1. Sharp Estimates for Geman–Yor Processes and applications to Arithmetic Average Asian options
- Author
-
Sergio Polidoro, Francesco Rossi, and Gennaro Cibelli
- Subjects
Hypoelliptic operators ,Pointwise ,Harnack inequality ,Partial differential equation ,Stochastic process ,Geman–Yor stochastic process ,Optimal control ,Mathematics (all) ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,010101 applied mathematics ,Fundamental solution ,Applied mathematics ,Asian option ,0101 mathematics ,Invariant (mathematics) ,Mathematics ,Harnack's inequality - Abstract
We prove the existence of the fundamental solution of the degenerate second order partial differential equation related to Geman–Yor stochastic processes, that arise in models for option pricing theory in finance. We then prove pointwise lower and upper bounds for such fundamental solution. Lower bounds are obtained by using repeatedly an invariant Harnack inequality and by solving an associated optimal control problem with quadratic cost. Upper bounds are obtained by the fact that the cost satisfies a specific Hamilton–Jacobi–Bellman equation.
- Published
- 2019