The stability of an equation from mathematical finance without the boundary value condition

Abstract

Abstract The strong degenerate parabolic equation ∂xxu+u∂yu−∂tu=f(x,y,t,u) $$ \partial _{xx}u+u\partial _{y}u-\partial _{t}u =f(x,y,t,u) $$ comes from the mathematics of finance. A kind of entropy solution is introduced. By Kružkov’s bi-variable method, the stability for entropy solutions only depending on the initial value is proved. The usual boundary value condition is replaced by the regularity of the domain in a special sense.

Keywords

• Mathematics finance
• Entropy solution
• Kružkov’s bi-variables method
• Boundary value condition