Abstract
This paper presents a closed-form solution for the joint probability of the endpoints and minimums of a multidimensional Wiener process for some correlation matrices. This is the only explicit expressions found in the literature for this joint probability. The analysis can only be carried out for special correlation structures as it is related to the fundamentals regions of irreducible spherical simplexes generated by reflections and the link to the method of images. This joint distribution can be used in financial mathematics to obtain prices of credit or market related products in high dimension. The solution could be generalized to account for stochastic volatility and other stylized features of the financial markets.
Highlights
The paper finds closed-form expressions for the joint density/distribution function of the endpoints and extrema of a n-dimensional Wiener process
The results found in this paper can be applied to processes that can be derived from a Wiener process using suitable transformations like log-normal processes and Ornstein-Uhlenbeck processes with suitable parameters
We show a method to create the image of a hyperplane, the image of a point reflected across a given hyperplane, and the sign of the new point as needed by the method of images; this uses standard concepts from geometry
Summary
The paper finds closed-form expressions for the joint density/distribution function of the endpoints and extrema of a n-dimensional Wiener process. In principle there could be as many as 5 different correlation structures, without permutations, in dimension 4 or as little as three in dimensions 3, 5, and higher than 8 This is derived using the fundamental regions for irreducible groups generated by reflections as presented in Table IV page 297 in [5] together with the finding in [6, 7]. It simplifies this PDE to a heat equation.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
