On a problem of optimal stopping in mathematical finance

Abstract

This note concerns a problem of optimally stopping a non-degenerate, two-dimensional geometric Lévy process Q_t  = (x_t , y_t ) initially starting at (x, y), with the goal of maximizing an expected cost over the class of all stopping times τ  Є  χ^Q with values in [0,∞), for which k  >  0 is a positive constant. It is proved that the maximal value is a logarithmic function, and the optimal stopping time τ* admits the form τ*  =  inf{t  >  0 : x_t   ≥  ψ(y_t } where ψ(.)  Є C ²(0,∞), positive solution of a certain second-order nonlinear, ordinary integro-differential equation. The result has several possible applications in mathematical finance.

Keywords and phrases:

• Cobb-Douglas function
• geometric Lévy processes
• optimal stopping problem
• smooth-fit principle