Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable

Abstract

The purpose of this article is to study the Burgers and Black–Merton–Scholes equations with real time variable and complex spatial variable. The complexification of the spatial variable in these equations is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding (real) solution by replacing the usual sum of variables (translation) by an exponential product (rotation) and secondly, one complexifies the spatial variable in the corresponding evolution equation and then one searches for analytic and non-analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two dimensional spatial variables are generated and their solutions are constructed.

Keywords:

• complex singular integrals
• semigroups of linear operators
• partial differential equations
• Burgers equation
• Black–Merton–Scholes equation
• complex spatial variables

AMS Subject Classifications:

• Primary: 47D06
• Secondary: 47D03, 47D60