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We prove the existence and pointwise lower and upper bounds for thefundamental solution of the degenerate second order partial differentialequation related to Geman-Yor stochastic processes, that arise in models foroption pricing theory in finance.  Lower bounds are obtained by using repeatedly an invariant Harnack inequalityand by solving an associated optimal control problem with quadratic cost. Upperbounds are obtained by the fact that the optimal cost satisfies a specificHamilton-Jacobi-Bellman equation.

Sharp Estimates for Geman-Yor Processes and applications to Arithmetic Average Asian options