima and endpoint values for a three dimensional (3D) Wiener process. In particular, the results provide the joint cumulative distributions for the maxima and/or minima of the components of the process. The method of images is used to derive explicit expressions for the densities; the analysis can only be carried out for special correlation structures and requires a detailed study of partitions of the sphere by means of spherical triangles. The joint densities obtained can be used in several applied fields such as financial mathematics to obtain analytical expressions for prices of options for the 3D geometric Brownian motion process.

Keywords:

• method of images
• 3D Wiener process
• distribution of minimum
• maximum

Keywords:

• 60E99
• 60H15
• 60G99